Optimal. Leaf size=583 \[ -\frac{8 b^2 e^{7/2} n^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g \sqrt{f+g x} (e f-d g)^3}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (f+g x)^{3/2} (e f-d g)^2}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (f+g x)^{5/2} (e f-d g)}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{128 b^2 e^3 n^2}{105 g \sqrt{f+g x} (e f-d g)^3}-\frac{16 b^2 e^2 n^2}{105 g (f+g x)^{3/2} (e f-d g)^2}+\frac{8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac{368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{105 g (e f-d g)^{7/2}}-\frac{16 b^2 e^{7/2} n^2 \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{7 g (e f-d g)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.84839, antiderivative size = 583, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 15, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.577, Rules used = {2398, 2411, 2347, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 51} \[ -\frac{8 b^2 e^{7/2} n^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g \sqrt{f+g x} (e f-d g)^3}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (f+g x)^{3/2} (e f-d g)^2}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (f+g x)^{5/2} (e f-d g)}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{128 b^2 e^3 n^2}{105 g \sqrt{f+g x} (e f-d g)^3}-\frac{16 b^2 e^2 n^2}{105 g (f+g x)^{3/2} (e f-d g)^2}+\frac{8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac{368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{105 g (e f-d g)^{7/2}}-\frac{16 b^2 e^{7/2} n^2 \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{7 g (e f-d g)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2398
Rule 2411
Rule 2347
Rule 63
Rule 208
Rule 2348
Rule 12
Rule 1587
Rule 6741
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 2319
Rule 51
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{(f+g x)^{9/2}} \, dx &=-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac{(4 b e n) \int \frac{a+b \log \left (c (d+e x)^n\right )}{(d+e x) (f+g x)^{7/2}} \, dx}{7 g}\\ &=-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac{(4 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{7/2}} \, dx,x,d+e x\right )}{7 g}\\ &=-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{(4 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{7/2}} \, dx,x,d+e x\right )}{7 (e f-d g)}+\frac{(4 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{5/2}} \, dx,x,d+e x\right )}{7 g (e f-d g)}\\ &=\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{(4 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{5/2}} \, dx,x,d+e x\right )}{7 (e f-d g)^2}+\frac{\left (4 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{7 g (e f-d g)^2}-\frac{\left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{5/2}} \, dx,x,d+e x\right )}{35 g (e f-d g)}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{\left (4 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{7 (e f-d g)^3}+\frac{\left (4 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \sqrt{\frac{e f-d g}{e}+\frac{g x}{e}}} \, dx,x,d+e x\right )}{7 g (e f-d g)^3}-\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{35 g (e f-d g)^2}-\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{e f-d g}{e}+\frac{g x}{e}\right )^{3/2}} \, dx,x,d+e x\right )}{21 g (e f-d g)^2}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt{f+g x}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{e f-d g}{e}+\frac{g x}{e}}} \, dx,x,d+e x\right )}{35 g (e f-d g)^3}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{e f-d g}{e}+\frac{g x}{e}}} \, dx,x,d+e x\right )}{21 g (e f-d g)^3}-\frac{\left (4 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int -\frac{2 \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f-\frac{d g}{e}+\frac{g x}{e}}}{\sqrt{e f-d g}}\right )}{\sqrt{e f-d g} x} \, dx,x,d+e x\right )}{7 g (e f-d g)^3}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{e f-d g}{e}+\frac{g x}{e}}} \, dx,x,d+e x\right )}{7 g (e f-d g)^3}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt{f+g x}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac{\left (8 b^2 e^{7/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f-\frac{d g}{e}+\frac{g x}{e}}}{\sqrt{e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{7 g (e f-d g)^{7/2}}-\frac{\left (16 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{e f-d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{35 g^2 (e f-d g)^3}-\frac{\left (16 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{e f-d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{21 g^2 (e f-d g)^3}-\frac{\left (16 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{e f-d g}{g}+\frac{e x^2}{g}} \, dx,x,\sqrt{f+g x}\right )}{7 g^2 (e f-d g)^3}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt{f+g