Optimal. Leaf size=344 \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}+\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g^2}+\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac{\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{b e^2 f n \log \left (f+g x^2\right )}{4 g^2 \left (d^2 g+e^2 f\right )}-\frac{b e^2 f n \log (d+e x)}{2 g^2 \left (d^2 g+e^2 f\right )}-\frac{b d e \sqrt{f} n \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 g^{3/2} \left (d^2 g+e^2 f\right )} \]
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Rubi [A] time = 0.407449, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {266, 43, 2416, 2413, 706, 31, 635, 205, 260, 2394, 2393, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}+\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g^2}+\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac{\log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{\log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{b e^2 f n \log \left (f+g x^2\right )}{4 g^2 \left (d^2 g+e^2 f\right )}-\frac{b e^2 f n \log (d+e x)}{2 g^2 \left (d^2 g+e^2 f\right )}-\frac{b d e \sqrt{f} n \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 g^{3/2} \left (d^2 g+e^2 f\right )} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 2416
Rule 2413
Rule 706
Rule 31
Rule 635
Rule 205
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 )}{\left (f+g x^2\right )^2} \, dx &=\int \left (-\frac{f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \left (f+g x^2\right )^2}+\frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{g}-\frac{f \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx}{g}\\ &=\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac{\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 f n) \int \frac{1}{(d+e x) \left (f+g x^2\right )} \, dx}{2 g^2}\\ &=\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}-\frac{\int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 g^{3/2}}+\frac{\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 f n) \int \frac{d g-e g x}{f+g x^2} \, dx}{2 g^2 \left (e^2 f+d^2 g\right )}-\frac{\left (b e^3 f n\right ) \int \frac{1}{d+e x} \, dx}{2 g^2 \left (e^2 f+d^2 g\right )}\\ &=-\frac{b e^2 f n \log (d+e x)}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac{\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{\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 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 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 e f n) \int \frac{1}{f+g x^2} \, dx}{2 g \left (e^2 f+d^2 g\right )}+\frac{\left (b e^2 f n\right ) \int \frac{x}{f+g x^2} \, dx}{2 g \left (e^2 f+d^2 g\right )}\\ &=-\frac{b d e \sqrt{f} n \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 g^{3/2} \left (e^2 f+d^2 g\right )}-\frac{b e^2 f n \log (d+e x)}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac{\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{\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^2 f n \log \left (f+g x^2\right )}{4 g^2 \left (e^2 f+d^2 g\right )}-\frac{(b 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 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 e \sqrt{f} n \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 g^{3/2} \left (e^2 f+d^2 g\right )}-\frac{b e^2 f n \log (d+e x)}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac{\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{\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^2 f n \log \left (f+g x^2\right )}{4 g^2 \left (e^2 f+d^2 g\right )}+\frac{b n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}+\frac{b n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}\\ \end{align*}
Mathematica [C] time = 0.881506, size = 455, normalized size = 1.32 \[ \frac{b n \left (2 \left (\text{PolyLog}\left (2,-\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}-i d \sqrt{g}}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt{f}+i \sqrt{g} x\right )}{e \sqrt{f}-i d \sqrt{g}}\right )\right )+2 \left (\text{PolyLog}\left (2,\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}+i d \sqrt{g}}\right )+\log (d+e x) \log \left (\frac{e \left (\sqrt{f}-i \sqrt{g} x\right )}{e \sqrt{f}+i d \sqrt{g}}\right )\right )+\frac{\sqrt{f} \left (e \left (\sqrt{f}+i \sqrt{g} x\right ) \log \left (-\sqrt{g} x+i \sqrt{f}\right )-i \sqrt{g} (d+e x) \log (d+e x)\right )}{\left (\sqrt{f}+i \sqrt{g} x\right ) \left (e \sqrt{f}-i d \sqrt{g}\right )}+\frac{\sqrt{f} \left (i \sqrt{g} (d+e x) \log (d+e x)+e \left (\sqrt{f}-i \sqrt{g} x\right ) \log \left (\sqrt{g} x+i \sqrt{f}\right )\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (e \sqrt{f}+i d \sqrt{g}\right )}\right )+2 \log \left (f+g x^2\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+\frac{2 f \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{f+g x^2}}{4 g^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.408, size = 726, normalized size = 2.1 \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{f}{g^{3} x^{2} + f g^{2}} + \frac{\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^{2} x^{4} + 2 \, f g x^{2} + f^{2}}\,{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^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, 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}}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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