Optimal. Leaf size=635 \[ \frac{8 b^2 p^2 q^2 (f g-e h)^{5/2} \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{5 f^{5/2} h}-\frac{8 b p q \sqrt{g+h x} (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}+\frac{8 b p q (f g-e h)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}-\frac{8 b p q (g+h x)^{3/2} (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{368 b^2 p^2 q^2 \sqrt{g+h x} (f g-e h)^2}{75 f^2 h}-\frac{8 b^2 p^2 q^2 (f g-e h)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{5 f^{5/2} h}-\frac{368 b^2 p^2 q^2 (f g-e h)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{75 f^{5/2} h}+\frac{16 b^2 p^2 q^2 (f g-e h)^{5/2} \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{5 f^{5/2} h}+\frac{128 b^2 p^2 q^2 (g+h x)^{3/2} (f g-e h)}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 4.34782, antiderivative size = 635, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 16, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {2398, 2411, 2346, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 50, 2445} \[ \frac{8 b^2 p^2 q^2 (f g-e h)^{5/2} \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{5 f^{5/2} h}-\frac{8 b p q \sqrt{g+h x} (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}+\frac{8 b p q (f g-e h)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}-\frac{8 b p q (g+h x)^{3/2} (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{368 b^2 p^2 q^2 \sqrt{g+h x} (f g-e h)^2}{75 f^2 h}-\frac{8 b^2 p^2 q^2 (f g-e h)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{5 f^{5/2} h}-\frac{368 b^2 p^2 q^2 (f g-e h)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{75 f^{5/2} h}+\frac{16 b^2 p^2 q^2 (f g-e h)^{5/2} \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{5 f^{5/2} h}+\frac{128 b^2 p^2 q^2 (g+h x)^{3/2} (f g-e h)}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2398
Rule 2411
Rule 2346
Rule 63
Rule 208
Rule 2348
Rule 12
Rule 1587
Rule 6741
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 2319
Rule 50
Rule 2445
Rubi steps
\begin{align*} \int (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx &=\operatorname{Subst}\left (\int (g+h x)^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2 \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\operatorname{Subst}\left (\frac{(4 b f p q) \int \frac{(g+h x)^{5/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x} \, dx}{5 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\operatorname{Subst}\left (\frac{(4 b p q) \operatorname{Subst}\left (\int \frac{\left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{5/2} \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, dx,x,e+f x\right )}{5 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\operatorname{Subst}\left (\frac{(4 b p q) \operatorname{Subst}\left (\int \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{3/2} \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{5 f},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(4 b (f g-e h) p q) \operatorname{Subst}\left (\int \frac{\left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{3/2} \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, dx,x,e+f x\right )}{5 f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\operatorname{Subst}\left (\frac{(4 b (f g-e h) p q) \operatorname{Subst}\left (\int \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}} \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{5 f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (4 b (f g-e h)^2 p q\right ) \operatorname{Subst}\left (\int \frac{\sqrt{\frac{f g-e h}{f}+\frac{h x}{f}} \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, dx,x,e+f x\right )}{5 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{5/2}}{x} \, dx,x,e+f x\right )}{25 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\operatorname{Subst}\left (\frac{\left (4 b (f g-e h)^2 p q\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{\sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{5 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (4 b (f g-e h)^3 p q\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{x \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{5 f^3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h) p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{3/2}}{x} \, dx,x,e+f x\right )}{25 f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h) p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{3/2}}{x} \, dx,x,e+f x\right )}{15 f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{8 b (f g-e h)^2 p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h)^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}}{x} \, dx,x,e+f x\right )}{25 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h)^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}}{x} \, dx,x,e+f x\right )}{15 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h)^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}}{x} \, dx,x,e+f x\right )}{5 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (4 b^2 (f g-e h)^3 p^2 q^2\right ) \operatorname{Subst}\left (\int -\frac{2 \sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g-\frac{e h}{f}+\frac{h x}{f}}}{\sqrt{f g-e h}}\right )}{\sqrt{f g-e h} x} \, dx,x,e+f x\right )}{5 f^3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{368 b^2 (f g-e h)^2 p^2 q^2 \sqrt{g+h x}}{75 f^2 