Integrand size = 28, antiderivative size = 432 \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\frac {e^{-\frac {a}{b p q}} (f g-e h)^2 (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^3 p^3 q^3}+\frac {4 e^{-\frac {2 a}{b p q}} h (f g-e h) (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^3 f^3 p^3 q^3}+\frac {9 e^{-\frac {3 a}{b p q}} h^2 (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{2 b^3 f^3 p^3 q^3}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac {(f g-e h) (e+f x) (g+h x)}{b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {3 (e+f x) (g+h x)^2}{2 b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \]
1/2*(-e*h+f*g)^2*(f*x+e)*Ei((a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)/b^3/exp(a/b /p/q)/f^3/p^3/q^3/((c*(d*(f*x+e)^p)^q)^(1/p/q))+4*h*(-e*h+f*g)*(f*x+e)^2*E i(2*(a+b*ln(c*(d*(f*x+e)^p)^q))/b/p/q)/b^3/exp(2*a/b/p/q)/f^3/p^3/q^3/((c* (d*(f*x+e)^p)^q)^(2/p/q))+9/2*h^2*(f*x+e)^3*Ei(3*(a+b*ln(c*(d*(f*x+e)^p)^q ))/b/p/q)/b^3/exp(3*a/b/p/q)/f^3/p^3/q^3/((c*(d*(f*x+e)^p)^q)^(3/p/q))-1/2 *(f*x+e)*(h*x+g)^2/b/f/p/q/(a+b*ln(c*(d*(f*x+e)^p)^q))^2+(-e*h+f*g)*(f*x+e )*(h*x+g)/b^2/f^2/p^2/q^2/(a+b*ln(c*(d*(f*x+e)^p)^q))-3/2*(f*x+e)*(h*x+g)^ 2/b^2/f/p^2/q^2/(a+b*ln(c*(d*(f*x+e)^p)^q))
Time = 1.28 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.01 \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\frac {e^{-\frac {3 a}{b p q}} (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \left (e^{\frac {2 a}{b p q}} (f g-e h)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {2}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-8 e^{\frac {a}{b p q}} h (-f g+e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+9 h^2 (e+f x)^2 \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-b e^{\frac {3 a}{b p q}} f p q \left (c \left (d (e+f x)^p\right )^q\right )^{\frac {3}{p q}} (g+h x) \left (b f p q (g+h x)+a (f g+2 e h+3 f h x)+b (2 e h+f (g+3 h x)) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{2 b^3 f^3 p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \]
((e + f*x)*(E^((2*a)/(b*p*q))*(f*g - e*h)^2*(c*(d*(e + f*x)^p)^q)^(2/(p*q) )*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)]*(a + b*Log[c*(d* (e + f*x)^p)^q])^2 - 8*E^(a/(b*p*q))*h*(-(f*g) + e*h)*(e + f*x)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/( b*p*q)]*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + 9*h^2*(e + f*x)^2*ExpIntegral Ei[(3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)]*(a + b*Log[c*(d*(e + f*x) ^p)^q])^2 - b*E^((3*a)/(b*p*q))*f*p*q*(c*(d*(e + f*x)^p)^q)^(3/(p*q))*(g + h*x)*(b*f*p*q*(g + h*x) + a*(f*g + 2*e*h + 3*f*h*x) + b*(2*e*h + f*(g + 3 *h*x))*Log[c*(d*(e + f*x)^p)^q])))/(2*b^3*E^((3*a)/(b*p*q))*f^3*p^3*q^3*(c *(d*(e + f*x)^p)^q)^(3/(p*q))*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)
Leaf count is larger than twice the leaf count of optimal. \(951\) vs. \(2(432)=864\).
Time = 3.25 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.20, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2895, 2847, 2847, 2836, 2737, 2609, 2846, 2009}
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.
