Integrand size = 28, antiderivative size = 86 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=-\frac {4 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}} \]
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Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2442, 65, 214, 2495} \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}}-\frac {4 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}} \]
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Rule 65
Rule 214
Rule 2442
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{(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 )}{h \sqrt {g+h x}}+\text {Subst}\left (\frac {(2 b f p q) \int \frac {1}{(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 )}{h \sqrt {g+h x}}+\text {Subst}\left (\frac {(4 b f p q) \text {Subst}\left (\int \frac {1}{e-\frac {f g}{h}+\frac {f x^2}{h}} \, dx,x,\sqrt {g+h x}\right )}{h^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\frac {4 b \sqrt {f} p q \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\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 )}{h \sqrt {g+h x}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\frac {-\frac {4 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {g+h x}}}{h} \]
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\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right )^{\frac {3}{2}}}d x\]
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none
Time = 0.34 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.79 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\left [\frac {2 \, {\left ({\left (b h p q x + b g p q\right )} \sqrt {\frac {f}{f g - e h}} \log \left (\frac {f h x + 2 \, f g - e h - 2 \, {\left (f g - e h\right )} \sqrt {h x + g} \sqrt {\frac {f}{f g - e h}}}{f x + e}\right ) - {\left (b p q \log \left (f x + e\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a\right )} \sqrt {h x + g}\right )}}{h^{2} x + g h}, -\frac {2 \, {\left (2 \, {\left (b h p q x + b g p q\right )} \sqrt {-\frac {f}{f g - e h}} \arctan \left (-\frac {{\left (f g - e h\right )} \sqrt {h x + g} \sqrt {-\frac {f}{f g - e h}}}{f h x + f g}\right ) + {\left (b p q \log \left (f x + e\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a\right )} \sqrt {h x + g}\right )}}{h^{2} x + g h}\right ] \]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\left (g + h x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\frac {4 \, b f p q \arctan \left (\frac {\sqrt {h x + g} f}{\sqrt {-f^{2} g + e f h}}\right )}{\sqrt {-f^{2} g + e f h} h} - \frac {2 \, b p q \log \left ({\left (h x + g\right )} f - f g + e h\right )}{\sqrt {h x + g} h} + \frac {2 \, {\left (b p q \log \left (h\right ) - b q \log \left (d\right ) - b \log \left (c\right ) - a\right )}}{\sqrt {h x + g} h} \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{{\left (g+h\,x\right )}^{3/2}} \,d x \]
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