Optimal. Leaf size=75 \[ \frac {(a+b x)^4 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-4/n} \text {Ei}\left (\frac {4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x)^4 (b c-a d)} \]
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Rubi [A] time = 0.08, 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} \[ \frac {(a+b x)^4 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-4/n} \text {Ei}\left (\frac {4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x)^4 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2510
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{(c+d x)^5 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=\frac {(a+b x)^4 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-4/n} \text {Ei}\left (\frac {4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 1.00 \[ \frac {(a+b x)^4 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-4/n} \text {Ei}\left (\frac {4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x)^4 (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 110, normalized size = 1.47 \[ \frac {\operatorname {log\_integral}\left (\frac {{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} e^{\frac {4}{n}}}{d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}}\right )}{{\left (b c - a d\right )} e^{\frac {4}{n}} n} \]
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 = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{3}}{\left (d x +c \right )^{5} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{3}}{{\left (d x + c\right )}^{5} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^3}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,{\left (c+d\,x\right )}^5} \,d x \]
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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