| \(\displaystyle \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}dx\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {(f g-e h) \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}dx}{b f p q}+\frac {3 \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}dx}{2 b p q}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {(f g-e h) \left (-\frac {(f g-e h) \int \frac {1}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b f p q}+\frac {2 \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{b f p q}+\frac {3 \left (-\frac {2 (f g-e h) \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b f p q}+\frac {3 \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{2 b p q}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\) |
| \(\Big \downarrow \) 2836 |
| \(\displaystyle -\frac {(f g-e h) \left (-\frac {(f g-e h) \int \frac {1}{a+b \log \left (c d^q (e+f x)^{p q}\right )}d(e+f x)}{b f^2 p q}+\frac {2 \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{b f p q}+\frac {3 \left (-\frac {2 (f g-e h) \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b f p q}+\frac {3 \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{2 b p q}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\) |
| \(\Big \downarrow \) 2737 |
| \(\displaystyle -\frac {(f g-e h) \left (-\frac {(e+f x) (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \frac {\left (c d^q (e+f x)^{p q}\right )^{\frac {1}{p q}}}{a+b \log \left (c d^q (e+f x)^{p q}\right )}d\log \left (c d^q (e+f x)^{p q}\right )}{b f^2 p^2 q^2}+\frac {2 \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{b f p q}+\frac {3 \left (-\frac {2 (f g-e h) \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b f p q}+\frac {3 \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{2 b p q}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\) |
| \(\Big \downarrow \) 2609 |
| \(\displaystyle -\frac {(f g-e h) \left (\frac {2 \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{b f p q}+\frac {3 \left (-\frac {2 (f g-e h) \int \frac {g+h x}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b f p q}+\frac {3 \int \frac {(g+h x)^2}{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}dx}{b p q}-\frac {(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{2 b p q}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\) |
| \(\Big \downarrow \) 2846 |
| \(\displaystyle -\frac {(f g-e h) \left (\frac {2 \int \left (\frac {f g-e h}{f \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\frac {h (e+f x)}{f \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )dx}{b p q}-\frac {(e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right )}{b^2 f^2 p^2 q^2}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{b f p q}+\frac {3 \left (\frac {3 \int \left (\frac {(f g-e h)^2}{f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\frac {2 h (e+f x) (f g-e h)}{f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\frac {h^2 (e+f x)^2}{f^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )dx}{b p q}-\frac {2 (f g-e h) \int \left (\frac {f g-e h}{f \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\frac {h (e+f x)}{f \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )dx}{b f p q}-\frac {(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{2 b p q}-\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(e+f x) (g+h x)^2}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}-\frac {(f g-e h) \left (-\frac {e^{-\frac {a}{b p q}} (f g-e h) (e+f x) \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}\right ) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}}{b^2 f^2 p^2 q^2}+\frac {2 \left (\frac {e^{-\frac {2 a}{b p q}} h (e+f x)^2 \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}}}{b f^2 p q}+\frac {e^{-\frac {a}{b p q}} (f g-e h) (e+f x) \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}}{b f^2 p q}\right )}{b p q}-\frac {(e+f x) (g+h x)}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\right )}{b f p q}+\frac {3 \left (-\frac {(e+f x) (g+h x)^2}{b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac {2 (f g-e h) \left (\frac {e^{-\frac {2 a}{b p q}} h (e+f x)^2 \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}}}{b f^2 p q}+\frac {e^{-\frac {a}{b p q}} (f g-e h) (e+f x) \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}}{b f^2 p q}\right )}{b f p q}+\frac {3 \left (\frac {e^{-\frac {3 a}{b p q}} h^2 (e+f x)^3 \operatorname {ExpIntegralEi}\left (\frac {3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}}}{b f^3 p q}+\frac {2 e^{-\frac {2 a}{b p q}} h (f g-e h) (e+f x)^2 \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}}}{b f^3 p q}+\frac {e^{-\frac {a}{b p q}} (f g-e h)^2 (e+f x) \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}}{b f^3 p q}\right )}{b p q}\right )}{2 b p q}\) |
-1/2*((e + f*x)*(g + h*x)^2)/(b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^2) - ((f*g - e*h)*(-(((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c*d^q*(e + f*x)^(p*q)])/(b*p*q)])/(b^2*E^(a/(b*p*q))*f^2*p^2*q^2*(c*d^q*(e + f*x)^ (p*q))^(1/(p*q)))) + (2*(((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c *(d*(e + f*x)^p)^q])/(b*p*q)])/(b*E^(a/(b*p*q))*f^2*p*q*(c*(d*(e + f*x)^p) ^q)^(1/(p*q))) + (h*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x) ^p)^q]))/(b*p*q)])/(b*E^((2*a)/(b*p*q))*f^2*p*q*(c*(d*(e + f*x)^p)^q)^(2/( p*q)))))/(b*p*q) - ((e + f*x)*(g + h*x))/(b*f*p*q*(a + b*Log[c*(d*(e + f*x )^p)^q]))))/(b*f*p*q) + (3*((-2*(f*g - e*h)*(((f*g - e*h)*(e + f*x)*ExpInt egralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(b*E^(a/(b*p*q))*f^2*p* q*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (h*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(b*E^((2*a)/(b*p*q))*f^2*p*q*(c*(d* (e + f*x)^p)^q)^(2/(p*q)))))/(b*f*p*q) + (3*(((f*g - e*h)^2*(e + f*x)*ExpI ntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(b*E^(a/(b*p*q))*f^3* p*q*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (2*h*(f*g - e*h)*(e + f*x)^2*ExpInt egralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(b*E^((2*a)/(b*p*q) )*f^3*p*q*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) + (h^2*(e + f*x)^3*ExpIntegralE i[(3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p*q)])/(b*E^((3*a)/(b*p*q))*f^3* p*q*(c*(d*(e + f*x)^p)^q)^(3/(p*q)))))/(b*p*q) - ((e + f*x)*(g + h*x)^2)/( b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q]))))/(2*b*p*q)
3.5.55.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x ^n)^(1/n)) Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ [{a, b, c, n, p}, x]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[1/e Subst[Int[(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{ a, b, c, d, e, n, p}, x]
Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.) ]*(b_.)), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e* x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] & & IGtQ[q, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e *x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1)) Int[( f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) /(b*e*n*(p + 1))) Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 ), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt Q[p, -1] && GtQ[q, 0]
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
\[\int \frac {\left (h x +g \right )^{2}}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{3}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1682 vs. \(2 (426) = 852\).
