Optimal. Leaf size=625 \[ -\frac{8 b^2 f^{7/2} p^2 q^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{7 h (f g-e h)^{7/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h \sqrt{g+h x} (f g-e h)^3}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (g+h x)^{3/2} (f g-e h)^2}-\frac{8 b f^{7/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^{7/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (g+h x)^{5/2} (f g-e h)}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h \sqrt{g+h x} (f g-e h)^3}-\frac{16 b^2 f^2 p^2 q^2}{105 h (g+h x)^{3/2} (f g-e h)^2}+\frac{8 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{7 h (f g-e h)^{7/2}}+\frac{368 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{105 h (f g-e h)^{7/2}}-\frac{16 b^2 f^{7/2} p^2 q^2 \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{7 h (f g-e h)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.92716, antiderivative size = 625, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 16, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {2398, 2411, 2347, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 51, 2445} \[ -\frac{8 b^2 f^{7/2} p^2 q^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{7 h (f g-e h)^{7/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h \sqrt{g+h x} (f g-e h)^3}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (g+h x)^{3/2} (f g-e h)^2}-\frac{8 b f^{7/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^{7/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (g+h x)^{5/2} (f g-e h)}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h \sqrt{g+h x} (f g-e h)^3}-\frac{16 b^2 f^2 p^2 q^2}{105 h (g+h x)^{3/2} (f g-e h)^2}+\frac{8 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{7 h (f g-e h)^{7/2}}+\frac{368 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{105 h (f g-e h)^{7/2}}-\frac{16 b^2 f^{7/2} p^2 q^2 \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{7 h (f g-e h)^{7/2}} \]
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
Rule 2445
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{9/2}} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(g+h x)^{9/2}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}+\operatorname{Subst}\left (\frac{(4 b f p q) \int \frac{a+b \log \left (c d^q (e+f x)^{p q}\right )}{(e+f x) (g+h x)^{7/2}} \, dx}{7 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}+\operatorname{Subst}\left (\frac{(4 b p q) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{x \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{7/2}} \, dx,x,e+f x\right )}{7 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\operatorname{Subst}\left (\frac{(4 b p q) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{\left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{7/2}} \, dx,x,e+f x\right )}{7 (f g-e h)},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 p q) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{x \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{5/2}} \, dx,x,e+f x\right )}{7 h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\operatorname{Subst}\left (\frac{(4 b f p q) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{\left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{5/2}} \, dx,x,e+f x\right )}{7 (f g-e 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 (4 b f^2 p q\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{x \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{3/2}} \, dx,x,e+f x\right )}{7 h (f g-e 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 (8 b^2 f p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{5/2}} \, dx,x,e+f x\right )}{35 h (f g-e h)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\operatorname{Subst}\left (\frac{\left (4 b f^2 p q\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c d^q x^{p q}\right )}{\left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{3/2}} \, dx,x,e+f x\right )}{7 (f g-e h)^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^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 )}{7 h (f g-e h)^3},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^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{3/2}} \, dx,x,e+f x\right )}{35 h (f g-e 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 (8 b^2 f^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (\frac{f g-e h}{f}+\frac{h x}{f}\right )^{3/2}} \, dx,x,e+f x\right )}{21 h (f g-e h)^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h (f g-e h)^3 \sqrt{g+h x}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^3 \sqrt{g+h x}}-\frac{8 b f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\operatorname{Subst}\left (\frac{\left (8 b^2 f^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 )}{35 h (f g-e h)^3},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^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 )}{21 h (f g-e h)^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^2 f^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 )}{7 h (f g-e h)^3},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^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 )}{7 h (f g-e h)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h (f g-e h)^3 \sqrt{g+h x}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^3 \sqrt{g+h x}}-\frac{8 b f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}+\operatorname{Subst}\left (\frac{\left (8 b^2 f^{7/2} 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 )}{7 h (f g-e h)^{7/2}},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^4 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 )}{35 h^2 (f g-e h)^3},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^4 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 )}{21 h^2 (f g-e h)^3},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^4 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 )}{7 h^2 (f g-e h)^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h (f g-e h)^3 \sqrt{g+h x}}+\frac{368 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{105 h (f g-e h)^{7/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^3 \sqrt{g+h x}}-\frac{8 b f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f^{9/2} 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 )}{7 h (f g-e h)^{7/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h (f g-e h)^3 \sqrt{g+h x}}+\frac{368 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{105 h (f g-e h)^{7/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^3 \sqrt{g+h x}}-\frac{8 b f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f^{9/2} 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 )}{7 h (f g-e h)^{7/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h (f g-e h)^3 \sqrt{g+h x}}+\frac{368 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{105 h (f g-e h)^{7/2}}+\frac{8 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{7 h (f g-e h)^{7/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^3 \sqrt{g+h x}}-\frac{8 b f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\operatorname{Subst}\left (\frac{\left (16 b^2 f^4 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 )}{7 h (f g-e h)^4},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h (f g-e h)^3 \sqrt{g+h x}}+\frac{368 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{105 h (f g-e h)^{7/2}}+\frac{8 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{7 h (f g-e h)^{7/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^3 \sqrt{g+h x}}-\frac{8 b f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\frac{16 b^2 f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f^4 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 )}{7 h (f g-e h)^4},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h (f g-e h)^3 \sqrt{g+h x}}+\frac{368 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{105 h (f g-e h)^{7/2}}+\frac{8 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{7 h (f g-e h)^{7/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^3 \sqrt{g+h x}}-\frac{8 b f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\frac{16 b^2 f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\operatorname{Subst}\left (\frac{\left (16 b^2 f^{7/2} 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 )}{7 h (f g-e h)^{7/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{16 b^2 f^2 p^2 q^2}{105 h (f g-e h)^2 (g+h x)^{3/2}}-\frac{128 b^2 f^3 p^2 q^2}{105 h (f g-e h)^3 \sqrt{g+h x}}+\frac{368 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{105 h (f g-e h)^{7/2}}+\frac{8 b^2 f^{7/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{7 h (f g-e h)^{7/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{35 h (f g-e h) (g+h x)^{5/2}}+\frac{8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{21 h (f g-e h)^2 (g+h x)^{3/2}}+\frac{8 b f^3 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{7 h (f g-e h)^3 \sqrt{g+h x}}-\frac{8 b f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{7 h (g+h x)^{7/2}}-\frac{16 b^2 f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}-\frac{8 b^2 f^{7/2} 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 )}{7 h (f g-e h)^{7/2}}\\ \end{align*}
Mathematica [C] time = 7.79691, size = 1249, normalized size = 2. \[ \frac{4 a b p q \left (\frac{\sqrt{f} \sqrt{\frac{f g-e h+h (e+f x)}{f}} \left (-15 \log (e+f x) (f g-e h)^3+6 (f g+f h x) (f g-e h)^2+10 (f g+f h x)^2 (f g-e h)+30 (f g+f h x)^3\right )}{(f g-e h)^3 (f g+f h x)^4}-\frac{30 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{\frac{f g-e h+h (e+f x)}{f}}}{\sqrt{f g-e h}}\right )}{(f g-e h)^{7/2}}\right ) f^{7/2}}{105 h}+\frac{4 b^2 p q^2 \left (\frac{\sqrt{f} \sqrt{\frac{f g-e h+h (e+f x)}{f}} \left (-15 \log (e+f x) (f g-e h)^3+6 (f g+f h x) (f g-e h)^2+10 (f g+f h x)^2 (f g-e h)+30 (f g+f h x)^3\right )}{(f g-e h)^3 (f g+f h x)^4}-\frac{30 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{\frac{f g-e h+h (e+f x)}{f}}}{\sqrt{f g-e h}}\right )}{(f g-e h)^{7/2}}\right ) \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right ) f^{7/2}}{105 h}+\frac{4 b^2 p q \left (\frac{\sqrt{f} \sqrt{\frac{f g-e h+h (e+f x)}{f}} \left (-15 \log (e+f x) (f g-e h)^3+6 (f g+f h x) (f g-e h)^2+10 (f g+f h x)^2 (f g-e h)+30 (f g+f h x)^3\right )}{(f g-e h)^3 (f g+f h x)^4}-\frac{30 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{\frac{f g-e h+h (e+f x)}{f}}}{\sqrt{f g-e h}}\right )}{(f g-e h)^{7/2}}\right ) \left (-q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )-\log \left (d (e+f x)^p\right ) \left (q-\frac{q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )}{\log \left (d (e+f x)^p\right )}\right )+\log \left (c e^{q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )} \left (d (e+f x)^p\right )^{q-\frac{q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )}{\log \left (d (e+f x)^p\right )}}\right )\right ) f^{7/2}}{105 h}-\frac{2 \left (a+b q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )+b \left (-q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )-\log \left (d (e+f x)^p\right ) \left (q-\frac{q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )}{\log \left (d (e+f x)^p\right )}\right )+\log \left (c e^{q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )} \left (d (e+f x)^p\right )^{q-\frac{q \left (\log \left (d (e+f x)^p\right )-p \log (e+f x)\right )}{\log \left (d (e+f x)^p\right )}}\right )\right )\right )^2}{7 h (g+h x)^{7/2}}+\frac{2 b^2 p^2 q^2 \left (\frac{h (e+f x)}{f g-e h}+1\right ) \left (f g \log ^2(e+f x) \left (\frac{f g-e h+h (e+f x)}{f g-e h}\right )^{7/2}-e h \log ^2(e+f x) \left (\frac{f g-e h+h (e+f x)}{f g-e h}\right )^{7/2}+7 h (e+f x) \, _4F_3\left (1,1,1,\frac{9}{2};2,2,2;\frac{h (e+f x)}{e h-f g}\right ) \left (\frac{f g-e h+h (e+f x)}{f g-e h}\right )^{7/2}-7 h (e+f x) \, _3F_2\left (1,1,\frac{9}{2};2,2;\frac{h (e+f x)}{e h-f g}\right ) \log (e+f x) \left (\frac{f g-e h+h (e+f x)}{f g-e h}\right )^{7/2}-f g \log ^2(e+f x)+e h \log ^2(e+f x)\right )}{7 h \left (\frac{f g-e h+h (e+f x)}{f}\right )^{9/2} f} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.623, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{2} \left ( hx+g \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{h x + g} b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, \sqrt{h x + g} a b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt{h x + g} a^{2}}{h^{5} x^{5} + 5 \, g h^{4} x^{4} + 10 \, g^{2} h^{3} x^{3} + 10 \, g^{3} h^{2} x^{2} + 5 \, g^{4} h x + g^{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 ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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