Optimal. Leaf size=635 \[ \frac {368 b^2 (f g-e h)^2 p^2 q^2 \sqrt {g+h x}}{75 f^2 h}+\frac {128 b^2 (f g-e h) p^2 q^2 (g+h x)^{3/2}}{225 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{5/2}}{125 h}-\frac {368 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{75 f^{5/2} h}-\frac {8 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 f^{5/2} h}-\frac {8 b (f g-e h)^2 p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^2 h}-\frac {8 b (f g-e h) p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 f h}-\frac {8 b p q (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{25 h}+\frac {8 b (f g-e h)^{5/2} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 f^{5/2} h}+\frac {2 (g+h x)^{5/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h}+\frac {16 b^2 (f g-e h)^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 f^{5/2} h}+\frac {8 b^2 (f g-e h)^{5/2} p^2 q^2 \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 f^{5/2} h} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.92, antiderivative size = 635, normalized size of antiderivative = 1.00, 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 = {2445,
2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52, 2495}
\begin {gather*} \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 (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 \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 (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 {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 {368 b^2 p^2 q^2 \sqrt {g+h x} (f g-e h)^2}{75 f^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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 52
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2388
Rule 2390
Rule 2445
Rule 2449
Rule 2458
Rule 2495
Rule 6055
Rule 6131
Rule 6873
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 &=\text {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}-\text {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}-\text {Subst}\left (\frac {(4 b p q) \text {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}-\text {Subst}\left (\frac {(4 b p q) \text {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 )-\text {Subst}\left (\frac {(4 b (f g-e h) p q) \text {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}-\text {Subst}\left (\frac {(4 b (f g-e h) p q) \text {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 )-\text {Subst}\left (\frac {\left (4 b (f g-e h)^2 p q\right ) \text {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 )+\text {Subst}\left (\frac {\left (8 b^2 p^2 q^2\right ) \text {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}-\text {Subst}\left (\frac {\left (4 b (f g-e h)^2 p q\right ) \text {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 )-\text {Subst}\left (\frac {\left (4 b (f g-e h)^3 p q\right ) \text {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 )+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h) p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h) p^2 q^2\right ) \text {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}+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^2 p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^2 p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^2 p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (4 b^2 (f g-e h)^3 p^2 q^2\right ) \text {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}-\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^{5/2} p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^3 p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^3 p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^3 p^2 q^2\right ) \text {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}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{5/2} p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^3 p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^3 p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^3 p^2 q^2\right ) \text {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}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{5/2} p^2 q^2\right ) \text {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}+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^2 p^2 q^2\right ) \text {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}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^2 p^2 q^2\right ) \text {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}+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{5/2} p^2 q^2\right ) \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 6.62, size = 1168, normalized size = 1.84 \begin {gather*} \frac {2 \left (-\frac {15 b^2 p^2 q^2 \sqrt {g+h x} \left (-10 h (-f g+e h) (e+f x) \, _4F_3\left (-\frac {3}{2},1,1,1;2,2,2;\frac {h (e+f x)}{-f g+e h}\right )+15 e^2 h^2 \, _4F_3\left (-\frac {1}{2},1,1,1;2,2,2;\frac {h (e+f x)}{-f g+e h}\right )+15 e f h^2 x \, _4F_3\left (-\frac {1}{2},1,1,1;2,2,2;\frac {h (e+f x)}{-f g+e h}\right )-4 f^2 g^2 \log (e+f x)+8 e f g h \log (e+f x)-4 e^2 h^2 \log (e+f x)+4 f^2 g^2 \sqrt {\frac {f (g+h x)}{f g-e h}} \log (e+f x)+8 f^2 g h x \sqrt {\frac {f (g+h x)}{f g-e h}} \log (e+f x)+4 f^2 h^2 x^2 \sqrt {\frac {f (g+h x)}{f g-e h}} \log (e+f x)-15 e^2 h^2 \, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {h (e+f x)}{-f g+e h}\right ) \log (e+f x)-15 e f h^2 x \, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {h (e+f x)}{-f g+e h}\right ) \log (e+f x)-2 f^2 g^2 \log ^2(e+f x)-e f g h \log ^2(e+f x)+3 e^2 h^2 \log ^2(e+f x)+2 f^2 g^2 \sqrt {\frac {f (g+h x)}{f g-e h}} \log ^2(e+f x)-f^2 g h x \sqrt {\frac {f (g+h x)}{f g-e h}} \log ^2(e+f x)-3 f^2 h^2 x^2 \sqrt {\frac {f (g+h x)}{f g-e h}} \log ^2(e+f x)+10 h (-f g+e h) (e+f x) \, _3F_2\left (-\frac {3}{2},1,1;2,2;\frac {h (e+f x)}{-f g+e h}\right ) (1+\log (e+f x))\right )}{f^2 \sqrt {\frac {f (g+h x)}{f g-e h}}}+\frac {75 b^2 g 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)}{-f g+e h}\right )+\log (e+f x) \left (-3 h (e+f x) \, _3F_2\left (-\frac {1}{2},1,1;2,2;\frac {h (e+f x)}{-f g+e h}\right )+\left (e h+f h x \sqrt {\frac {f (g+h x)}{f g-e h}}+f g \left (-1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right ) \log (e+f x)\right )\right )}{f \sqrt {\frac {f (g+h x)}{f g-e h}}}-\frac {50 b g p q \left (6 (f g-e h)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )+\sqrt {f} \sqrt {g+h x} (6 e h-2 f (4 g+h x)+3 f (g+h x) \log (e+f x))\right ) \left (-a+b p q \log (e+f x)-b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^{3/2}}+\frac {2 b p q \left (30 \sqrt {f g-e h} \left (2 f^2 g^2+e f g h-3 e^2 h^2\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )+\sqrt {f} \sqrt {g+h x} \left (90 e^2 h^2-30 e f h (2 g+h x)+2 f^2 \left (-31 g^2+8 g h x+9 h^2 x^2\right )+15 f^2 \left (2 g^2-g h x-3 h^2 x^2\right ) \log (e+f x)\right )\right ) \left (-a+b p q \log (e+f x)-b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^{5/2}}+45 (g+h x)^{5/2} \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2\right )}{225 h} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \left (h x +g \right )^{\frac {3}{2}} \left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )^{2}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (g+h\,x\right )}^{3/2}\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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