Optimal. Leaf size=75 \[ \frac {(a+b x)^3 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-3/n} \text {Ei}\left (\frac {3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2561, 2347,
2209} \begin {gather*} \frac {(a+b 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 (c+d x)^3 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2347
Rule 2561
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{(c+d x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=\frac {(a+b 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 )}{(b c-a d) n (c+d x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 75, normalized size = 1.00 \begin {gather*} \frac {(a+b 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 )}{(b c-a d) n (c+d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{2}}{\left (d x +c \right )^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 84, normalized size = 1.12 \begin {gather*} \frac {e^{\left (-\frac {3}{n}\right )} \operatorname {log\_integral}\left (\frac {{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} e^{\frac {3}{n}}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{{\left (b c - a d\right )} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^2}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,{\left (c+d\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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