Optimal. Leaf size=346 \[ \frac{2 b^2 n^2 x^{1-m} (f x)^{m-1} \text{PolyLog}\left (2,-\frac{d x^{-m}}{e}\right )}{3 d^3 e m^3}-\frac{2 b n x^{1-m} (f x)^{m-1} \log \left (\frac{d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e m^2}-\frac{2 b n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}+\frac{b n x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac{x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac{b^2 n^2 x^{1-m} (f x)^{m-1}}{3 d^2 e m^3 \left (d+e x^m\right )}-\frac{b^2 n^2 x^{1-m} \log (x) (f x)^{m-1}}{3 d^3 e m^2}+\frac{b^2 n^2 x^{1-m} (f x)^{m-1} \log \left (d+e x^m\right )}{d^3 e m^3} \]
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Rubi [A] time = 0.714242, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2339, 2338, 2349, 2345, 2391, 2335, 260, 266, 44} \[ \frac{2 b^2 n^2 x^{1-m} (f x)^{m-1} \text{PolyLog}\left (2,-\frac{d x^{-m}}{e}\right )}{3 d^3 e m^3}-\frac{2 b n x^{1-m} (f x)^{m-1} \log \left (\frac{d x^{-m}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^3 e m^2}-\frac{2 b n x (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}+\frac{b n x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac{x^{1-m} (f x)^{m-1} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac{b^2 n^2 x^{1-m} (f x)^{m-1}}{3 d^2 e m^3 \left (d+e x^m\right )}-\frac{b^2 n^2 x^{1-m} \log (x) (f x)^{m-1}}{3 d^3 e m^2}+\frac{b^2 n^2 x^{1-m} (f x)^{m-1} \log \left (d+e x^m\right )}{d^3 e m^3} \]
Antiderivative was successfully verified.
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Rule 2339
Rule 2338
Rule 2349
Rule 2345
Rule 2391
Rule 2335
Rule 260
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int \frac{x^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^4} \, dx\\ &=-\frac{x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}+\frac{\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )^3} \, dx}{3 e m}\\ &=-\frac{x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac{\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^3} \, dx}{3 d m}+\frac{\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )^2} \, dx}{3 d e m}\\ &=\frac{b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac{x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac{\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^m\right )^2} \, dx}{3 d^2 m}+\frac{\left (2 b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^m\right )} \, dx}{3 d^2 e m}-\frac{\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac{1}{x \left (d+e x^m\right )^2} \, dx}{3 d e m^2}\\ &=\frac{b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac{2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}-\frac{x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac{2 b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-m}}{e}\right )}{3 d^3 e m^2}-\frac{\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \operatorname{Subst}\left (\int \frac{1}{x (d+e x)^2} \, dx,x,x^m\right )}{3 d e m^3}+\frac{\left (2 b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac{x^{-1+m}}{d+e x^m} \, dx}{3 d^3 m^2}+\frac{\left (2 b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \int \frac{\log \left (1+\frac{d x^{-m}}{e}\right )}{x} \, dx}{3 d^3 e m^2}\\ &=\frac{b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac{2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}-\frac{x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac{2 b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-m}}{e}\right )}{3 d^3 e m^2}+\frac{2 