Time = 0.37 (sec) , antiderivative size = 1682, normalized size of antiderivative = 3.89 \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\text {Too large to display} \]
1/2*(8*((b^2*f*g*h - b^2*e*h^2)*p^2*q^2*log(f*x + e)^2 + a^2*f*g*h - a^2*e *h^2 + (b^2*f*g*h - b^2*e*h^2)*q^2*log(d)^2 + (b^2*f*g*h - b^2*e*h^2)*log( c)^2 + 2*((b^2*f*g*h - b^2*e*h^2)*p*q^2*log(d) + (b^2*f*g*h - b^2*e*h^2)*p *q*log(c) + (a*b*f*g*h - a*b*e*h^2)*p*q)*log(f*x + e) + 2*(a*b*f*g*h - a*b *e*h^2)*log(c) + 2*((b^2*f*g*h - b^2*e*h^2)*q*log(c) + (a*b*f*g*h - a*b*e* h^2)*q)*log(d))*e^((b*q*log(d) + b*log(c) + a)/(b*p*q))*log_integral((f^2* x^2 + 2*e*f*x + e^2)*e^(2*(b*q*log(d) + b*log(c) + a)/(b*p*q))) + ((b^2*f^ 2*g^2 - 2*b^2*e*f*g*h + b^2*e^2*h^2)*p^2*q^2*log(f*x + e)^2 + a^2*f^2*g^2 - 2*a^2*e*f*g*h + a^2*e^2*h^2 + (b^2*f^2*g^2 - 2*b^2*e*f*g*h + b^2*e^2*h^2 )*q^2*log(d)^2 + (b^2*f^2*g^2 - 2*b^2*e*f*g*h + b^2*e^2*h^2)*log(c)^2 + 2* ((b^2*f^2*g^2 - 2*b^2*e*f*g*h + b^2*e^2*h^2)*p*q^2*log(d) + (b^2*f^2*g^2 - 2*b^2*e*f*g*h + b^2*e^2*h^2)*p*q*log(c) + (a*b*f^2*g^2 - 2*a*b*e*f*g*h + a*b*e^2*h^2)*p*q)*log(f*x + e) + 2*(a*b*f^2*g^2 - 2*a*b*e*f*g*h + a*b*e^2* h^2)*log(c) + 2*((b^2*f^2*g^2 - 2*b^2*e*f*g*h + b^2*e^2*h^2)*q*log(c) + (a *b*f^2*g^2 - 2*a*b*e*f*g*h + a*b*e^2*h^2)*q)*log(d))*e^(2*(b*q*log(d) + b* log(c) + a)/(b*p*q))*log_integral((f*x + e)*e^((b*q*log(d) + b*log(c) + a) /(b*p*q))) - (b^2*e*f^2*g^2*p^2*q^2 + (b^2*f^3*h^2*p^2*q^2 + 3*a*b*f^3*h^2 *p*q)*x^3 + (a*b*e*f^2*g^2 + 2*a*b*e^2*f*g*h)*p*q + ((2*b^2*f^3*g*h + b^2* e*f^2*h^2)*p^2*q^2 + (4*a*b*f^3*g*h + 5*a*b*e*f^2*h^2)*p*q)*x^2 + ((b^2*f^ 3*g^2 + 2*b^2*e*f^2*g*h)*p^2*q^2 + (a*b*f^3*g^2 + 6*a*b*e*f^2*g*h + 2*a...