h}+\frac{128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{8 b (f g-e h)^2 p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h)^{5/2} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g-\frac{e h}{f}+\frac{h x}{f}}}{\sqrt{f g-e h}}\right )}{x} \, dx,x,e+f x\right )}{5 f^{5/2} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h)^3 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{25 f^3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h)^3 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{15 f^3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (8 b^2 (f g-e h)^3 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{5 f^3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{368 b^2 (f g-e h)^2 p^2 q^2 \sqrt{g+h x}}{75 f^2 h}+\frac{128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{8 b (f g-e h)^2 p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h)^{5/2} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{e h+f \left (-g+x^2\right )} \, dx,x,\sqrt{g+h x}\right )}{5 f^{3/2} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h)^3 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{f g-e h}{h}+\frac{f x^2}{h}} \, dx,x,\sqrt{g+h x}\right )}{25 f^2 h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h)^3 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{f g-e h}{h}+\frac{f x^2}{h}} \, dx,x,\sqrt{g+h x}\right )}{15 f^2 h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h)^3 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{f g-e h}{h}+\frac{f x^2}{h}} \, dx,x,\sqrt{g+h x}\right )}{5 f^2 h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{368 b^2 (f g-e h)^2 p^2 q^2 \sqrt{g+h x}}{75 f^2 h}+\frac{128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{75 f^{5/2} h}-\frac{8 b (f g-e h)^2 p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}-\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h)^{5/2} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{x \tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{-f g+e h+f x^2} \, dx,x,\sqrt{g+h x}\right )}{5 f^{3/2} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{368 b^2 (f g-e h)^2 p^2 q^2 \sqrt{g+h x}}{75 f^2 h}+\frac{128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{75 f^{5/2} h}-\frac{8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{5 f^{5/2} h}-\frac{8 b (f g-e h)^2 p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h)^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{f g-e h}}\right )}{1-\frac{\sqrt{f} x}{\sqrt{f g-e h}}} \, dx,x,\sqrt{g+h x}\right )}{5 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{368 b^2 (f g-e h)^2 p^2 q^2 \sqrt{g+h x}}{75 f^2 h}+\frac{128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{75 f^{5/2} h}-\frac{8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{5 f^{5/2} h}-\frac{8 b (f g-e h)^2 p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\frac{16 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{5 f^{5/2} h}-\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h)^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-\frac{\sqrt{f} x}{\sqrt{f g-e h}}}\right )}{1-\frac{f x^2}{f g-e h}} \, dx,x,\sqrt{g+h x}\right )}{5 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{368 b^2 (f g-e h)^2 p^2 q^2 \sqrt{g+h x}}{75 f^2 h}+\frac{128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{75 f^{5/2} h}-\frac{8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{5 f^{5/2} h}-\frac{8 b (f g-e h)^2 p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\frac{16 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{5 f^{5/2} h}+\operatorname{Subst}\left (\frac{\left (16 b^2 (f g-e h)^{5/2} p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{5 f^{5/2} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{368 b^2 (f g-e h)^2 p^2 q^2 \sqrt{g+h x}}{75 f^2 h}+\frac{128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac{16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac{368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{75 f^{5/2} h}-\frac{8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{5 f^{5/2} h}-\frac{8 b (f g-e h)^2 p q \sqrt{g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac{8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac{8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac{8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac{2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\frac{16 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{5 f^{5/2} h}+\frac{8 b^2 (f g-e h)^{5/2} p^2 q^2 \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{5 f^{5/2} h}\\ \end{align*}
Mathematica [C] time = 8.82927, size = 2450, normalized size = 3.86 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.703, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{{\frac{3}{2}}} \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{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 ({\left (b^{2} h x + b^{2} g\right )} \sqrt{h x + g} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \,{\left (a b h x + a b g\right )} \sqrt{h x + g} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) +{\left (a^{2} h x + a^{2} g\right )} \sqrt{h x + g}, 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{\left (h x + g\right )}^{\frac{3}{2}}{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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