b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{3 d^3 e m^3}+\frac{2 b^2 n^2 x^{1-m} (f x)^{-1+m} \text{Li}_2\left (-\frac{d x^{-m}}{e}\right )}{3 d^3 e m^3}-\frac{\left (b^2 n^2 x^{1-m} (f x)^{-1+m}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx,x,x^m\right )}{3 d e m^3}\\ &=-\frac{b^2 n^2 x^{1-m} (f x)^{-1+m}}{3 d^2 e m^3 \left (d+e x^m\right )}-\frac{b^2 n^2 x^{1-m} (f x)^{-1+m} \log (x)}{3 d^3 e m^2}+\frac{b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d e m^2 \left (d+e x^m\right )^2}-\frac{2 b n x (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{3 d^3 m^2 \left (d+e x^m\right )}-\frac{x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{3 e m \left (d+e x^m\right )^3}-\frac{2 b n x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-m}}{e}\right )}{3 d^3 e m^2}+\frac{b^2 n^2 x^{1-m} (f x)^{-1+m} \log \left (d+e x^m\right )}{d^3 e m^3}+\frac{2 b^2 n^2 x^{1-m} (f x)^{-1+m} \text{Li}_2\left (-\frac{d x^{-m}}{e}\right )}{3 d^3 e m^3}\\ \end{align*}
Mathematica [A] time = 0.53747, size = 240, normalized size = 0.69 \[ \frac{x^{-m} (f x)^m \left (\frac{2 b^2 n^2 \left (\text{PolyLog}\left (2,\frac{e x^m}{d}+1\right )+\left (\log \left (-\frac{e x^m}{d}\right )-m \log (x)\right ) \log \left (d+e x^m\right )+\frac{1}{2} m^2 \log ^2(x)\right )}{d^3}+\frac{b n \left (2 a m+2 b m \log \left (c x^n\right )-b n\right )}{d^2 \left (d+e x^m\right )}-\frac{m^2 \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^m\right )^3}+\frac{b m n \left (a+b \log \left (c x^n\right )\right )}{d \left (d+e x^m\right )^2}-\frac{2 a b m n \log \left (d-d x^m\right )}{d^3}+\frac{2 b^2 m n \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^m\right )}{d^3}+\frac{3 b^2 n^2 \log \left (d-d x^m\right )}{d^3}\right )}{3 e f m^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.921, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{-1+m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{ \left ( d+e{x}^{m} \right ) ^{4}}}\, 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{1}{3} \, a b f^{m} n{\left (\frac{2 \, e x^{m} + 3 \, d}{{\left (d^{2} e^{3} f m x^{2 \, m} + 2 \, d^{3} e^{2} f m x^{m} + d^{4} e f m\right )} m} + \frac{2 \, \log \left (x\right )}{d^{3} e f m} - \frac{2 \, \log \left (e x^{m} + d\right )}{d^{3} e f m^{2}}\right )} - \frac{1}{3} \,{\left (\frac{f^{m} \log \left (x^{n}\right )^{2}}{e^{4} f m x^{3 \, m} + 3 \, d e^{3} f m x^{2 \, m} + 3 \, d^{2} e^{2} f m x^{m} + d^{3} e f m} - 3 \, \int \frac{3 \, e f^{m} m x^{m} \log \left (c\right )^{2} + 2 \,{\left (d f^{m} n +{\left (3 \, e f^{m} m \log \left (c\right ) + e f^{m} n\right )} x^{m}\right )} \log \left (x^{n}\right )}{3 \,{\left (e^{5} f m x x^{4 \, m} + 4 \, d e^{4} f m x x^{3 \, m} + 6 \, d^{2} e^{3} f m x x^{2 \, m} + 4 \, d^{3} e^{2} f m x x^{m} + d^{4} e f m x\right )}}\,{d x}\right )} b^{2} - \frac{2 \, a b f^{m} \log \left (c x^{n}\right )}{3 \,{\left (e^{4} f m x^{3 \, m} + 3 \, d e^{3} f m x^{2 \, m} + 3 \, d^{2} e^{2} f m x^{m} + d^{3} e f m\right )}} - \frac{a^{2} f^{m}}{3 \,{\left (e^{4} f m x^{3 \, m} + 3 \, d e^{3} f m x^{2 \, m} + 3 \, d^{2} e^{2} f m x^{m} + d^{3} e f m\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46645, size = 1823, normalized size = 5.27 \begin{align*} \text{result too large to display} \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 (c x^{n}\right ) + a\right )}^{2} \left (f x\right )^{m - 1}}{{\left (e x^{m} + d\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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