Integrand size = 59, antiderivative size = 352 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx=\frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (6 a q^7+42 a p q^6 x^3-2 a p q^6 x^4+126 a p^2 q^5 x^6-10 a p^2 q^5 x^7-5 a p^2 q^5 x^8+210 a p^3 q^4 x^9-20 a p^3 q^4 x^{10}-15 a p^3 q^4 x^{11}+24 b q x^{12}+210 a p^4 q^3 x^{12}-15 a p^3 q^4 x^{12}-20 a p^4 q^3 x^{13}-15 a p^4 q^3 x^{14}+24 b p x^{15}+126 a p^5 q^2 x^{15}-15 a p^4 q^3 x^{15}-10 a p^5 q^2 x^{16}-5 a p^5 q^2 x^{17}+42 a p^6 q x^{18}-2 a p^6 q x^{19}+6 a p^7 x^{21}\right )}{48 x^{16}}+\frac {1}{4} \left (8 b p q+5 a p^4 q^4\right ) \log (x)+\frac {1}{8} \left (-8 b p q-5 a p^4 q^4\right ) \log \left (q+p x^3+\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \]
1/48*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(6*a*p^7*x^21-2*a*p^6*q*x^19+ 42*a*p^6*q*x^18-5*a*p^5*q^2*x^17-10*a*p^5*q^2*x^16+126*a*p^5*q^2*x^15-15*a *p^4*q^3*x^15-15*a*p^4*q^3*x^14-20*a*p^4*q^3*x^13+210*a*p^4*q^3*x^12-15*a* p^3*q^4*x^12-15*a*p^3*q^4*x^11-20*a*p^3*q^4*x^10+210*a*p^3*q^4*x^9+24*b*p* x^15-5*a*p^2*q^5*x^8-10*a*p^2*q^5*x^7+126*a*p^2*q^5*x^6+24*b*q*x^12-2*a*p* q^6*x^4+42*a*p*q^6*x^3+6*a*q^7)/x^16+1/4*(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^4+2*p*q*x^3+q^2)^(1/2))
Time = 0.48 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.63 \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx=\frac {1}{48} \left (\frac {\left (q+p x^3\right ) \sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6} \left (24 b x^{12}+a \left (6 q^6-2 p q^5 (-18+x) x^3-2 p^5 q (-18+x) x^{15}+6 p^6 x^{18}+p^2 q^4 x^6 \left (90-8 x-5 x^2\right )+p^4 q^2 x^{12} \left (90-8 x-5 x^2\right )-p^3 q^3 x^9 \left (-120+12 x+10 x^2+15 x^3\right )\right )\right )}{x^{16}}+12 \left (8 b p q+5 a p^4 q^4\right ) \log (x)-6 \left (8 b p q+5 a p^4 q^4\right ) \log \left (q+p x^3+\sqrt {q^2-2 p q (-1+x) x^3+p^2 x^6}\right )\right ) \]
Integrate[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x ^12 + a*(q + p*x^3)^6))/x^17,x]
(((q + p*x^3)*Sqrt[q^2 - 2*p*q*(-1 + x)*x^3 + p^2*x^6]*(24*b*x^12 + a*(6*q ^6 - 2*p*q^5*(-18 + x)*x^3 - 2*p^5*q*(-18 + x)*x^15 + 6*p^6*x^18 + p^2*q^4 *x^6*(90 - 8*x - 5*x^2) + p^4*q^2*x^12*(90 - 8*x - 5*x^2) - p^3*q^3*x^9*(- 120 + 12*x + 10*x^2 + 15*x^3))))/x^16 + 12*(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^3 + 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.
