Integrand size = 30, antiderivative size = 537 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=-\frac {16 b^2 f^2 p^2 q^2}{15 h (f g-e h)^2 \sqrt {g+h x}}+\frac {64 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}+\frac {8 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 h (f g-e h)^{5/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{5/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}-\frac {16 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 h (f g-e h)^{5/2}}-\frac {8 b^2 f^{5/2} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 h (f g-e h)^{5/2}} \]
[Out]
Time = 2.14 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2445, 2458, 2389, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 53, 2495} \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=-\frac {8 b f^{5/2} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^{5/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h \sqrt {g+h x} (f g-e h)^2}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (g+h x)^{3/2} (f g-e h)}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}+\frac {8 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 h (f g-e h)^{5/2}}+\frac {64 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}-\frac {16 b^2 f^{5/2} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 h (f g-e h)^{5/2}}-\frac {8 b^2 f^{5/2} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{5 h (f g-e h)^{5/2}}-\frac {16 b^2 f^2 p^2 q^2}{15 h \sqrt {g+h x} (f g-e h)^2} \]
[In]
[Out]
Rule 12
Rule 53
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2389
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 \frac {\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2}{(g+h x)^{7/2}} \, 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}{5 h (g+h x)^{5/2}}+\text {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)^{5/2}} \, dx}{5 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}{5 h (g+h x)^{5/2}}+\text {Subst}\left (\frac {(4 b p q) \text {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 )}{5 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}{5 h (g+h x)^{5/2}}-\text {Subst}\left (\frac {(4 b p q) \text {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 )}{5 (f g-e h)},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 p q) \text {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 )}{5 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 )}{15 h (f g-e h) (g+h x)^{3/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}-\text {Subst}\left (\frac {(4 b f p q) \text {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 )}{5 (f g-e 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 (4 b f^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 )}{5 h (f g-e 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 (8 b^2 f p^2 q^2\right ) \text {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 )}{15 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}{15 h (f g-e h)^2 \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 )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{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 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}-\text {Subst}\left (\frac {\left (8 b^2 f^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 )}{15 h (f g-e 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 (4 b^2 f^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 )}{5 h (f g-e 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 (8 b^2 f^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 )}{5 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}{15 h (f g-e h)^2 \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 )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{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 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}+\text {Subst}\left (\frac {\left (8 b^2 f^{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 h (f g-e h)^{5/2}},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^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 h^2 (f g-e 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^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 h^2 (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}{15 h (f g-e h)^2 \sqrt {g+h x}}+\frac {64 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{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 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}+\text {Subst}\left (\frac {\left (16 b^2 f^{7/2} 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 h (f g-e h)^{5/2}},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}{15 h (f g-e h)^2 \sqrt {g+h x}}+\frac {64 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{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 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}+\text {Subst}\left (\frac {\left (16 b^2 f^{7/2} 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 h (f g-e h)^{5/2}},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}{15 h (f g-e h)^2 \sqrt {g+h x}}+\frac {64 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}+\frac {8 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 h (f g-e h)^{5/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{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 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}-\text {Subst}\left (\frac {\left (16 b^2 f^3 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 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}{15 h (f g-e h)^2 \sqrt {g+h x}}+\frac {64 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}+\frac {8 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 h (f g-e h)^{5/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{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 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}-\frac {16 b^2 f^{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 h (f g-e h)^{5/2}}+\text {Subst}\left (\frac {\left (16 b^2 f^3 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 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}{15 h (f g-e h)^2 \sqrt {g+h x}}+\frac {64 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}+\frac {8 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 h (f g-e h)^{5/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{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 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}-\frac {16 b^2 f^{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 h (f g-e h)^{5/2}}-\text {Subst}\left (\frac {\left (16 b^2 f^{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 h (f g-e h)^{5/2}},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}{15 h (f g-e h)^2 \sqrt {g+h x}}+\frac {64 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{15 h (f g-e h)^{5/2}}+\frac {8 b^2 f^{5/2} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{5 h (f g-e h)^{5/2}}+\frac {8 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{15 h (f g-e h) (g+h x)^{3/2}}+\frac {8 b f^2 p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{5 h (f g-e h)^2 \sqrt {g+h x}}-\frac {8 b f^{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 h (f g-e h)^{5/2}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{5 h (g+h x)^{5/2}}-\frac {16 b^2 f^{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 h (f g-e h)^{5/2}}-\frac {8 b^2 f^{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 h (f g-e h)^{5/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1349\) vs. \(2(537)=1074\).
