Integrand size = 30, antiderivative size = 514 \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx=\frac {4 e^{-\frac {a}{b p q}} (f g-e h)^2 \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{3 b^{5/2} f^3 p^{5/2} q^{5/2}}+\frac {16 e^{-\frac {2 a}{b p q}} h (f g-e h) \sqrt {2 \pi } (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{3 b^{5/2} f^3 p^{5/2} q^{5/2}}+\frac {4 e^{-\frac {3 a}{b p q}} h^2 \sqrt {3 \pi } (e+f x)^3 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{5/2} f^3 p^{5/2} q^{5/2}}-\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}+\frac {8 (f g-e h) (e+f x) (g+h x)}{3 b^2 f^2 p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {4 (e+f x) (g+h x)^2}{b^2 f p^2 q^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \] Output:
4/3*(-e*h+f*g)^2*Pi^(1/2)*(f*x+e)*erfi((a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/b ^(1/2)/p^(1/2)/q^(1/2))/b^(5/2)/exp(a/b/p/q)/f^3/p^(5/2)/q^(5/2)/((c*(d*(f *x+e)^p)^q)^(1/p/q))+16/3*h*(-e*h+f*g)*2^(1/2)*Pi^(1/2)*(f*x+e)^2*erfi(2^( 1/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/b^(1/2)/p^(1/2)/q^(1/2))/b^(5/2)/ex p(2*a/b/p/q)/f^3/p^(5/2)/q^(5/2)/((c*(d*(f*x+e)^p)^q)^(2/p/q))+4*h^2*3^(1/ 2)*Pi^(1/2)*(f*x+e)^3*erfi(3^(1/2)*(a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/b^(1/ 2)/p^(1/2)/q^(1/2))/b^(5/2)/exp(3*a/b/p/q)/f^3/p^(5/2)/q^(5/2)/((c*(d*(f*x +e)^p)^q)^(3/p/q))-2/3*(f*x+e)*(h*x+g)^2/b/f/p/q/(a+b*ln(c*(d*(f*x+e)^p)^q ))^(3/2)+8/3*(-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))^(1/2)-4*(f*x+e)*(h*x+g)^2/b^2/f/p^2/q^2/(a+b*ln(c*(d*(f*x+e)^p)^ q))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(6370\) vs. \(2(514)=1028\).
Time = 6.29 (sec) , antiderivative size = 6370, normalized size of antiderivative = 12.39 \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(g + h*x)^2/(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2),x]
Output:
Result too large to show
Leaf count is larger than twice the leaf count of optimal. \(1154\) vs. \(2(514)=1028\).
Time = 9.37 (sec) , antiderivative size = 1154, normalized size of antiderivative = 2.25, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2895, 2847, 2847, 2836, 2737, 2611, 2633, 2848, 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.
| \(\displaystyle \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {4 (f g-e h) \int \frac {g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}dx}{3 b f p q}+\frac {2 \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}dx}{b p q}-\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {4 (f g-e h) \left (-\frac {2 (f g-e h) \int \frac {1}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b f p q}+\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{3 b f p q}+\frac {2 \left (-\frac {4 (f g-e h) \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b f p q}+\frac {6 \int \frac {(g+h x)^2}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{b p q}-\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2836 |
| \(\displaystyle -\frac {4 (f g-e h) \left (-\frac {2 (f g-e h) \int \frac {1}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}d(e+f x)}{b f^2 p q}+\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{3 b f p q}+\frac {2 \left (-\frac {4 (f g-e h) \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b f p q}+\frac {6 \int \frac {(g+h x)^2}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{b p q}-\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2737 |
| \(\displaystyle -\frac {4 (f g-e h) \left (-\frac {2 (e+f x) (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \frac {\left (c d^q (e+f x)^{p q}\right )^{\frac {1}{p q}}}{\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}d\log \left (c d^q (e+f x)^{p q}\right )}{b f^2 p^2 q^2}+\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{3 b f p q}+\frac {2 \left (-\frac {4 (f g-e h) \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b f p q}+\frac {6 \int \frac {(g+h x)^2}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{b p q}-\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {4 (f g-e h) \left (-\frac {4 (e+f x) (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \int \exp \left (\frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{b p q}-\frac {a}{b p q}\right )d\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{b^2 f^2 p^2 q^2}+\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{3 b f p q}+\frac {2 \left (-\frac {4 (f g-e h) \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b f p q}+\frac {6 \int \frac {(g+h x)^2}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{b p q}-\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {4 (f g-e h) \left (\frac {4 \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{3 b f p q}+\frac {2 \left (-\frac {4 (f g-e h) \int \frac {g+h x}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b f p q}+\frac {6 \int \frac {(g+h x)^2}{\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}dx}{b p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{b p q}-\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle -\frac {4 (f g-e h) \left (\frac {4 \int \left (\frac {f g-e h}{f \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\frac {h (e+f x)}{f \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )dx}{b p q}-\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b p q}} (f g-e h) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right )}{b^{3/2} f^2 p^{3/2} q^{3/2}}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{3 b f p q}+\frac {2 \left (\frac {6 \int \left (\frac {(f g-e h)^2}{f^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\frac {2 h (e+f x) (f g-e h)}{f^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\frac {h^2 (e+f x)^2}{f^2 \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )dx}{b p q}-\frac {4 (f g-e h) \int \left (\frac {f g-e h}{f \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}+\frac {h (e+f x)}{f \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )dx}{b f p q}-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{b p q}-\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (e+f x) (g+h x)^2}{3 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}-\frac {4 (f g-e h) \left (-\frac {2 e^{-\frac {a}{b p q}} (f g-e h) \sqrt {\pi } (e+f x) \text {erfi}\left (\frac {\sqrt {a+b \log \left (c d^q (e+f x)^{p q}\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \left (c d^q (e+f x)^{p q}\right )^{-\frac {1}{p q}}}{b^{3/2} f^2 p^{3/2} q^{3/2}}+\frac {4 \left (\frac {e^{-\frac {2 a}{b p q}} h \sqrt {\frac {\pi }{2}} (e+f x)^2 \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}}}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}+\frac {e^{-\frac {a}{b p q}} (f g-e h) \sqrt {\pi } (e+f x) \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}\right )}{b p q}-\frac {2 (e+f x) (g+h x)}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}\right )}{3 b f p q}+\frac {2 \left (-\frac {2 (e+f x) (g+h x)^2}{b f p q \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}-\frac {4 (f g-e h) \left (\frac {e^{-\frac {2 a}{b p q}} h \sqrt {\frac {\pi }{2}} (e+f x)^2 \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}}}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}+\frac {e^{-\frac {a}{b p q}} (f g-e h) \sqrt {\pi } (e+f x) \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}}{\sqrt {b} f^2 \sqrt {p} \sqrt {q}}\right )}{b f p q}+\frac {6 \left (\frac {e^{-\frac {3 a}{b p q}} h^2 \sqrt {\frac {\pi }{3}} (e+f x)^3 \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {3}{p q}}}{\sqrt {b} f^3 \sqrt {p} \sqrt {q}}+\frac {e^{-\frac {2 a}{b p q}} h (f g-e h) \sqrt {2 \pi } (e+f x)^2 \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {2}{p q}}}{\sqrt {b} f^3 \sqrt {p} \sqrt {q}}+\frac {e^{-\frac {a}{b p q}} (f g-e h)^2 \sqrt {\pi } (e+f x) \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{\sqrt {b} \sqrt {p} \sqrt {q}}\right ) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac {1}{p q}}}{\sqrt {b} f^3 \sqrt {p} \sqrt {q}}\right )}{b p q}\right )}{b p q}\) |
Input:
Int[(g + h*x)^2/(a + b*Log[c*(d*(e + f*x)^p)^q])^(5/2),x]
Output:
(-2*(e + f*x)*(g + h*x)^2)/(3*b*f*p*q*(a + b*Log[c*(d*(e + f*x)^p)^q])^(3/ 2)) - (4*(f*g - e*h)*((-2*(f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*L og[c*d^q*(e + f*x)^(p*q)]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(b^(3/2)*E^(a/(b*p* q))*f^2*p^(3/2)*q^(3/2)*(c*d^q*(e + f*x)^(p*q))^(1/(p*q))) + (4*(((f*g - e *h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]* Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^(a/(b*p*q))*f^2*Sqrt[p]*Sqrt[q]*(c*(d*(e + f *x)^p)^q)^(1/(p*q))) + (h*Sqrt[Pi/2]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b* Log[c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^((2*a)/( b*p*q))*f^2*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(2/(p*q)))))/(b*p*q) - ( 2*(e + f*x)*(g + h*x))/(b*f*p*q*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])))/(3 *b*f*p*q) + (2*((-4*(f*g - e*h)*(((f*g - e*h)*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt [a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^(a /(b*p*q))*f^2*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (h*Sqrt[P i/2]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]])/(Sqrt [b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^((2*a)/(b*p*q))*f^2*Sqrt[p]*Sqrt[q]*(c*( d*(e + f*x)^p)^q)^(2/(p*q)))))/(b*f*p*q) + (6*(((f*g - e*h)^2*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]/(Sqrt[b]*Sqrt[p]*Sqrt[q]) ])/(Sqrt[b]*E^(a/(b*p*q))*f^3*Sqrt[p]*Sqrt[q]*(c*(d*(e + f*x)^p)^q)^(1/(p* q))) + (h*(f*g - e*h)*Sqrt[2*Pi]*(e + f*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[ c*(d*(e + f*x)^p)^q]])/(Sqrt[b]*Sqrt[p]*Sqrt[q])])/(Sqrt[b]*E^((2*a)/(b...
