Integrand size = 30, antiderivative size = 547 \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {64 b^2 (f g-e h) p^2 q^2 \sqrt {g+h x}}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h}-\frac {64 b^2 (f g-e h)^{3/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}-\frac {8 b^2 (f g-e h)^{3/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {8 b (f g-e h) p q \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}+\frac {8 b (f g-e h)^{3/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^{3/2} h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}+\frac {16 b^2 (f g-e h)^{3/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 f^{3/2} h}+\frac {8 b^2 (f g-e h)^{3/2} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 f^{3/2} h} \]
[Out]
Time = 2.11 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2445, 2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52, 2495} \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {8 b p q (f g-e h)^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^{3/2} h}-\frac {8 b p q (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 h}-\frac {8 b p q \sqrt {g+h x} (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f h}+\frac {2 (g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 h}-\frac {8 b^2 p^2 q^2 (f g-e h)^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{3 f^{3/2} h}-\frac {64 b^2 p^2 q^2 (f g-e h)^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{9 f^{3/2} h}+\frac {16 b^2 p^2 q^2 (f g-e h)^{3/2} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 f^{3/2} h}+\frac {8 b^2 p^2 q^2 (f g-e h)^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{3 f^{3/2} h}+\frac {64 b^2 p^2 q^2 \sqrt {g+h x} (f g-e h)}{9 f h}+\frac {16 b^2 p^2 q^2 (g+h x)^{3/2}}{27 h} \]
[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*} \text {integral}& = \text {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}-\text {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}-\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 )^{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}-\text {Subst}\left (\frac {(4 b 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 )}{3 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 {\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}-\text {Subst}\left (\frac {(4 b (f g-e h) p q) \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 )}{3 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 {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 )+\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 )^{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}+\text {Subst}\left (\frac {\left (8 b^2 (f g-e h) 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 )}{9 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 {\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 )+\text {Subst}\left (\frac {\left (4 b^2 (f g-e h)^2 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 )}{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}-\text {Subst}\left (\frac {\left (8 b^2 (f g-e h)^{3/2} 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 )}{3 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 (8 b^2 (f g-e h)^2 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 )}{9 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 {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}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{3/2} 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 )}{3 \sqrt {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 (16 b^2 (f g-e h)^2 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 )}{9 f 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)^2 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 )}{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}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{3/2} 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 )}{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}+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h) 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 )}{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}-\text {Subst}\left (\frac {\left (16 b^2 (f g-e h) 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 )}{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}+\text {Subst}\left (\frac {\left (16 b^2 (f g-e h)^{3/2} 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 )}{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*}
Leaf count is larger than twice the leaf count of optimal. \(1323\) vs. \(2(547)=1094\).
Time = 11.77 (sec) , antiderivative size = 1323, normalized size of antiderivative = 2.42 \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {2 \left (-\frac {6 b p q \left (6 (f g-e h)^{3/2} \text {arctanh}\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}}+9 (g+h x)^{3/2} \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-\frac {b^2 p^2 q^2 \left (96 f^2 g^2 h^{3/2} \sqrt {f g-e h} (e+f x)^{3/2} \left (\frac {f (g+h x)}{h (e+f x)}\right )^{3/2} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )-192 e f g h^{5/2} \sqrt {f g-e h} (e+f x)^{3/2} \left (\frac {f (g+h x)}{h (e+f x)}\right )^{3/2} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )+96 e^2 h^{7/2} \sqrt {f g-e h} (e+f x)^{3/2} \left (\frac {f (g+h x)}{h (e+f x)}\right )^{3/2} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )-f^2 \sqrt {-(f g-e h)^2} (g+h x)^2 \left (8 (13 f g-12 e h+f h x)-12 (4 f g-3 e h+f h x) \log (e+f x)+9 f (g+h x) \log ^2(e+f x)\right )-9 f^3 g^2 \sqrt {-f g+e h} (g+h x) \left (4 \sqrt {f (g+h x)} \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \left (\log (e+f x)-\log \left (\frac {h (e+f x)}{-f g+e h}\right )\right )-\sqrt {f g-e h} \sqrt {\frac {f (g+h x)}{f g-e h}} \left (\log ^2\left (\frac {h (e+f x)}{-f g+e h}\right )-4 \log \left (\frac {h (e+f x)}{-f g+e h}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )\right )+18 e f^2 g h \sqrt {-f g+e h} (g+h x) \left (4 \sqrt {f (g+h x)} \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \left (\log (e+f x)-\log \left (\frac {h (e+f x)}{-f g+e h}\right )\right )-\sqrt {f g-e h} \sqrt {\frac {f (g+h x)}{f g-e h}} \left (\log ^2\left (\frac {h (e+f x)}{-f g+e h}\right )-4 \log \left (\frac {h (e+f x)}{-f g+e h}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )\right )-9 e^2 f h^2 \sqrt {-f g+e h} (g+h x) \left (4 \sqrt {f (g+h x)} \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \left (\log (e+f x)-\log \left (\frac {h (e+f x)}{-f g+e h}\right )\right )-\sqrt {f g-e h} \sqrt {\frac {f (g+h x)}{f g-e h}} \left (\log ^2\left (\frac {h (e+f x)}{-f g+e h}\right )-4 \log \left (\frac {h (e+f x)}{-f g+e h}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )\right )\right )}{f^3 \sqrt {-(f g-e h)^2} (g+h x)^{3/2}}\right )}{27 h} \]
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\[\int \sqrt {h x +g}\, {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}d x\]
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\[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\int { \sqrt {h x + g} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2} \,d x } \]
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\[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\int \left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2} \sqrt {g + h x}\, dx \]
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Exception generated. \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Exception raised: ValueError} \]
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\[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\int { \sqrt {h x + g} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \sqrt {g+h x} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\int \sqrt {g+h\,x}\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2 \,d x \]
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