Optimal. Leaf size=75 \[ \frac{(c+d x)^3 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{3/n} \text{Ei}\left (-\frac{3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^3 (b c-a d)} \]
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Rubi [A] time = 0.0698958, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {2510} \[ \frac{(c+d x)^3 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{3/n} \text{Ei}\left (-\frac{3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2510
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{(a+b x)^4 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\frac{\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{3/n} (c+d x)^3 \text{Ei}\left (-\frac{3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0182585, size = 75, normalized size = 1. \[ \frac{(c+d x)^3 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{3/n} \text{Ei}\left (-\frac{3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.67, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{2}}{ \left ( bx+a \right ) ^{4}} \left ( \ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-1}}\, 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*} \int \frac{{\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{4} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.520909, size = 180, normalized size = 2.4 \begin{align*} \frac{e^{\frac{3}{n}} \logintegral \left (\frac{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}{{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} e^{\frac{3}{n}}}\right )}{{\left (b c - a d\right )} n} \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 (d x + c\right )}^{2}}{{\left (b x + a\right )}^{4} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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