Integrand size = 28, antiderivative size = 249 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}+\frac {b p q \operatorname {PolyLog}\left (2,\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}} \]
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Time = 0.35 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2456, 2441, 2440, 2438, 2495} \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx=\frac {\log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{e \sqrt {h}+f \sqrt {-g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {\log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}+\frac {b p q \operatorname {PolyLog}\left (2,\frac {\sqrt {h} (e+f x)}{\sqrt {h} e+f \sqrt {-g}}\right )}{2 \sqrt {-g} \sqrt {h}} \]
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Rule 2438
Rule 2440
Rule 2441
Rule 2456
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^2} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \text {Subst}\left (\int \left (\frac {\sqrt {-g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{2 g \left (\sqrt {-g}-\sqrt {h} x\right )}+\frac {\sqrt {-g} \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{2 g \left (\sqrt {-g}+\sqrt {h} x\right )}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = -\text {Subst}\left (\frac {\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {-g}-\sqrt {h} x} \, dx}{2 \sqrt {-g}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {-g}+\sqrt {h} x} \, dx}{2 \sqrt {-g}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\text {Subst}\left (\frac {(b f p q) \int \frac {\log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{f \sqrt {-g}+e \sqrt {h}}\right )}{e+f x} \, dx}{2 \sqrt {-g} \sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\text {Subst}\left (\frac {(b f p q) \int \frac {\log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right )}{e+f x} \, dx}{2 \sqrt {-g} \sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}+\text {Subst}\left (\frac {(b p q) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {h} x}{f \sqrt {-g}-e \sqrt {h}}\right )}{x} \, dx,x,e+f x\right )}{2 \sqrt {-g} \sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\text {Subst}\left (\frac {(b p q) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {h} x}{f \sqrt {-g}+e \sqrt {h}}\right )}{x} \, dx,x,e+f x\right )}{2 \sqrt {-g} \sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right ) \\ & = \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}-\frac {b p q \text {Li}_2\left (-\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}-e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}}+\frac {b p q \text {Li}_2\left (\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx=\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \left (\log \left (\frac {f \left (\sqrt {-g}-\sqrt {h} x\right )}{f \sqrt {-g}+e \sqrt {h}}\right )-\log \left (\frac {f \left (\sqrt {-g}+\sqrt {h} x\right )}{f \sqrt {-g}-e \sqrt {h}}\right )\right )-b p q \operatorname {PolyLog}\left (2,-\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}-e \sqrt {h}}\right )+b p q \operatorname {PolyLog}\left (2,\frac {\sqrt {h} (e+f x)}{f \sqrt {-g}+e \sqrt {h}}\right )}{2 \sqrt {-g} \sqrt {h}} \]
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\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{h \,x^{2}+g}d x\]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x^{2} + g} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x^{2} + g} \,d x } \]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x^{2} + g} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{h\,x^2+g} \,d x \]
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