Integrand size = 26, antiderivative size = 421 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\frac {4 e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac {16 e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{3 b^{5/2} e^3 n^{5/2}}+\frac {4 e^{-\frac {3 a}{b n}} g^2 \sqrt {3 \pi } (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{5/2} e^3 n^{5/2}}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}+\frac {8 (e f-d g) (d+e x) (f+g x)}{3 b^2 e^2 n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x) (f+g x)^2}{b^2 e n^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
-2/3*(e*x+d)*(g*x+f)^2/b/e/n/(a+b*ln(c*(e*x+d)^n))^(3/2)+4/3*(-d*g+e*f)^2* (e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*Pi^(1/2)/b^(5/2) /e^3/exp(a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(1/n))+16/3*g*(-d*g+e*f)*(e*x+d)^2* erfi(2^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*2^(1/2)*Pi^(1/2) /b^(5/2)/e^3/exp(2*a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(2/n))+4*g^2*(e*x+d)^3*er fi(3^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*3^(1/2)*Pi^(1/2)/b ^(5/2)/e^3/exp(3*a/b/n)/n^(5/2)/((c*(e*x+d)^n)^(3/n))+8/3*(-d*g+e*f)*(e*x+ d)*(g*x+f)/b^2/e^2/n^2/(a+b*ln(c*(e*x+d)^n))^(1/2)-4*(e*x+d)*(g*x+f)^2/b^2 /e/n^2/(a+b*ln(c*(e*x+d)^n))^(1/2)
Time = 2.64 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.25 \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=-\frac {2 e^{-\frac {3 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-3/n} \left (2 d e^{\frac {2 a}{b n}} g (8 e f+d g) \sqrt {\pi } \left (c (d+e x)^n\right )^{2/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}+8 e^{\frac {a}{b n}} g (-e f+d g) \sqrt {2 \pi } (d+e x) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}-6 g^2 \sqrt {3 \pi } (d+e x)^2 \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}+\sqrt {b} e^{\frac {2 a}{b n}} \sqrt {n} \left (c (d+e x)^n\right )^{2/n} \left (2 b \left (e^2 f^2+6 d e f g+2 d^2 g^2\right ) n \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{3/2}+e e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} (f+g x) \left (b e n (f+g x)+2 a (e f+2 d g+3 e g x)+2 b (2 d g+e (f+3 g x)) \log \left (c (d+e x)^n\right )\right )\right )\right )}{3 b^{5/2} e^3 n^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \]
(-2*(d + e*x)*(2*d*E^((2*a)/(b*n))*g*(8*e*f + d*g)*Sqrt[Pi]*(c*(d + e*x)^n )^(2/n)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])]*(a + b*Log[ c*(d + e*x)^n])^(3/2) + 8*E^(a/(b*n))*g*(-(e*f) + d*g)*Sqrt[2*Pi]*(d + e*x )*(c*(d + e*x)^n)^n^(-1)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sq rt[b]*Sqrt[n])]*(a + b*Log[c*(d + e*x)^n])^(3/2) - 6*g^2*Sqrt[3*Pi]*(d + e *x)^2*Erfi[(Sqrt[3]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])]*(a + b*Log[c*(d + e*x)^n])^(3/2) + Sqrt[b]*E^((2*a)/(b*n))*Sqrt[n]*(c*(d + e* x)^n)^(2/n)*(2*b*(e^2*f^2 + 6*d*e*f*g + 2*d^2*g^2)*n*Gamma[1/2, -((a + b*L og[c*(d + e*x)^n])/(b*n))]*(-((a + b*Log[c*(d + e*x)^n])/(b*n)))^(3/2) + e *E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(f + g*x)*(b*e*n*(f + g*x) + 2*a*(e*f + 2*d*g + 3*e*g*x) + 2*b*(2*d*g + e*(f + 3*g*x))*Log[c*(d + e*x)^n]))))/(3 *b^(5/2)*e^3*E^((3*a)/(b*n))*n^(5/2)*(c*(d + e*x)^n)^(3/n)*(a + b*Log[c*(d + e*x)^n])^(3/2))
Leaf count is larger than twice the leaf count of optimal. \(924\) vs. \(2(421)=842\).
Time = 2.60 (sec) , antiderivative size = 924, normalized size of antiderivative = 2.19, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {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.
