Optimal. Leaf size=249 \[ -\frac{b p q \text{PolyLog}\left (2,-\frac{\sqrt{h} (e+f x)}{f \sqrt{-g}-e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{b p q \text{PolyLog}\left (2,\frac{\sqrt{h} (e+f x)}{e \sqrt{h}+f \sqrt{-g}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{\log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{e \sqrt{h}+f \sqrt{-g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}}-\frac{\log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}} \]
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Rubi [A] time = 0.503994, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2409, 2394, 2393, 2391, 2445} \[ -\frac{b p q \text{PolyLog}\left (2,-\frac{\sqrt{h} (e+f x)}{f \sqrt{-g}-e \sqrt{h}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{b p q \text{PolyLog}\left (2,\frac{\sqrt{h} (e+f x)}{e \sqrt{h}+f \sqrt{-g}}\right )}{2 \sqrt{-g} \sqrt{h}}+\frac{\log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{e \sqrt{h}+f \sqrt{-g}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}}-\frac{\log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}} \]
Antiderivative was successfully verified.
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Rule 2409
Rule 2394
Rule 2393
Rule 2391
Rule 2445
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{g+h x^2} \, dx &=\operatorname{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 )\\ &=\operatorname{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 )\\ &=-\operatorname{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 )-\operatorname{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}}-\operatorname{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 )+\operatorname{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}}+\operatorname{Subst}\left (\frac{(b p q) \operatorname{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 )-\operatorname{Subst}\left (\frac{(b p q) \operatorname{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*}
Mathematica [A] time = 0.137004, size = 190, normalized size = 0.76 \[ \frac{-b p q \text{PolyLog}\left (2,-\frac{\sqrt{h} (e+f x)}{f \sqrt{-g}-e \sqrt{h}}\right )+b p q \text{PolyLog}\left (2,\frac{\sqrt{h} (e+f x)}{e \sqrt{h}+f \sqrt{-g}}\right )+\left (\log \left (\frac{f \left (\sqrt{-g}-\sqrt{h} x\right )}{e \sqrt{h}+f \sqrt{-g}}\right )-\log \left (\frac{f \left (\sqrt{-g}+\sqrt{h} x\right )}{f \sqrt{-g}-e \sqrt{h}}\right )\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 \sqrt{-g} \sqrt{h}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.731, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) }{h{x}^{2}+g}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x^{2} + g}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{h x^{2} + g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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