Optimal. Leaf size=313 \[ -\frac{d^2 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^3}-\frac{d^2 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )}{e^3}+\frac{d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac{d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}-\frac{d^2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )}{e^3}-\frac{d^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^3}-\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} e^2}+\frac{2 d p x}{e^2}-\frac{p x^2}{2 e} \]
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Rubi [A] time = 0.334236, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {2466, 2448, 321, 205, 2454, 2389, 2295, 2462, 260, 2416, 2394, 2393, 2391} \[ -\frac{d^2 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^3}-\frac{d^2 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )}{e^3}+\frac{d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac{d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}-\frac{d^2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )}{e^3}-\frac{d^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^3}-\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} e^2}+\frac{2 d p x}{e^2}-\frac{p x^2}{2 e} \]
Antiderivative was successfully verified.
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Rule 2466
Rule 2448
Rule 321
Rule 205
Rule 2454
Rule 2389
Rule 2295
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx &=\int \left (-\frac{d \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{x \log \left (c \left (a+b x^2\right )^p\right )}{e}+\frac{d^2 \log \left (c \left (a+b x^2\right )^p\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{d \int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{e^2}+\frac{d^2 \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx}{e^2}+\frac{\int x \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{e}\\ &=-\frac{d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac{\operatorname{Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,x^2\right )}{2 e}-\frac{\left (2 b d^2 p\right ) \int \frac{x \log (d+e x)}{a+b x^2} \, dx}{e^3}+\frac{(2 b d p) \int \frac{x^2}{a+b x^2} \, dx}{e^2}\\ &=\frac{2 d p x}{e^2}-\frac{d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac{\operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x^2\right )}{2 b e}-\frac{\left (2 b d^2 p\right ) \int \left (-\frac{\log (d+e x)}{2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\log (d+e x)}{2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{e^3}-\frac{(2 a d p) \int \frac{1}{a+b x^2} \, dx}{e^2}\\ &=\frac{2 d p x}{e^2}-\frac{p x^2}{2 e}-\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} e^2}-\frac{d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac{d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac{\left (\sqrt{b} d^2 p\right ) \int \frac{\log (d+e x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{e^3}-\frac{\left (\sqrt{b} d^2 p\right ) \int \frac{\log (d+e x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{e^3}\\ &=\frac{2 d p x}{e^2}-\frac{p x^2}{2 e}-\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} e^2}-\frac{d^2 p \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{b} d+\sqrt{-a} e}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right ) \log (d+e x)}{e^3}-\frac{d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac{d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac{\left (d^2 p\right ) \int \frac{\log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{b} d+\sqrt{-a} e}\right )}{d+e x} \, dx}{e^2}+\frac{\left (d^2 p\right ) \int \frac{\log \left (\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{-\sqrt{b} d+\sqrt{-a} e}\right )}{d+e x} \, dx}{e^2}\\ &=\frac{2 d p x}{e^2}-\frac{p x^2}{2 e}-\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} e^2}-\frac{d^2 p \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{b} d+\sqrt{-a} e}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right ) \log (d+e x)}{e^3}-\frac{d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac{d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac{\left (d^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{b} x}{-\sqrt{b} d+\sqrt{-a} 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{b} x}{\sqrt{b} d+\sqrt{-a} e}\right )}{x} \, dx,x,d+e x\right )}{e^3}\\ &=\frac{2 d p x}{e^2}-\frac{p x^2}{2 e}-\frac{2 \sqrt{a} d p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} e^2}-\frac{d^2 p \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{b} d+\sqrt{-a} e}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right ) \log (d+e x)}{e^3}-\frac{d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac{d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac{d^2 p \text{Li}_2\left (\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^3}-\frac{d^2 p \text{Li}_2\left (\frac{\sqrt{b} (d+e x)}{\sqrt{b} d+\sqrt{-a} e}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.161251, size = 271, normalized size = 0.87 \[ \frac{-2 d^2 p \left (\text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )+\log (d+e x) \left (\log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )+\log \left (\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{-a} e-\sqrt{b} d}\right )\right )\right )+2 d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )-2 d e x \log \left (c \left (a+b x^2\right )^p\right )+\frac{e^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}+4 d e p \left (x-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}\right )-e^2 p x^2}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.465, size = 825, normalized size = 2.6 \begin{align*} \text{result too large to display} \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 ({\left (b x^{2} + a\right )}^{p} c\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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