The only 3 answers for this are

• 1,3,15,25,81
• 5,7,15,35,63
• 49,65,73,75,81

I used a simple C++ program to find these values:

https://drive.google.com/open?id=0BxZUVcBYkmCHS0JXSDRSeGtCb1k

The only 3 answers for this are

• 1,3,15,25,81
• 5,7,15,35,63
• 49,65,73,75,81

I used a simple C++ program to find these values:

#include <iostream>
#include <cmath>
bool integer(float k)
{
    if( k == (int) k) return true;
    return false;
}
int main()
{
    int a,b,c,d,e;
    for(a=1;a<100;a++)
    {
        for(b=1;b<100;b++)
        {
            for(c=1;c<100;c++)
            {
                for(d=1;d<100;d++)
                {   
                    for(e=1;e<100;e++)
                    {
                        if((a*b*c*d*e)%2 == 1 && integer(cbrt(a*b*c*d*e)) && integer(cbrt(a+b+c+d+e)) && (a!=b && a!=c && a!=d && a!=e && b!=c && b!=d && b!=e && c!=d && c!=e && d!=e) && (a<b && a<c && a<d && a<e && b<c && b<d && b<e && c<d && c<e && d<e))
                        {
                            printf("%d %d %d %d %d\n",a,b,c,d,e);
                        }
                    }
                }     
            }
        }    
    }
   return 0;
}