Optimal. Leaf size=278 \[ -\frac{b f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}-\frac{b f n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g^2}-\frac{f \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac{f \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac{b d^2 n \log (d+e x)}{2 e^2 g}+\frac{b d n x}{2 e g}-\frac{b n x^2}{4 g} \]
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Rubi [A] time = 0.325094, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {266, 43, 2416, 2395, 260, 2394, 2393, 2391} \[ -\frac{b f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}-\frac{b f n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g^2}-\frac{f \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac{f \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac{b d^2 n \log (d+e x)}{2 e^2 g}+\frac{b d n x}{2 e g}-\frac{b n x^2}{4 g} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2416
Rule 2395
Rule 260
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx &=\int \left (\frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac{f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int x \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}-\frac{f \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{g}\\ &=\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac{f \int \left (-\frac{a+b \log \left (c (d+e x)^n\right )}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{a+b \log \left (c (d+e x)^n\right )}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{g}-\frac{(b e n) \int \frac{x^2}{d+e x} \, dx}{2 g}\\ &=\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}+\frac{f \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 g^{3/2}}-\frac{f \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 g^{3/2}}-\frac{(b e n) \int \left (-\frac{d}{e^2}+\frac{x}{e}+\frac{d^2}{e^2 (d+e x)}\right ) \, dx}{2 g}\\ &=\frac{b d n x}{2 e g}-\frac{b n x^2}{4 g}-\frac{b d^2 n \log (d+e x)}{2 e^2 g}+\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}+\frac{(b e f n) \int \frac{\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{d+e x} \, dx}{2 g^2}+\frac{(b e f n) \int \frac{\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{d+e x} \, dx}{2 g^2}\\ &=\frac{b d n x}{2 e g}-\frac{b n x^2}{4 g}-\frac{b d^2 n \log (d+e x)}{2 e^2 g}+\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}+\frac{(b f n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{2 g^2}+\frac{(b f n) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{2 g^2}\\ &=\frac{b d n x}{2 e g}-\frac{b n x^2}{4 g}-\frac{b d^2 n \log (d+e x)}{2 e^2 g}+\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}-\frac{b f n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}-\frac{b f n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}\\ \end{align*}
Mathematica [A] time = 0.172062, size = 243, normalized size = 0.87 \[ -\frac{2 b f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )+2 b f n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )+2 f \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+2 f \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-2 g x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b g n \left (2 d^2 \log (d+e x)+e x (e x-2 d)\right )}{e^2}}{4 g^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.456, size = 631, normalized size = 2.3 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{x^{2}}{g} - \frac{f \log \left (g x^{2} + f\right )}{g^{2}}\right )} + b \int \frac{x^{3} \log \left ({\left (e x + d\right )}^{n}\right ) + x^{3} \log \left (c\right )}{g x^{2} + f}\,{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{b x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + a x^{3}}{g x^{2} + f}, 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{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{3}}{g x^{2} + f}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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