Optimal. Leaf size=449 \[ -\frac{8 b^2 f^{3/2} p^2 q^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{3 h (f g-e h)^{3/2}}-\frac{8 b f^{3/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h \sqrt{g+h x} (f g-e h)}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\frac{8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac{16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}}-\frac{16 b^2 f^{3/2} p^2 q^2 \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}} \]
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Rubi [A] time = 2.37791, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 15, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2398, 2411, 2347, 63, 208, 2348, 12, 1587, 6741, 5984, 5918, 2402, 2315, 2319, 2445} \[ -\frac{8 b^2 f^{3/2} p^2 q^2 \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right )}{3 h (f g-e h)^{3/2}}-\frac{8 b f^{3/2} p q \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h \sqrt{g+h x} (f g-e h)}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\frac{8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac{16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}}-\frac{16 b^2 f^{3/2} p^2 q^2 \log \left (\frac{2}{1-\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}}\right ) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}} \]
Antiderivative was successfully verified.
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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 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)^{5/2}} \, 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)^{5/2}} \, 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}{3 h (g+h x)^{3/2}}+\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)^{3/2}} \, dx}{3 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}{3 h (g+h x)^{3/2}}+\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 )^{3/2}} \, dx,x,e+f x\right )}{3 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}{3 h (g+h x)^{3/2}}-\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 )^{3/2}} \, dx,x,e+f x\right )}{3 (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 \sqrt{\frac{f g-e h}{f}+\frac{h x}{f}}} \, dx,x,e+f x\right )}{3 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 )}{3 h (f g-e h) \sqrt{g+h x}}-\frac{8 b f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\operatorname{Subst}\left (\frac{\left (4 b^2 f 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 )}{3 h (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{\left (8 b^2 f 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 )}{3 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 )}{3 h (f g-e h) \sqrt{g+h x}}-\frac{8 b f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\operatorname{Subst}\left (\frac{\left (8 b^2 f^{3/2} 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 )}{3 h (f g-e h)^{3/2}},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^2 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 )}{3 h^2 (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^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt{g+h x}}-\frac{8 b f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f^{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 )}{3 h (f g-e h)^{3/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt{g+h x}}-\frac{8 b f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f^{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 )}{3 h (f g-e h)^{3/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac{8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt{g+h x}}-\frac{8 b f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\operatorname{Subst}\left (\frac{\left (16 b^2 f^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 )}{3 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^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac{8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt{g+h x}}-\frac{8 b f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\frac{16 b^2 f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}+\operatorname{Subst}\left (\frac{\left (16 b^2 f^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 )}{3 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^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac{8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt{g+h x}}-\frac{8 b f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\frac{16 b^2 f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\operatorname{Subst}\left (\frac{\left (16 b^2 f^{3/2} 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 )}{3 h (f g-e h)^{3/2}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{16 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )}{3 h (f g-e h)^{3/2}}+\frac{8 b^2 f^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )^2}{3 h (f g-e h)^{3/2}}+\frac{8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 h (f g-e h) \sqrt{g+h x}}-\frac{8 b f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h (g+h x)^{3/2}}-\frac{16 b^2 f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}-\frac{8 b^2 f^{3/2} 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 )}{3 h (f g-e h)^{3/2}}\\ \end{align*}
Mathematica [C] time = 6.17957, size = 657, normalized size = 1.46 \[ \frac{2 \left (3 b^2 f h p^2 q^2 (e+f x) (g+h x) \sqrt{\frac{f (g+h x)}{f g-e h}} \, _4F_3\left (1,1,1,\frac{5}{2};2,2,2;\frac{h (e+f x)}{e h-f g}\right )-(f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2-4 a b f^{3/2} p q (g+h x)^{3/2} \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )+2 a b p q (f g-e h) ((e h-f g) \log (e+f x)+2 f (g+h x))-4 b^2 f^{3/2} p q (g+h x)^{3/2} \sqrt{f g-e h} \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right ) \log \left (c \left (d (e+f x)^p\right )^q\right )+2 b^2 p q (f g-e h) ((e h-f g) \log (e+f x)+2 f (g+h x)) \log \left (c \left (d (e+f x)^p\right )^q\right )+4 b^2 f^{3/2} p^2 q^2 (g+h x)^{3/2} \sqrt{f g-e h} \log (e+f x) \tanh ^{-1}\left (\frac{\sqrt{f} \sqrt{g+h x}}{\sqrt{f g-e h}}\right )-2 b^2 p^2 q^2 (f g-e h) \log (e+f x) ((e h-f g) \log (e+f x)+2 f (g+h x))+b^2 p^2 q^2 (f g-e h) \log (e+f x) \left (\log (e+f x) \left (f h x \sqrt{\frac{f (g+h x)}{f g-e h}}+f g \left (\sqrt{\frac{f (g+h x)}{f g-e h}}-1\right )+e h\right )-4 f (g+h x) \left (\sqrt{\frac{f (g+h x)}{f g-e h}}+\sqrt{\frac{f (g+h x)}{f g-e h}} \log \left (\frac{1}{2} \left (\sqrt{\frac{f (g+h x)}{f g-e h}}+1\right )\right )-1\right )\right )\right )}{3 h (g+h x)^{3/2} (f g-e h)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.674, 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{5}{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^{3} x^{3} + 3 \, g h^{2} x^{2} + 3 \, g^{2} h x + g^{3}}, 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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