Time = 9.70 (sec) , antiderivative size = 1349, normalized size of antiderivative = 2.51 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\frac {4 a b f^{5/2} p q \left (-\frac {6 \text {arctanh}\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)^{5/2}}+\frac {\sqrt {f} \sqrt {\frac {f g-e h+h (e+f x)}{f}} \left (2 (f g-e h) (f g+f h x)+6 (f g+f h x)^2-3 (f g-e h)^2 \log (e+f x)\right )}{(f g-e h)^2 (f g+f h x)^3}\right )}{15 h}+\frac {4 b^2 f^{5/2} p q^2 \left (-\frac {6 \text {arctanh}\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)^{5/2}}+\frac {\sqrt {f} \sqrt {\frac {f g-e h+h (e+f x)}{f}} \left (2 (f g-e h) (f g+f h x)+6 (f g+f h x)^2-3 (f g-e h)^2 \log (e+f x)\right )}{(f g-e h)^2 (f g+f h x)^3}\right ) \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{15 h}+\frac {4 b^2 f^{5/2} p q \left (-\frac {6 \text {arctanh}\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)^{5/2}}+\frac {\sqrt {f} \sqrt {\frac {f g-e h+h (e+f x)}{f}} \left (2 (f g-e h) (f g+f h x)+6 (f g+f h x)^2-3 (f g-e h)^2 \log (e+f x)\right )}{(f g-e h)^2 (f g+f h x)^3}\right ) \left (-q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )-\log \left (d (e+f x)^p\right ) \left (q-\frac {q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{\log \left (d (e+f x)^p\right )}\right )+\log \left (c e^{q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )} \left (d (e+f x)^p\right )^{q-\frac {q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{\log \left (d (e+f x)^p\right )}}\right )\right )}{15 h}-\frac {2 \left (a+b q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )+b \left (-q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )-\log \left (d (e+f x)^p\right ) \left (q-\frac {q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{\log \left (d (e+f x)^p\right )}\right )+\log \left (c e^{q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )} \left (d (e+f x)^p\right )^{q-\frac {q \left (-p \log (e+f x)+\log \left (d (e+f x)^p\right )\right )}{\log \left (d (e+f x)^p\right )}}\right )\right )\right )^2}{5 h (g+h x)^{5/2}}+\frac {2 b^2 f^3 p^2 q^2 \sqrt {\frac {f g-e h+h (e+f x)}{f}} \left (\frac {32 h^{5/2} (e+f x)^{5/2} \left (\frac {f g+f h x}{h (e+f x)}\right )^{5/2} \arcsin \left (\frac {\sqrt {-f g+e h}}{\sqrt {h} \sqrt {e+f x}}\right )}{(-f g+e h)^{5/2}}+4 \left (1+\frac {h (e+f x)}{f g-e h}\right ) \log (e+f x)-3 \log ^2(e+f x)+\frac {4 (f g+f h x)^2 (-2+3 \log (e+f x))}{(f g-e h)^2}-\frac {12 (f g+f h x)^{5/2} \text {arctanh}\left (\frac {\sqrt {f g-e h+h (e+f 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 )}{(f g-e h)^{5/2}}+\frac {3 (f g+f h x)^2 \sqrt {1+\frac {h (e+f 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 {1+\frac {h (e+f x)}{f g-e h}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1+\frac {h (e+f x)}{f g-e h}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {1+\frac {h (e+f x)}{f g-e h}}\right )\right )\right )}{(f g-e h)^2}\right )}{15 h (f g+f h x)^3} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\left (h x +g \right )^{\frac {7}{2}}}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{7/2}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{{\left (g+h\,x\right )}^{7/2}} \,d x \]
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