Optimal. Leaf size=107 \[ -\frac{\text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 d^2 e^2}-\frac{\log (x) \log \left (\frac{e x}{d}+1\right )}{3 d^2 e^2}+\frac{x^2 \log ^2(x) (3 d+e x)}{6 d^2 (d+e x)^3}-\frac{x}{3 d^2 e (d+e x)}+\frac{x \log (x)}{3 d e (d+e x)^2} \]
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Rubi [A] time = 0.406657, antiderivative size = 157, normalized size of antiderivative = 1.47, number of steps used = 22, number of rules used = 10, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.769, Rules used = {2353, 2319, 2347, 2344, 2301, 2317, 2391, 2314, 31, 44} \[ -\frac{\text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 d^2 e^2}+\frac{\log ^2(x)}{6 d^2 e^2}-\frac{\log (x) \log \left (\frac{e x}{d}+1\right )}{3 d^2 e^2}+\frac{\log (x)}{3 d^2 e^2}-\frac{x \log (x)}{3 d^2 e (d+e x)}+\frac{1}{3 d e^2 (d+e x)}-\frac{\log ^2(x)}{2 e^2 (d+e x)^2}+\frac{d \log ^2(x)}{3 e^2 (d+e x)^3}-\frac{\log (x)}{3 e^2 (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 2353
Rule 2319
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 44
Rubi steps
\begin{align*} \int \frac{x \log ^2(x)}{(d+e x)^4} \, dx &=\int \left (-\frac{d \log ^2(x)}{e (d+e x)^4}+\frac{\log ^2(x)}{e (d+e x)^3}\right ) \, dx\\ &=\frac{\int \frac{\log ^2(x)}{(d+e x)^3} \, dx}{e}-\frac{d \int \frac{\log ^2(x)}{(d+e x)^4} \, dx}{e}\\ &=\frac{d \log ^2(x)}{3 e^2 (d+e x)^3}-\frac{\log ^2(x)}{2 e^2 (d+e x)^2}+\frac{\int \frac{\log (x)}{x (d+e x)^2} \, dx}{e^2}-\frac{(2 d) \int \frac{\log (x)}{x (d+e x)^3} \, dx}{3 e^2}\\ &=\frac{d \log ^2(x)}{3 e^2 (d+e x)^3}-\frac{\log ^2(x)}{2 e^2 (d+e x)^2}-\frac{2 \int \frac{\log (x)}{x (d+e x)^2} \, dx}{3 e^2}+\frac{\int \frac{\log (x)}{x (d+e x)} \, dx}{d e^2}+\frac{2 \int \frac{\log (x)}{(d+e x)^3} \, dx}{3 e}-\frac{\int \frac{\log (x)}{(d+e x)^2} \, dx}{d e}\\ &=-\frac{\log (x)}{3 e^2 (d+e x)^2}-\frac{x \log (x)}{d^2 e (d+e x)}+\frac{d \log ^2(x)}{3 e^2 (d+e x)^3}-\frac{\log ^2(x)}{2 e^2 (d+e x)^2}+\frac{\int \frac{1}{x (d+e x)^2} \, dx}{3 e^2}+\frac{\int \frac{\log (x)}{x} \, dx}{d^2 e^2}-\frac{2 \int \frac{\log (x)}{x (d+e x)} \, dx}{3 d e^2}+\frac{\int \frac{1}{d+e x} \, dx}{d^2 e}-\frac{\int \frac{\log (x)}{d+e x} \, dx}{d^2 e}+\frac{2 \int \frac{\log (x)}{(d+e x)^2} \, dx}{3 d e}\\ &=-\frac{\log (x)}{3 e^2 (d+e x)^2}-\frac{x \log (x)}{3 d^2 e (d+e x)}+\frac{\log ^2(x)}{2 d^2 e^2}+\frac{d \log ^2(x)}{3 e^2 (d+e x)^3}-\frac{\log ^2(x)}{2 e^2 (d+e x)^2}+\frac{\log (d+e x)}{d^2 e^2}-\frac{\log (x) \log \left (1+\frac{e x}{d}\right )}{d^2 e^2}+\frac{\int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx}{3 e^2}-\frac{2 \int \frac{\log (x)}{x} \, dx}{3 d^2 e^2}+\frac{\int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^2 e^2}-\frac{2 \int \frac{1}{d+e x} \, dx}{3 d^2 e}+\frac{2 \int \frac{\log (x)}{d+e x} \, dx}{3 d^2 e}\\ &=\frac{1}{3 d e^2 (d+e x)}+\frac{\log (x)}{3 d^2 e^2}-\frac{\log (x)}{3 e^2 (d+e x)^2}-\frac{x \log (x)}{3 d^2 e (d+e x)}+\frac{\log ^2(x)}{6 d^2 e^2}+\frac{d \log ^2(x)}{3 e^2 (d+e x)^3}-\frac{\log ^2(x)}{2 e^2 (d+e x)^2}-\frac{\log (x) \log \left (1+\frac{e x}{d}\right )}{3 d^2 e^2}-\frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{d^2 e^2}-\frac{2 \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{3 d^2 e^2}\\ &=\frac{1}{3 d e^2 (d+e x)}+\frac{\log (x)}{3 d^2 e^2}-\frac{\log (x)}{3 e^2 (d+e x)^2}-\frac{x \log (x)}{3 d^2 e (d+e x)}+\frac{\log ^2(x)}{6 d^2 e^2}+\frac{d \log ^2(x)}{3 e^2 (d+e x)^3}-\frac{\log ^2(x)}{2 e^2 (d+e x)^2}-\frac{\log (x) \log \left (1+\frac{e x}{d}\right )}{3 d^2 e^2}-\frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{3 d^2 e^2}\\ \end{align*}
Mathematica [A] time = 0.126762, size = 96, normalized size = 0.9 \[ \frac{-2 (d+e x)^3 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+e^2 x^2 \log ^2(x) (3 d+e x)+2 d (d+e x)^2-2 \log (x) (d+e x) \left ((d+e x)^2 \log \left (\frac{e x}{d}+1\right )-d e x\right )}{6 d^2 e^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.345, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( \ln \left ( x \right ) \right ) ^{2}}{ \left ( ex+d \right ) ^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18798, size = 178, normalized size = 1.66 \begin{align*} -\frac{d^{2} \log \left (x\right )^{2} - 2 \,{\left (e^{2} \log \left (x\right ) + e^{2}\right )} x^{2} - 2 \, d^{2} +{\left (3 \, d e \log \left (x\right )^{2} - 2 \, d e \log \left (x\right ) - 4 \, d e\right )} x}{6 \,{\left (d e^{5} x^{3} + 3 \, d^{2} e^{4} x^{2} + 3 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} + \frac{\log \left (x\right )^{2}}{6 \, d^{2} e^{2}} - \frac{\log \left (\frac{e x}{d} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{e x}{d}\right )}{3 \, d^{2} e^{2}} \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 \log \left (x\right )^{2}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.4248, size = 357, normalized size = 3.34 \begin{align*} \text{result too large to display} \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 \log \left (x\right )^{2}}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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