Optimal. Leaf size=287 \[ \frac{p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{d}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{d}-\frac{p \text{PolyLog}\left (2,\frac{b}{a x^2}+1\right )}{2 d}-\frac{2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d}-\frac{\log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d}-\frac{\log \left (-\frac{b}{a x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 d}+\frac{p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{d}+\frac{p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{-a} d-\sqrt{b} e}\right )}{d}-\frac{2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d} \]
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Rubi [A] time = 0.460966, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {2466, 2454, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ \frac{p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{d}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{d}-\frac{p \text{PolyLog}\left (2,\frac{b}{a x^2}+1\right )}{2 d}-\frac{2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{d}-\frac{\log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d}-\frac{\log \left (-\frac{b}{a x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 d}+\frac{p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{d}+\frac{p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{-a} d-\sqrt{b} e}\right )}{d}-\frac{2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d} \]
Antiderivative was successfully verified.
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Rule 2466
Rule 2454
Rule 2394
Rule 2315
Rule 2462
Rule 260
Rule 2416
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{x (d+e x)} \, dx &=\int \left (\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d x}-\frac{e \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{x} \, dx}{d}-\frac{e \int \frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d+e x} \, dx}{d}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac{1}{x^2}\right )}{2 d}-\frac{(2 b p) \int \frac{\log (d+e x)}{\left (a+\frac{b}{x^2}\right ) x^3} \, dx}{d}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log \left (-\frac{b}{a x^2}\right )}{2 d}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{d}+\frac{(b p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{b x}{a}\right )}{a+b x} \, dx,x,\frac{1}{x^2}\right )}{2 d}-\frac{(2 b p) \int \left (\frac{\log (d+e x)}{b x}-\frac{a x \log (d+e x)}{b \left (b+a x^2\right )}\right ) \, dx}{d}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log \left (-\frac{b}{a x^2}\right )}{2 d}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac{p \text{Li}_2\left (1+\frac{b}{a x^2}\right )}{2 d}-\frac{(2 p) \int \frac{\log (d+e x)}{x} \, dx}{d}+\frac{(2 a p) \int \frac{x \log (d+e x)}{b+a x^2} \, dx}{d}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log \left (-\frac{b}{a x^2}\right )}{2 d}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac{2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d}-\frac{p \text{Li}_2\left (1+\frac{b}{a x^2}\right )}{2 d}+\frac{(2 a p) \int \left (-\frac{\sqrt{-a} \log (d+e x)}{2 a \left (\sqrt{b}-\sqrt{-a} x\right )}+\frac{\sqrt{-a} \log (d+e x)}{2 a \left (\sqrt{b}+\sqrt{-a} x\right )}\right ) \, dx}{d}+\frac{(2 e p) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{d}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log \left (-\frac{b}{a x^2}\right )}{2 d}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac{2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d}-\frac{p \text{Li}_2\left (1+\frac{b}{a x^2}\right )}{2 d}-\frac{2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d}-\frac{\left (\sqrt{-a} p\right ) \int \frac{\log (d+e x)}{\sqrt{b}-\sqrt{-a} x} \, dx}{d}+\frac{\left (\sqrt{-a} p\right ) \int \frac{\log (d+e x)}{\sqrt{b}+\sqrt{-a} x} \, dx}{d}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log \left (-\frac{b}{a x^2}\right )}{2 d}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac{2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d}+\frac{p \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right ) \log (d+e x)}{d}+\frac{p \log \left (-\frac{e \left (\sqrt{b}+\sqrt{-a} x\right )}{\sqrt{-a} d-\sqrt{b} e}\right ) \log (d+e x)}{d}-\frac{p \text{Li}_2\left (1+\frac{b}{a x^2}\right )}{2 d}-\frac{2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d}-\frac{(e p) \int \frac{\log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{d+e x} \, dx}{d}-\frac{(e p) \int \frac{\log \left (\frac{e \left (\sqrt{b}+\sqrt{-a} x\right )}{-\sqrt{-a} d+\sqrt{b} e}\right )}{d+e x} \, dx}{d}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log \left (-\frac{b}{a x^2}\right )}{2 d}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac{2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d}+\frac{p \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right ) \log (d+e x)}{d}+\frac{p \log \left (-\frac{e \left (\sqrt{b}+\sqrt{-a} x\right )}{\sqrt{-a} d-\sqrt{b} e}\right ) \log (d+e x)}{d}-\frac{p \text{Li}_2\left (1+\frac{b}{a x^2}\right )}{2 d}-\frac{2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d}-\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-a} x}{-\sqrt{-a} d+\sqrt{b} e}\right )}{x} \, dx,x,d+e x\right )}{d}-\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-a} x}{\sqrt{-a} d+\sqrt{b} e}\right )}{x} \, dx,x,d+e x\right )}{d}\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log \left (-\frac{b}{a x^2}\right )}{2 d}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{d}-\frac{2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{d}+\frac{p \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right ) \log (d+e x)}{d}+\frac{p \log \left (-\frac{e \left (\sqrt{b}+\sqrt{-a} x\right )}{\sqrt{-a} d-\sqrt{b} e}\right ) \log (d+e x)}{d}-\frac{p \text{Li}_2\left (1+\frac{b}{a x^2}\right )}{2 d}+\frac{p \text{Li}_2\left (\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{d}+\frac{p \text{Li}_2\left (\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{d}-\frac{2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.116015, size = 264, normalized size = 0.92 \[ -\frac{-2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )-2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )+p \text{PolyLog}\left (2,\frac{b}{a x^2}+1\right )+4 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\log \left (-\frac{b}{a x^2}\right ) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )-2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )-2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{b} e-\sqrt{-a} d}\right )+4 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.785, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( ex+d \right ) }\ln \left ( c \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (a + \frac{b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x}\,{d x} \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{\log \left (c \left (\frac{a x^{2} + b}{x^{2}}\right )^{p}\right )}{e x^{2} + d x}, 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{\log \left ({\left (a + \frac{b}{x^{2}}\right )}^{p} c\right )}{{\left (e x + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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