Optimal. Leaf size=939 \[ \text{result too large to display} \]
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Rubi [F] time = 0.0295654, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx &=\int \frac{\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx\\ \end{align*}
Mathematica [A] time = 1.17526, size = 781, normalized size = 0.83 \[ \frac{-2 b m n \left (e x \left (-2 e x \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log (x) \log \left (\frac{e x}{d}+1\right )\right )+2 d (\log (x)+1)+e x \log ^2(x)\right )+\left (d^2-e^2 x^2\right ) \log (d+e x)+2 d^2 \log (x) \log (d+e x)+e^2 x^2 \log \left (-\frac{e x}{d}\right )+e x (d+e x)\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )+b^2 n^2 \left (-2 e^2 x^2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right ) \left (2 m \log (d+e x)+2 \log \left (f x^m\right )-2 m \log (x)+m\right )-4 e^2 m x^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )+4 e^2 m x^2 \text{PolyLog}\left (3,\frac{e x}{d}+1\right )+4 e^2 m x^2 (\log (x)-1) \text{PolyLog}\left (2,-\frac{e x}{d}\right )-2 d^2 \log ^2(d+e x) \log \left (f x^m\right )-d^2 m \log ^2(d+e x)+2 e^2 x^2 \log ^2(d+e x) \log \left (f x^m\right )-4 e^2 x^2 \log (d+e x) \log \left (f x^m\right )-4 e^2 x^2 \log \left (-\frac{e x}{d}\right ) \log (d+e x) \log \left (f x^m\right )+e^2 m x^2 \log ^2(d+e x)-2 e^2 m x^2 \log \left (-\frac{e x}{d}\right ) \log ^2(d+e x)-2 e^2 m x^2 \log ^2(x) \log (d+e x)+2 e^2 m x^2 \log ^2(x) \log \left (\frac{e x}{d}+1\right )+2 e^2 m x^2 \log \left (-\frac{e x}{d}\right )-6 e^2 m x^2 \log (d+e x)+4 e^2 m x^2 \log (x) \log (d+e x)-2 e^2 m x^2 \log \left (-\frac{e x}{d}\right ) \log (d+e x)+4 e^2 m x^2 \log (x) \log \left (-\frac{e x}{d}\right ) \log (d+e x)-4 e^2 m x^2 \log (x) \log \left (\frac{e x}{d}+1\right )-4 d e x \log (d+e x) \log \left (f x^m\right )-6 d e m x \log (d+e x)+4 e^2 x^2 \log (x) \log \left (f x^m\right )-2 e^2 m x^2 \log ^2(x)+4 e^2 m x^2 \log (x)\right )+d^2 \left (-2 \log \left (f x^m\right )+2 m \log (x)-m\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2-2 d^2 m \log (x) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+4 b n \left (e^2 x^2 \log \left (-\frac{e x}{d}\right )+(d+e x) ((d-e x) \log (d+e x)+e x)\right ) \left (m \log (x)-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )}{4 d^2 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.454, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( f{x}^{m} \right ) \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{{x}^{3}}}\, 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*} -\frac{{\left (b^{2}{\left (m + 2 \, \log \left (f\right )\right )} + 2 \, b^{2} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right )^{2}}{4 \, x^{2}} + \int \frac{2 \, b^{2} d \log \left (c\right )^{2} \log \left (f\right ) + 4 \, a b d \log \left (c\right ) \log \left (f\right ) + 2 \, a^{2} d \log \left (f\right ) + 2 \,{\left (b^{2} e \log \left (c\right )^{2} \log \left (f\right ) + 2 \, a b e \log \left (c\right ) \log \left (f\right ) + a^{2} e \log \left (f\right )\right )} x +{\left (4 \, b^{2} d \log \left (c\right ) \log \left (f\right ) + 4 \, a b d \log \left (f\right ) +{\left (4 \, a b e \log \left (f\right ) +{\left (4 \, e \log \left (c\right ) \log \left (f\right ) +{\left (m n + 2 \, n \log \left (f\right )\right )} e\right )} b^{2}\right )} x + 2 \,{\left (2 \, b^{2} d \log \left (c\right ) + 2 \, a b d +{\left ({\left (e n + 2 \, e \log \left (c\right )\right )} b^{2} + 2 \, a b e\right )} x\right )} \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) + 2 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d +{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x\right )} \log \left (x^{m}\right )}{2 \,{\left (e x^{4} + d x^{3}\right )}}\,{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^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} \log \left (f x^{m}\right ) + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a^{2} \log \left (f x^{m}\right )}{x^{3}}, 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 )}^{2} \log \left (f x^{m}\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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