\[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int \frac {\left (g + h x\right )^{2}}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{3}}\, dx \]
\[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int { \frac {{\left (h x + g\right )}^{2}}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{3}} \,d x } \]
-1/2*((3*a*f^2*h^2 + (f^2*h^2*p*q + 3*f^2*h^2*q*log(d) + 3*f^2*h^2*log(c)) *b)*x^3 + ((4*f^2*g*h + 5*e*f*h^2)*a + (2*f^2*g*h*p*q + e*f*h^2*p*q + (4*f ^2*g*h + 5*e*f*h^2)*log(c) + (4*f^2*g*h*q + 5*e*f*h^2*q)*log(d))*b)*x^2 + (e*f*g^2 + 2*e^2*g*h)*a + (e*f*g^2*p*q + (e*f*g^2 + 2*e^2*g*h)*log(c) + (e *f*g^2*q + 2*e^2*g*h*q)*log(d))*b + ((f^2*g^2 + 6*e*f*g*h + 2*e^2*h^2)*a + (f^2*g^2*p*q + 2*e*f*g*h*p*q + (f^2*g^2 + 6*e*f*g*h + 2*e^2*h^2)*log(c) + (f^2*g^2*q + 6*e*f*g*h*q + 2*e^2*h^2*q)*log(d))*b)*x + (3*b*f^2*h^2*x^3 + (4*f^2*g*h + 5*e*f*h^2)*b*x^2 + (f^2*g^2 + 6*e*f*g*h + 2*e^2*h^2)*b*x + ( e*f*g^2 + 2*e^2*g*h)*b)*log(((f*x + e)^p)^q))/(b^4*f^2*p^2*q^2*log(((f*x + e)^p)^q)^2 + a^2*b^2*f^2*p^2*q^2 + 2*(f^2*p^2*q^3*log(d) + f^2*p^2*q^2*lo g(c))*a*b^3 + (f^2*p^2*q^4*log(d)^2 + 2*f^2*p^2*q^3*log(c)*log(d) + f^2*p^ 2*q^2*log(c)^2)*b^4 + 2*(a*b^3*f^2*p^2*q^2 + (f^2*p^2*q^3*log(d) + f^2*p^2 *q^2*log(c))*b^4)*log(((f*x + e)^p)^q)) + integrate(1/2*(9*f^2*h^2*x^2 + f ^2*g^2 + 6*e*f*g*h + 2*e^2*h^2 + 2*(4*f^2*g*h + 5*e*f*h^2)*x)/(b^3*f^2*p^2 *q^2*log(((f*x + e)^p)^q) + a*b^2*f^2*p^2*q^2 + (f^2*p^2*q^3*log(d) + f^2* p^2*q^2*log(c))*b^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 5889 vs. \(2 (426) = 852\).
Time = 0.54 (sec) , antiderivative size = 5889, normalized size of antiderivative = 13.63 \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\text {Too large to display} \]
-1/2*((f*x + e)*b^2*f^2*g^2*p^2*q^2*log(f*x + e) + 4*(f*x + e)^2*b^2*f*g*h *p^2*q^2*log(f*x + e) - 2*(f*x + e)*b^2*e*f*g*h*p^2*q^2*log(f*x + e) + 3*( f*x + e)^3*b^2*h^2*p^2*q^2*log(f*x + e) - 4*(f*x + e)^2*b^2*e*h^2*p^2*q^2* log(f*x + e) + (f*x + e)*b^2*e^2*h^2*p^2*q^2*log(f*x + e) - b^2*f^2*g^2*p^ 2*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) )*log(f*x + e)^2/(c^(1/(p*q))*d^(1/p)) + 2*b^2*e*f*g*h*p^2*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(f*x + e)^2/ (c^(1/(p*q))*d^(1/p)) - b^2*e^2*h^2*p^2*q^2*Ei(log(d)/p + log(c)/(p*q) + a /(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(f*x + e)^2/(c^(1/(p*q))*d^(1/p )) + (f*x + e)*b^2*f^2*g^2*p^2*q^2 + 2*(f*x + e)^2*b^2*f*g*h*p^2*q^2 - 2*( f*x + e)*b^2*e*f*g*h*p^2*q^2 + (f*x + e)^3*b^2*h^2*p^2*q^2 - 2*(f*x + e)^2 *b^2*e*h^2*p^2*q^2 + (f*x + e)*b^2*e^2*h^2*p^2*q^2 - 8*b^2*f*g*h*p^2*q^2*E i(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p *q))*log(f*x + e)^2/(c^(2/(p*q))*d^(2/p)) + 8*b^2*e*h^2*p^2*q^2*Ei(2*log(d )/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log( f*x + e)^2/(c^(2/(p*q))*d^(2/p)) + (f*x + e)*b^2*f^2*g^2*p*q^2*log(d) + 4* (f*x + e)^2*b^2*f*g*h*p*q^2*log(d) - 2*(f*x + e)*b^2*e*f*g*h*p*q^2*log(d) + 3*(f*x + e)^3*b^2*h^2*p*q^2*log(d) - 4*(f*x + e)^2*b^2*e*h^2*p*q^2*log(d ) + (f*x + e)*b^2*e^2*h^2*p*q^2*log(d) - 2*b^2*f^2*g^2*p*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(f*x + e)*l...
Timed out. \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx=\int \frac {{\left (g+h\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^3} \,d x \]