Integrand size = 60, antiderivative size = 352 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx=\frac {\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (6 a q^7-2 a p q^6 x^2+42 a p q^6 x^3-5 a p^2 q^5 x^4-10 a p^2 q^5 x^5+24 b q x^6-15 a p^3 q^4 x^6+126 a p^2 q^5 x^6-15 a p^3 q^4 x^7-20 a p^3 q^4 x^8+24 b p x^9-15 a p^4 q^3 x^9+210 a p^3 q^4 x^9-15 a p^4 q^3 x^{10}-20 a p^4 q^3 x^{11}+210 a p^4 q^3 x^{12}-5 a p^5 q^2 x^{13}-10 a p^5 q^2 x^{14}+126 a p^5 q^2 x^{15}-2 a p^6 q x^{17}+42 a p^6 q x^{18}+6 a p^7 x^{21}\right )}{48 x^8}+\frac {1}{8} \left (8 b p q+5 a p^4 q^4\right ) \log (x)+\frac {1}{8} \left (-8 b p q-5 a p^4 q^4\right ) \log \left (q+p x^3+\sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6}\right ) \]
1/48*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(6*a*p^7*x^21+42*a*p^6*q*x^18 -2*a*p^6*q*x^17+126*a*p^5*q^2*x^15-10*a*p^5*q^2*x^14-5*a*p^5*q^2*x^13+210* a*p^4*q^3*x^12-20*a*p^4*q^3*x^11-15*a*p^4*q^3*x^10-15*a*p^4*q^3*x^9+210*a* p^3*q^4*x^9-20*a*p^3*q^4*x^8-15*a*p^3*q^4*x^7-15*a*p^3*q^4*x^6+126*a*p^2*q ^5*x^6-10*a*p^2*q^5*x^5-5*a*p^2*q^5*x^4+42*a*p*q^6*x^3+24*b*p*x^9-2*a*p*q^ 6*x^2+6*a*q^7+24*b*q*x^6)/x^8+1/8*(5*a*p^4*q^4+8*b*p*q)*ln(x)+1/8*(-5*a*p^ 4*q^4-8*b*p*q)*ln(q+p*x^3+(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2))
Time = 0.43 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.64 \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx=\frac {1}{48} \left (\frac {\left (q+p x^3\right ) \sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6} \left (24 b x^6+a \left (6 q^6+6 p^6 x^{18}+2 p q^5 x^2 (-1+18 x)+2 p^5 q x^{14} (-1+18 x)+p^2 q^4 x^4 \left (-5-8 x+90 x^2\right )+p^4 q^2 x^{10} \left (-5-8 x+90 x^2\right )+p^3 q^3 x^6 \left (-15-10 x-12 x^2+120 x^3\right )\right )\right )}{x^8}+6 \left (8 b p q+5 a p^4 q^4\right ) \log (x)-6 \left (8 b p q+5 a p^4 q^4\right ) \log \left (q+p x^3+\sqrt {q^2+2 p q (-1+x) x^2+p^2 x^6}\right )\right ) \]
Integrate[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*x ^6 + a*(q + p*x^3)^6))/x^9,x]
(((q + p*x^3)*Sqrt[q^2 + 2*p*q*(-1 + x)*x^2 + p^2*x^6]*(24*b*x^6 + a*(6*q^ 6 + 6*p^6*x^18 + 2*p*q^5*x^2*(-1 + 18*x) + 2*p^5*q*x^14*(-1 + 18*x) + p^2* q^4*x^4*(-5 - 8*x + 90*x^2) + p^4*q^2*x^10*(-5 - 8*x + 90*x^2) + p^3*q^3*x ^6*(-15 - 10*x - 12*x^2 + 120*x^3))))/x^8 + 6*(8*b*p*q + 5*a*p^4*q^4)*Log[ x] - 6*(8*b*p*q + 5*a*p^4*q^4)*Log[q + p*x^3 + Sqrt[q^2 + 2*p*q*(-1 + x)*x ^2 + p^2*x^6]])/48
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (2 p x^3-q\right ) \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (a \left (p x^3+q\right )^6+b x^6\right )}{x^9} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 p \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (5 a p^2 q^4+b\right )-\frac {q \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2} \left (3 a p^2 q^4+b\right )}{x^3}-\frac {a q^7 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{x^9}-\frac {4 a p q^6 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{x^6}+2 a p^7 x^{12} \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}+11 a p^6 q x^9 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}+24 a p^5 q^2 x^6 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}+25 a p^4 q^3 x^3 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 p \left (5 a p^2 q^4+b\right ) \int \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}dx-q \left (3 a p^2 q^4+b\right ) \int \frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{x^3}dx-a q^7 \int \frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{x^9}dx-4 a p q^6 \int \frac {\sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}}{x^6}dx+2 a p^7 \int x^{12} \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}dx+11 a p^6 q \int x^9 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}dx+24 a p^5 q^2 \int x^6 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}dx+25 a p^4 q^3 \int x^3 \sqrt {p^2 x^6+2 p q x^3-2 p q x^2+q^2}dx\) |