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
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[((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_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[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]]
\[\int \frac {\left (h x +g \right )^{2}}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{\frac {5}{2}}}d x\]
Input:
int((h*x+g)^2/(a+b*ln(c*(d*(f*x+e)^p)^q))^(5/2),x)
Output:
int((h*x+g)^2/(a+b*ln(c*(d*(f*x+e)^p)^q))^(5/2),x)
Exception generated. \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((h*x+g)^2/(a+b*log(c*(d*(f*x+e)^p)^q))^(5/2),x, algorithm="frica s")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx=\int \frac {\left (g + h x\right )^{2}}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((h*x+g)**2/(a+b*ln(c*(d*(f*x+e)**p)**q))**(5/2),x)
Output:
Integral((g + h*x)**2/(a + b*log(c*(d*(e + f*x)**p)**q))**(5/2), x)
\[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx=\int { \frac {{\left (h x + g\right )}^{2}}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((h*x+g)^2/(a+b*log(c*(d*(f*x+e)^p)^q))^(5/2),x, algorithm="maxim a")
Output:
integrate((h*x + g)^2/(b*log(((f*x + e)^p*d)^q*c) + a)^(5/2), x)
\[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx=\int { \frac {{\left (h x + g\right )}^{2}}{{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((h*x+g)^2/(a+b*log(c*(d*(f*x+e)^p)^q))^(5/2),x, algorithm="giac" )
Output:
integrate((h*x + g)^2/(b*log(((f*x + e)^p*d)^q*c) + a)^(5/2), x)
Timed out. \[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx=\int \frac {{\left (g+h\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^{5/2}} \,d x \] Input:
int((g + h*x)^2/(a + b*log(c*(d*(e + f*x)^p)^q))^(5/2),x)
Output:
int((g + h*x)^2/(a + b*log(c*(d*(e + f*x)^p)^q))^(5/2), x)
\[ \int \frac {(g+h x)^2}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((h*x+g)^2/(a+b*log(c*(d*(f*x+e)^p)^q))^(5/2),x)
Output:
( - 2*sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*e*g**2 + 3*int((sqrt(log(d* *q*(e + f*x)**(p*q)*c)*b + a)*x**3)/(log(d**q*(e + f*x)**(p*q)*c)**3*b**3* e + log(d**q*(e + f*x)**(p*q)*c)**3*b**3*f*x + 3*log(d**q*(e + f*x)**(p*q) *c)**2*a*b**2*e + 3*log(d**q*(e + f*x)**(p*q)*c)**2*a*b**2*f*x + 3*log(d** q*(e + f*x)**(p*q)*c)*a**2*b*e + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*f*x + a**3*e + a**3*f*x),x)*log(d**q*(e + f*x)**(p*q)*c)**2*b**3*f**2*h**2*p* q + 6*int((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*x**3)/(log(d**q*(e + f *x)**(p*q)*c)**3*b**3*e + log(d**q*(e + f*x)**(p*q)*c)**3*b**3*f*x + 3*log (d**q*(e + f*x)**(p*q)*c)**2*a*b**2*e + 3*log(d**q*(e + f*x)**(p*q)*c)**2* a*b**2*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*e + 3*log(d**q*(e + f*x )**(p*q)*c)*a**2*b*f*x + a**3*e + a**3*f*x),x)*log(d**q*(e + f*x)**(p*q)*c )*a*b**2*f**2*h**2*p*q + 3*int((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*x **3)/(log(d**q*(e + f*x)**(p*q)*c)**3*b**3*e + log(d**q*(e + f*x)**(p*q)*c )**3*b**3*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)**2*a*b**2*e + 3*log(d**q*(e + f*x)**(p*q)*c)**2*a*b**2*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*e + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*f*x + a**3*e + a**3*f*x),x)*a**2*b *f**2*h**2*p*q + 3*int((sqrt(log(d**q*(e + f*x)**(p*q)*c)*b + a)*x**2)/(lo g(d**q*(e + f*x)**(p*q)*c)**3*b**3*e + log(d**q*(e + f*x)**(p*q)*c)**3*b** 3*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)**2*a*b**2*e + 3*log(d**q*(e + f*x)* *(p*q)*c)**2*a*b**2*f*x + 3*log(d**q*(e + f*x)**(p*q)*c)*a**2*b*e + 3*l...