Integrand size = 30, antiderivative size = 330 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\frac {8 b^2 \sqrt {f} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\frac {16 b^2 \sqrt {f} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}-\frac {8 b^2 \sqrt {f} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}} \]
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Time = 1.23 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2445, 2458, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2495} \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=-\frac {8 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}+\frac {8 b^2 \sqrt {f} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {16 b^2 \sqrt {f} p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}}-\frac {8 b^2 \sqrt {f} p^2 q^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}}\right )}{h \sqrt {f g-e h}} \]
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Rule 12
Rule 65
Rule 214
Rule 1601
Rule 2352
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)^{3/2}} \, 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}{h \sqrt {g+h x}}+\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) \sqrt {g+h x}} \, dx}{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}{h \sqrt {g+h x}}+\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 \sqrt {\frac {f g-e h}{f}+\frac {h x}{f}}} \, dx,x,e+f x\right )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {8 b \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\text {Subst}\left (\frac {\left (4 b^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 )}{h},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {8 b \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}+\text {Subst}\left (\frac {\left (8 b^2 \sqrt {f} 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 )}{h \sqrt {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 \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}+\text {Subst}\left (\frac {\left (16 b^2 f^{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 )}{h \sqrt {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 \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}+\text {Subst}\left (\frac {\left (16 b^2 f^{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 )}{h \sqrt {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^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\text {Subst}\left (\frac {\left (16 b^2 f 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 )}{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^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\frac {16 b^2 \sqrt {f} 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 )}{h \sqrt {f g-e h}}+\text {Subst}\left (\frac {\left (16 b^2 f 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 )}{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^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\frac {16 b^2 \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\text {Subst}\left (\frac {\left (16 b^2 \sqrt {f} 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 )}{h \sqrt {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^2 \sqrt {f} p^2 q^2 \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )^2}{h \sqrt {f g-e h}}-\frac {8 b \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{h \sqrt {g+h x}}-\frac {16 b^2 \sqrt {f} 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 )}{h \sqrt {f g-e h}}-\frac {8 b^2 \sqrt {f} 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 )}{h \sqrt {f g-e h}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(785\) vs. \(2(330)=660\).
Time = 2.54 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=-\frac {2 \left (a^2 \sqrt {f g-e h} \sqrt {f (g+h x)}+4 b^2 f g p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \log (e+f x)+4 b^2 f h p^2 q^2 x \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \log (e+f x)-4 b^2 f g p^2 q^2 \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \log \left (\frac {h (e+f x)}{-f g+e h}\right )-4 b^2 f h p^2 q^2 x \text {arctanh}\left (\frac {\sqrt {f (g+h x)}}{\sqrt {f g-e h}}\right ) \log \left (\frac {h (e+f x)}{-f g+e h}\right )-b^2 \sqrt {f g-e h} p^2 q^2 \sqrt {f (g+h x)} \sqrt {\frac {f (g+h x)}{f g-e h}} \log ^2\left (\frac {h (e+f x)}{-f g+e h}\right )+2 a b \sqrt {f g-e h} \sqrt {f (g+h x)} \log \left (c \left (d (e+f x)^p\right )^q\right )+b^2 \sqrt {f g-e h} \sqrt {f (g+h x)} \log ^2\left (c \left (d (e+f x)^p\right )^q\right )+4 b \sqrt {f} p q \sqrt {g+h x} \sqrt {f (g+h x)} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right ) \left (a-b p q \log (e+f x)+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+4 b^2 \sqrt {f g-e h} p^2 q^2 \sqrt {f (g+h x)} \sqrt {\frac {f (g+h x)}{f g-e h}} \log \left (\frac {h (e+f x)}{-f g+e h}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )-2 b^2 \sqrt {f g-e h} p^2 q^2 \sqrt {f (g+h x)} \sqrt {\frac {f (g+h x)}{f g-e h}} \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )+4 b^2 \sqrt {f g-e h} p^2 q^2 \sqrt {f (g+h x)} \sqrt {\frac {f (g+h x)}{f g-e h}} \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {\frac {f (g+h x)}{f g-e h}}\right )\right )}{h \sqrt {f g-e h} \sqrt {g+h x} \sqrt {f (g+h x)}} \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{2}}{\left (h x +g \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\int \frac {\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right )^{2}}{\left (g + h x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{2}}{{\left (h x + g\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{(g+h x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^2}{{\left (g+h\,x\right )}^{3/2}} \,d x \]
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