x}}+\frac{368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac{\left (16 b^2 e^{9/2} n^2\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt{f+g x}\right )}{7 g (e f-d g)^{7/2}}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt{f+g x}}+\frac{368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}+\frac{\left (16 b^2 e^{9/2} n^2\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt{f+g x}\right )}{7 g (e f-d g)^{7/2}}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt{f+g x}}+\frac{368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac{8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{\left (16 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{e f-d g}}\right )}{1-\frac{\sqrt{e} x}{\sqrt{e f-d g}}} \, dx,x,\sqrt{f+g x}\right )}{7 g (e f-d g)^4}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt{f+g x}}+\frac{368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac{8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{16 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}+\frac{\left (16 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{\sqrt{e} x}{\sqrt{e f-d g}}}\right )}{1-\frac{e x^2}{e f-d g}} \, dx,x,\sqrt{f+g x}\right )}{7 g (e f-d g)^4}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt{f+g x}}+\frac{368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac{8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{16 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}-\frac{\left (16 b^2 e^{7/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}\\ &=-\frac{16 b^2 e^2 n^2}{105 g (e f-d g)^2 (f+g x)^{3/2}}-\frac{128 b^2 e^3 n^2}{105 g (e f-d g)^3 \sqrt{f+g x}}+\frac{368 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{105 g (e f-d g)^{7/2}}+\frac{8 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )^2}{7 g (e f-d g)^{7/2}}+\frac{8 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )}{35 g (e f-d g) (f+g x)^{5/2}}+\frac{8 b e^2 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{21 g (e f-d g)^2 (f+g x)^{3/2}}+\frac{8 b e^3 n \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^3 \sqrt{f+g x}}-\frac{8 b e^{7/2} n \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{7 g (e f-d g)^{7/2}}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{7 g (f+g x)^{7/2}}-\frac{16 b^2 e^{7/2} n^2 \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}-\frac{8 b^2 e^{7/2} n^2 \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}}\right )}{7 g (e f-d g)^{7/2}}\\ \end{align*}
Mathematica [C] time = 3.9823, size = 728, normalized size = 1.25 \[ \frac{2 \left (\frac{b e n (f+g x) \left (-15 b e^{5/2} n (f+g x)^{5/2} \left (2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{e} \sqrt{f+g x}}{2 \sqrt{e f-d g}}\right )+\log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right ) \left (\log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right )+2 \log \left (\frac{1}{2} \left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}+1\right )\right )\right )\right )+15 b e^{5/2} n (f+g x)^{5/2} \left (2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}+1\right )\right )+\log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right ) \left (\log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right )+2 \log \left (\frac{1}{2}-\frac{\sqrt{e} \sqrt{f+g x}}{2 \sqrt{e f-d g}}\right )\right )\right )+60 e^2 (f+g x)^2 \sqrt{e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right )+30 e^{5/2} (f+g x)^{5/2} \log \left (\sqrt{e f-d g}-\sqrt{e} \sqrt{f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-30 e^{5/2} (f+g x)^{5/2} \log \left (\sqrt{e f-d g}+\sqrt{e} \sqrt{f+g x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+12 (e f-d g)^{5/2} \left (a+b \log \left (c (d+e x)^n\right )\right )+20 e (f+g x) (e f-d g)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )-40 b e^2 n (f+g x)^2 \sqrt{e f-d g} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right )+120 b e^{5/2} n (f+g x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )-8 b e n (f+g x) (e f-d g)^{3/2} \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{e (f+g x)}{e f-d g}\right )\right )}{(e f-d g)^{7/2}}-15 \left (a+b \log \left (c (d+e x)^n\right )\right )^2\right )}{105 g (f+g x)^{7/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.92, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2} \left ( gx+f \right ) ^{-{\frac{9}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{g x + f} b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, \sqrt{g x + f} a b \log \left ({\left (e x + d\right )}^{n} c\right ) + \sqrt{g x + f} a^{2}}{g^{5} x^{5} + 5 \, f g^{4} x^{4} + 10 \, f^{2} g^{3} x^{3} + 10 \, f^{3} g^{2} x^{2} + 5 \, f^{4} g x + f^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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