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 = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, size = 75, normalized size = 1.00 \[ \frac {(c+d x)^3 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{3/n} \text {Ei}\left (-\frac {3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 88, normalized size = 1.17 \[ \frac {e^{\frac {3}{n}} \operatorname {log\_integral}\left (\frac {d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}{{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} e^{\frac {3}{n}}}\right )}{{\left (b c - a d\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{2}}{\left (b x +a \right )^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{4} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^2}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,{\left (a+b\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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