| \(\displaystyle \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {4 (e f-d g) \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}dx}{3 b e n}+\frac {2 \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}dx}{b n}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {4 (e f-d g) \left (-\frac {2 (e f-d g) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b e n}+\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b e n}+\frac {2 \left (-\frac {4 (e f-d g) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b e n}+\frac {6 \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{b n}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2836 |
| \(\displaystyle -\frac {4 (e f-d g) \left (-\frac {2 (e f-d g) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}d(d+e x)}{b e^2 n}+\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b e n}+\frac {2 \left (-\frac {4 (e f-d g) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b e n}+\frac {6 \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{b n}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2737 |
| \(\displaystyle -\frac {4 (e f-d g) \left (-\frac {2 (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \int \frac {\left (c (d+e x)^n\right )^{\frac {1}{n}}}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}d\log \left (c (d+e x)^n\right )}{b e^2 n^2}+\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b e n}+\frac {2 \left (-\frac {4 (e f-d g) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b e n}+\frac {6 \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{b n}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {4 (e f-d g) \left (-\frac {4 (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \int e^{\frac {a+b \log \left (c (d+e x)^n\right )}{b n}-\frac {a}{b n}}d\sqrt {a+b \log \left (c (d+e x)^n\right )}}{b^2 e^2 n^2}+\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b e n}+\frac {2 \left (-\frac {4 (e f-d g) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b e n}+\frac {6 \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{b n}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {4 (e f-d g) \left (\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b e n}+\frac {2 \left (-\frac {4 (e f-d g) \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b e n}+\frac {6 \int \frac {(f+g x)^2}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{b n}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle -\frac {4 (e f-d g) \left (\frac {4 \int \left (\frac {e f-d g}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g (d+e x)}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )dx}{b n}-\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b e n}+\frac {2 \left (\frac {6 \int \left (\frac {(e f-d g)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {2 g (d+e x) (e f-d g)}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g^2 (d+e x)^2}{e^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )dx}{b n}-\frac {4 (e f-d g) \int \left (\frac {e f-d g}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g (d+e x)}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )dx}{b e n}-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{b n}-\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (d+e x) (f+g x)^2}{3 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}-\frac {4 (e f-d g) \left (-\frac {2 e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (c (d+e x)^n\right )^{-1/n}}{b^{3/2} e^2 n^{3/2}}+\frac {4 \left (\frac {e^{-\frac {2 a}{b n}} g \sqrt {\frac {\pi }{2}} (d+e x)^2 \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (c (d+e x)^n\right )^{-2/n}}{\sqrt {b} e^2 \sqrt {n}}+\frac {e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (c (d+e x)^n\right )^{-1/n}}{\sqrt {b} e^2 \sqrt {n}}\right )}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )}{3 b e n}+\frac {2 \left (-\frac {2 (d+e x) (f+g x)^2}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (e f-d g) \left (\frac {e^{-\frac {2 a}{b n}} g \sqrt {\frac {\pi }{2}} (d+e x)^2 \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (c (d+e x)^n\right )^{-2/n}}{\sqrt {b} e^2 \sqrt {n}}+\frac {e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (c (d+e x)^n\right )^{-1/n}}{\sqrt {b} e^2 \sqrt {n}}\right )}{b e n}+\frac {6 \left (\frac {e^{-\frac {3 a}{b n}} g^2 \sqrt {\frac {\pi }{3}} (d+e x)^3 \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (c (d+e x)^n\right )^{-3/n}}{\sqrt {b} e^3 \sqrt {n}}+\frac {e^{-\frac {2 a}{b n}} g (e f-d g) \sqrt {2 \pi } (d+e x)^2 \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (c (d+e x)^n\right )^{-2/n}}{\sqrt {b} e^3 \sqrt {n}}+\frac {e^{-\frac {a}{b n}} (e f-d g)^2 \sqrt {\pi } (d+e x) \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \left (c (d+e x)^n\right )^{-1/n}}{\sqrt {b} e^3 \sqrt {n}}\right )}{b n}\right )}{b n}\) |
(-2*(d + e*x)*(f + g*x)^2)/(3*b*e*n*(a + b*Log[c*(d + e*x)^n])^(3/2)) - (4 *(e*f - d*g)*((-2*(e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(b^(3/2)*e^2*E^(a/(b*n))*n^(3/2)*(c*(d + e* x)^n)^n^(-1)) + (4*(((e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c* (d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^2*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)) + (g*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[ c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^2*E^((2*a)/(b*n))*Sqrt[n]* (c*(d + e*x)^n)^(2/n))))/(b*n) - (2*(d + e*x)*(f + g*x))/(b*e*n*Sqrt[a + b *Log[c*(d + e*x)^n]])))/(3*b*e*n) + (2*((-4*(e*f - d*g)*(((e*f - d*g)*Sqrt [Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])])/(Sq rt[b]*e^2*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)) + (g*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/ (Sqrt[b]*e^2*E^((2*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^(2/n))))/(b*e*n) + (6 *(((e*f - d*g)^2*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(S qrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1)) + (g*(e*f - d*g)*Sqrt[2*Pi]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^((2*a)/(b*n))*Sqrt[n]*(c*( d + e*x)^n)^(2/n)) + (g^2*Sqrt[Pi/3]*(d + e*x)^3*Erfi[(Sqrt[3]*Sqrt[a + b* Log[c*(d + e*x)^n]])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^3*E^((3*a)/(b*n))*Sqrt [n]*(c*(d + e*x)^n)^(3/n))))/(b*n) - (2*(d + e*x)*(f + g*x)^2)/(b*e*n*S...
3.2.34.3.1 Defintions of rubi rules used
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]
\[\int \frac {\left (g x +f \right )^{2}}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {\left (f + g x\right )^{2}}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{2}}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(f+g x)^2}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{5/2}} \,d x \]