Optimal. Leaf size=353 \[ -\frac{d^2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^3}-\frac{d^2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^3}+\frac{2 d^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e^3}+\frac{d^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^3}-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}-\frac{d^2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^3}-\frac{d^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^3}-\frac{2 \sqrt{b} d p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^2}+\frac{b p \log \left (a x^2+b\right )}{2 a e}+\frac{2 d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.492004, antiderivative size = 353, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {2466, 2448, 263, 205, 2455, 260, 2462, 2416, 2394, 2315, 2393, 2391} \[ -\frac{d^2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^3}-\frac{d^2 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^3}+\frac{2 d^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e^3}+\frac{d^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^3}-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}-\frac{d^2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^3}-\frac{d^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^3}-\frac{2 \sqrt{b} d p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^2}+\frac{b p \log \left (a x^2+b\right )}{2 a e}+\frac{2 d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
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Rule 2466
Rule 2448
Rule 263
Rule 205
Rule 2455
Rule 260
Rule 2462
Rule 2416
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d+e x} \, dx &=\int \left (-\frac{d \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{d \int \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \, dx}{e^2}+\frac{d^2 \int \frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{d+e x} \, dx}{e^2}+\frac{\int x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \, dx}{e}\\ &=-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac{\left (2 b d^2 p\right ) \int \frac{\log (d+e x)}{\left (a+\frac{b}{x^2}\right ) x^3} \, dx}{e^3}-\frac{(2 b d p) \int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^2} \, dx}{e^2}+\frac{(b p) \int \frac{1}{\left (a+\frac{b}{x^2}\right ) x} \, dx}{e}\\ &=-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac{\left (2 b d^2 p\right ) \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}{e^3}-\frac{(2 b d p) \int \frac{1}{b+a x^2} \, dx}{e^2}+\frac{(b p) \int \frac{x}{b+a x^2} \, dx}{e}\\ &=-\frac{2 \sqrt{b} d p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^2}-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac{b p \log \left (b+a x^2\right )}{2 a e}+\frac{\left (2 d^2 p\right ) \int \frac{\log (d+e x)}{x} \, dx}{e^3}-\frac{\left (2 a d^2 p\right ) \int \frac{x \log (d+e x)}{b+a x^2} \, dx}{e^3}\\ &=-\frac{2 \sqrt{b} d p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^2}-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac{2 d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3}+\frac{b p \log \left (b+a x^2\right )}{2 a e}-\frac{\left (2 a d^2 p\right ) \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}{e^3}-\frac{\left (2 d^2 p\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{e^2}\\ &=-\frac{2 \sqrt{b} d p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^2}-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac{2 d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3}+\frac{b p \log \left (b+a x^2\right )}{2 a e}+\frac{2 d^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{e^3}+\frac{\left (\sqrt{-a} d^2 p\right ) \int \frac{\log (d+e x)}{\sqrt{b}-\sqrt{-a} x} \, dx}{e^3}-\frac{\left (\sqrt{-a} d^2 p\right ) \int \frac{\log (d+e x)}{\sqrt{b}+\sqrt{-a} x} \, dx}{e^3}\\ &=-\frac{2 \sqrt{b} d p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^2}-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac{2 d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e \left (\sqrt{b}+\sqrt{-a} x\right )}{\sqrt{-a} d-\sqrt{b} e}\right ) \log (d+e x)}{e^3}+\frac{b p \log \left (b+a x^2\right )}{2 a e}+\frac{2 d^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{e^3}+\frac{\left (d^2 p\right ) \int \frac{\log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{d+e x} \, dx}{e^2}+\frac{\left (d^2 p\right ) \int \frac{\log \left (\frac{e \left (\sqrt{b}+\sqrt{-a} x\right )}{-\sqrt{-a} d+\sqrt{b} e}\right )}{d+e x} \, dx}{e^2}\\ &=-\frac{2 \sqrt{b} d p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^2}-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac{2 d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e \left (\sqrt{b}+\sqrt{-a} x\right )}{\sqrt{-a} d-\sqrt{b} e}\right ) \log (d+e x)}{e^3}+\frac{b p \log \left (b+a x^2\right )}{2 a e}+\frac{2 d^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{e^3}+\frac{\left (d^2 p\right ) \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 )}{e^3}+\frac{\left (d^2 p\right ) \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 )}{e^3}\\ &=-\frac{2 \sqrt{b} d p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^2}-\frac{d x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac{2 d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e \left (\sqrt{b}+\sqrt{-a} x\right )}{\sqrt{-a} d-\sqrt{b} e}\right ) \log (d+e x)}{e^3}+\frac{b p \log \left (b+a x^2\right )}{2 a e}-\frac{d^2 p \text{Li}_2\left (\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^3}-\frac{d^2 p \text{Li}_2\left (\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^3}+\frac{2 d^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.223409, size = 319, normalized size = 0.9 \[ \frac{2 d^2 p \left (-\text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )-\text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )+2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )-\log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )-\log (d+e x) \log \left (\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{b} e-\sqrt{-a} d}\right )+2 \log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )+2 d^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )-2 d e x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+e^2 x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{4 \sqrt{b} d e p \tan ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} x}\right )}{\sqrt{a}}+\frac{b e^2 p \left (\log \left (a+\frac{b}{x^2}\right )+2 \log (x)\right )}{a}}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.741, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ex+d}\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(-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{x^{2} \log \left (c \left (\frac{a x^{2} + b}{x^{2}}\right )^{p}\right )}{e x + d}, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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