Optimal. Leaf size=547 \[ \frac {8 b p q (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^{3/2} h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}-\frac {8 b p q \sqrt {g+h x} (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac {8 b^2 p^2 q^2 (f g-e h)^{3/2} \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 f^{3/2} h}-\frac {8 b^2 p^2 q^2 (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {64 b^2 p^2 q^2 (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}+\frac {16 b^2 p^2 q^2 (f g-e h)^{3/2} \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 f^{3/2} h}+\frac {64 b^2 p^2 q^2 \sqrt {g+h x} (f g-e h)}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.98, antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 22, 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} \[ \frac {8 b^2 p^2 q^2 (f g-e h)^{3/2} \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 f^{3/2} h}+\frac {8 b p q (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^{3/2} h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}-\frac {8 b p q \sqrt {g+h x} (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {8 b^2 p^2 q^2 (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {64 b^2 p^2 q^2 (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}+\frac {16 b^2 p^2 q^2 (f g-e h)^{3/2} \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{3 f^{3/2} h}+\frac {64 b^2 p^2 q^2 \sqrt {g+h x} (f g-e h)}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 50
Rule 63
Rule 208
Rule 1587
Rule 2315
Rule 2319
Rule 2346
Rule 2348
Rule 2398
Rule 2402
Rule 2411
Rule 2445
Rule 5918
Rule 5984
Rule 6741
Rubi steps
\begin {align*} \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx &=\operatorname {Subst}\left (\int \sqrt {g+h x} \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)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\operatorname {Subst}\left (\frac {(4 b f p q) \int \frac {(g+h x)^{3/2} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x} \, dx}{3 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 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 )^{3/2} \left (a+b \log \left (c d^q x^{p q}\right )\right )}{x} \, 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 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\operatorname {Subst}\left (\frac {(4 b 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 )}{3 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 {\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 )}{3 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)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\operatorname {Subst}\left (\frac {(4 b (f g-e h) p q) \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 )}{3 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 {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 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 )^{3/2}}{x} \, dx,x,e+f x\right )}{9 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)^{3/2}}{27 h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{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 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\operatorname {Subst}\left (\frac {\left (8 b^2 (f g-e h) 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 )}{9 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 {\sqrt {\frac {f g-e h}{f}+\frac {h x}{f}}}{x} \, dx,x,e+f x\right )}{3 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 (4 b^2 (f g-e h)^2 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 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{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 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\operatorname {Subst}\left (\frac {\left (8 b^2 (f g-e h)^{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 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 (8 b^2 (f g-e h)^2 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 )}{9 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 {1}{x \sqrt {\frac {f g-e h}{f}+\frac {h x}{f}}} \, dx,x,e+f x\right )}{3 f^2 h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{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 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\operatorname {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{3/2} 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 \sqrt {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 (16 b^2 (f g-e h)^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 )}{9 f 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)^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 f h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {64 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{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 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\operatorname {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{3/2} 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 \sqrt {f} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {64 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}-\frac {8 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{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 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\operatorname {Subst}\left (\frac {\left (16 b^2 (f g-e h) 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 f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {64 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}-\frac {8 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{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 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac {16 b^2 (f g-e h)^{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 f^{3/2} h}-\operatorname {Subst}\left (\frac {\left (16 b^2 (f g-e h) 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 f h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {64 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}-\frac {8 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{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 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac {16 b^2 (f g-e h)^{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 f^{3/2} h}+\operatorname {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{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 f^{3/2} h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {64 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}-\frac {8 b^2 (f g-e h)^{3/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{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 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac {16 b^2 (f g-e h)^{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 f^{3/2} h}+\frac {8 b^2 (f g-e h)^{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 f^{3/2} h}\\ \end {align*}
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Mathematica [C] time = 2.16, size = 365, normalized size = 0.67 \[ \frac {2 \left (\frac {3 b^2 p^2 q^2 \sqrt {g+h x} \left (3 h (e+f x) \, _4F_3\left (-\frac {1}{2},1,1,1;2,2,2;\frac {h (e+f x)}{e h-f g}\right )+\log (e+f x) \left (\log (e+f x) \left (f h x \sqrt {\frac {f (g+h x)}{f g-e h}}+f g \left (\sqrt {\frac {f (g+h x)}{f g-e h}}-1\right )+e h\right )-3 h (e+f x) \, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {h (e+f x)}{e h-f g}\right )\right )\right )}{f \sqrt {\frac {f (g+h x)}{f g-e h}}}-\frac {2 b p q \left (\sqrt {f} \sqrt {g+h x} (3 f (g+h x) \log (e+f x)+6 e h-2 f (4 g+h x))+6 (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )\right ) \left (-a-b \log \left (c \left (d (e+f x)^p\right )^q\right )+b p q \log (e+f x)\right )}{f^{3/2}}+3 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )-b p q \log (e+f x)\right )^2\right )}{9 h} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {h x + g} b^{2} \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right )^{2} + 2 \, \sqrt {h x + g} a b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt {h x + g} a^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {h x + g} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \sqrt {h x +g}\, \left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {g+h\,x}\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2} \sqrt {g + h x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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