Int[((-q + 2*p*x^3)*Sqrt[q^2 - 2*p*q*x^2 + 2*p*q*x^3 + p^2*x^6]*(b*x^6 + a *(q + p*x^3)^6))/x^9,x]
3.30.47.3.1 Defintions of rubi rules used
Time = 0.28 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.58
| method | result | size |
| pseudoelliptic | \(\frac {\left (p \,x^{3}+q \right ) \left (a \,p^{6} x^{18}+6 \left (x -\frac {1}{18}\right ) a \,x^{14} q \,p^{5}+15 \left (x^{2}-\frac {4}{45} x -\frac {1}{18}\right ) a \,x^{10} q^{2} p^{4}+20 a \left (x^{3}-\frac {1}{10} x^{2}-\frac {1}{12} x -\frac {1}{8}\right ) x^{6} q^{3} p^{3}+15 \left (x^{2}-\frac {4}{45} x -\frac {1}{18}\right ) a \,x^{4} q^{4} p^{2}+6 \left (x -\frac {1}{18}\right ) a \,x^{2} q^{5} p +a \,q^{6}+4 b \,x^{6}\right ) \sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}-5 p \,x^{8} q \left (a \,p^{3} q^{3}+\frac {8 b}{5}\right ) \ln \left (\frac {q +p \,x^{3}+\sqrt {p^{2} x^{6}+2 p q \,x^{2} \left (-1+x \right )+q^{2}}}{x}\right )}{8 x^{8}}\) | \(203\) |
int((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*x^6+a*(p*x^3+q) ^6)/x^9,x,method=_RETURNVERBOSE)
1/8*((p*x^3+q)*(a*p^6*x^18+6*(x-1/18)*a*x^14*q*p^5+15*(x^2-4/45*x-1/18)*a* x^10*q^2*p^4+20*a*(x^3-1/10*x^2-1/12*x-1/8)*x^6*q^3*p^3+15*(x^2-4/45*x-1/1 8)*a*x^4*q^4*p^2+6*(x-1/18)*a*x^2*q^5*p+a*q^6+4*b*x^6)*(p^2*x^6+2*p*q*x^2* (-1+x)+q^2)^(1/2)-5*p*x^8*q*(a*p^3*q^3+8/5*b)*ln((q+p*x^3+(p^2*x^6+2*p*q*x ^2*(-1+x)+q^2)^(1/2))/x))/x^8
Timed out. \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx=\text {Timed out} \]
integrate((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*x^6+a*(p* x^3+q)^6)/x^9,x, algorithm="fricas")
\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx=\int \frac {\left (2 p x^{3} - q\right ) \sqrt {p^{2} x^{6} + 2 p q x^{3} - 2 p q x^{2} + q^{2}} \left (a p^{6} x^{18} + 6 a p^{5} q x^{15} + 15 a p^{4} q^{2} x^{12} + 20 a p^{3} q^{3} x^{9} + 15 a p^{2} q^{4} x^{6} + 6 a p q^{5} x^{3} + a q^{6} + b x^{6}\right )}{x^{9}}\, dx \]
integrate((2*p*x**3-q)*(p**2*x**6+2*p*q*x**3-2*p*q*x**2+q**2)**(1/2)*(b*x* *6+a*(p*x**3+q)**6)/x**9,x)
Integral((2*p*x**3 - q)*sqrt(p**2*x**6 + 2*p*q*x**3 - 2*p*q*x**2 + q**2)*( a*p**6*x**18 + 6*a*p**5*q*x**15 + 15*a*p**4*q**2*x**12 + 20*a*p**3*q**3*x* *9 + 15*a*p**2*q**4*x**6 + 6*a*p*q**5*x**3 + a*q**6 + b*x**6)/x**9, x)
\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx=\int { \frac {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left ({\left (p x^{3} + q\right )}^{6} a + b x^{6}\right )} {\left (2 \, p x^{3} - q\right )}}{x^{9}} \,d x } \]
integrate((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*x^6+a*(p* x^3+q)^6)/x^9,x, algorithm="maxima")
integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*((p*x^3 + q)^6*a + b *x^6)*(2*p*x^3 - q)/x^9, x)
\[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx=\int { \frac {\sqrt {p^{2} x^{6} + 2 \, p q x^{3} - 2 \, p q x^{2} + q^{2}} {\left ({\left (p x^{3} + q\right )}^{6} a + b x^{6}\right )} {\left (2 \, p x^{3} - q\right )}}{x^{9}} \,d x } \]
integrate((2*p*x^3-q)*(p^2*x^6+2*p*q*x^3-2*p*q*x^2+q^2)^(1/2)*(b*x^6+a*(p* x^3+q)^6)/x^9,x, algorithm="giac")
integrate(sqrt(p^2*x^6 + 2*p*q*x^3 - 2*p*q*x^2 + q^2)*((p*x^3 + q)^6*a + b *x^6)*(2*p*x^3 - q)/x^9, x)
Timed out. \[ \int \frac {\left (-q+2 p x^3\right ) \sqrt {q^2-2 p q x^2+2 p q x^3+p^2 x^6} \left (b x^6+a \left (q+p x^3\right )^6\right )}{x^9} \, dx=\text {Hanged} \]