| \(\displaystyle \int \frac {\left (p x^3-2 q\right ) \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (a \left (p x^3+q\right )^6+b x^{12}\right )}{x^{17}} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 q \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (5 a p^4 q^2+b\right )}{x^5}+\frac {p \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (3 a p^4 q^2+b\right )}{x^2}-\frac {2 a q^7 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^{17}}-\frac {11 a p q^6 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^{14}}-\frac {24 a p^2 q^5 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^{11}}+a p^7 x^4 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+4 a p^6 q x \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}-\frac {25 a p^3 q^4 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 q \left (5 a p^4 q^2+b\right ) \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^5}dx+p \left (3 a p^4 q^2+b\right ) \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^2}dx-2 a q^7 \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^{17}}dx-11 a p q^6 \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^{14}}dx-24 a p^2 q^5 \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^{11}}dx+a p^7 \int x^4 \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}dx+4 a p^6 q \int x \sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}dx-25 a p^3 q^4 \int \frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}}{x^8}dx\) |
Int[((-2*q + p*x^3)*Sqrt[q^2 + 2*p*q*x^3 - 2*p*q*x^4 + p^2*x^6]*(b*x^12 + a*(q + p*x^3)^6))/x^17,x]
3.30.48.3.1 Defintions of rubi rules used
Time = 0.32 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(\frac {-5 p \,x^{15} q \left (a \,p^{3} q^{3}+\frac {8 b}{5}\right ) \ln \left (\frac {p \,x^{3}+\sqrt {\frac {p^{2} x^{6}-2 x^{3} p q \left (-1+x \right )+q^{2}}{x^{2}}}\, x +q}{x^{2}}\right )+\left (p \,x^{3}+q \right ) \left (a \,p^{6} x^{18}-\frac {a \,p^{5} q \,x^{16}}{3}+6 a \,p^{5} q \,x^{15}-\frac {5 a \,p^{4} q^{2} x^{14}}{6}-\frac {4 a \,p^{4} q^{2} x^{13}}{3}+\left (15 a \,p^{4} q^{2}-\frac {5}{2} a \,p^{3} q^{3}+4 b \right ) x^{12}-\frac {5 a \,p^{3} q^{3} x^{11}}{3}-2 a \,p^{3} q^{3} x^{10}+20 a \,p^{3} q^{3} x^{9}-\frac {5 a \,p^{2} q^{4} x^{8}}{6}-\frac {4 a \,p^{2} q^{4} x^{7}}{3}+15 a \,p^{2} q^{4} x^{6}-\frac {a p \,q^{5} x^{4}}{3}+6 a p \,q^{5} x^{3}+a \,q^{6}\right ) \sqrt {\frac {p^{2} x^{6}-2 x^{3} p q \left (-1+x \right )+q^{2}}{x^{2}}}}{8 x^{15}}\) | \(278\) |
int((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p*x^3+q )^6)/x^17,x,method=_RETURNVERBOSE)
1/8*(-5*p*x^15*q*(a*p^3*q^3+8/5*b)*ln((p*x^3+((p^2*x^6-2*x^3*p*q*(-1+x)+q^ 2)/x^2)^(1/2)*x+q)/x^2)+(p*x^3+q)*(a*p^6*x^18-1/3*a*p^5*q*x^16+6*a*p^5*q*x ^15-5/6*a*p^4*q^2*x^14-4/3*a*p^4*q^2*x^13+(15*a*p^4*q^2-5/2*a*p^3*q^3+4*b) *x^12-5/3*a*p^3*q^3*x^11-2*a*p^3*q^3*x^10+20*a*p^3*q^3*x^9-5/6*a*p^2*q^4*x ^8-4/3*a*p^2*q^4*x^7+15*a*p^2*q^4*x^6-1/3*a*p*q^5*x^4+6*a*p*q^5*x^3+a*q^6) *((p^2*x^6-2*x^3*p*q*(-1+x)+q^2)/x^2)^(1/2))/x^15
Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx=\text {Timed out} \]
integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p *x^3+q)^6)/x^17,x, algorithm="fricas")
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx=\int \frac {\left (p x^{3} - 2 q\right ) \sqrt {p^{2} x^{6} - 2 p q x^{4} + 2 p q x^{3} + q^{2}} \left (a p^{6} x^{18} + 6 a p^{5} q x^{15} + 15 a p^{4} q^{2} x^{12} + 20 a p^{3} q^{3} x^{9} + 15 a p^{2} q^{4} x^{6} + 6 a p q^{5} x^{3} + a q^{6} + b x^{12}\right )}{x^{17}}\, dx \]
integrate((p*x**3-2*q)*(p**2*x**6-2*p*q*x**4+2*p*q*x**3+q**2)**(1/2)*(b*x* *12+a*(p*x**3+q)**6)/x**17,x)
Integral((p*x**3 - 2*q)*sqrt(p**2*x**6 - 2*p*q*x**4 + 2*p*q*x**3 + 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**12)/x**17, x)
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx=\int { \frac {{\left (b x^{12} + {\left (p x^{3} + q\right )}^{6} a\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{17}} \,d x } \]
integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p *x^3+q)^6)/x^17,x, algorithm="maxima")
integrate((b*x^12 + (p*x^3 + q)^6*a)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^17, x)
\[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx=\int { \frac {{\left (b x^{12} + {\left (p x^{3} + q\right )}^{6} a\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{17}} \,d x } \]
integrate((p*x^3-2*q)*(p^2*x^6-2*p*q*x^4+2*p*q*x^3+q^2)^(1/2)*(b*x^12+a*(p *x^3+q)^6)/x^17,x, algorithm="giac")
integrate((b*x^12 + (p*x^3 + q)^6*a)*sqrt(p^2*x^6 - 2*p*q*x^4 + 2*p*q*x^3 + q^2)*(p*x^3 - 2*q)/x^17, x)
Timed out. \[ \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx=\text {Hanged} \]