Optimal. Leaf size=76 \[ \frac {4 x^4 e^{-\frac {4 a}{b n}} \left (c x^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^4}{b n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2306, 2310, 2178} \[ \frac {4 x^4 e^{-\frac {4 a}{b n}} \left (c x^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^4}{b n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2306
Rule 2310
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac {x^4}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac {4 \int \frac {x^3}{a+b \log \left (c x^n\right )} \, dx}{b n}\\ &=-\frac {x^4}{b n \left (a+b \log \left (c x^n\right )\right )}+\frac {\left (4 x^4 \left (c x^n\right )^{-4/n}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {4 x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{b n^2}\\ &=\frac {4 e^{-\frac {4 a}{b n}} x^4 \left (c x^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{b^2 n^2}-\frac {x^4}{b n \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 70, normalized size = 0.92 \[ \frac {x^4 \left (4 e^{-\frac {4 a}{b n}} \left (c x^n\right )^{-4/n} \text {Ei}\left (\frac {4 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {b n}{a+b \log \left (c x^n\right )}\right )}{b^2 n^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 101, normalized size = 1.33 \[ -\frac {{\left (b n x^{4} e^{\left (\frac {4 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )} - 4 \, {\left (b n \log \relax (x) + b \log \relax (c) + a\right )} \operatorname {log\_integral}\left (x^{4} e^{\left (\frac {4 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {4 \, {\left (b \log \relax (c) + a\right )}}{b n}\right )}}{b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 261, normalized size = 3.43 \[ -\frac {b n x^{4}}{b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}} + \frac {4 \, b n {\rm Ei}\left (\frac {4 \, \log \relax (c)}{n} + \frac {4 \, a}{b n} + 4 \, \log \relax (x)\right ) e^{\left (-\frac {4 \, a}{b n}\right )} \log \relax (x)}{{\left (b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} c^{\frac {4}{n}}} + \frac {4 \, b {\rm Ei}\left (\frac {4 \, \log \relax (c)}{n} + \frac {4 \, a}{b n} + 4 \, \log \relax (x)\right ) e^{\left (-\frac {4 \, a}{b n}\right )} \log \relax (c)}{{\left (b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} c^{\frac {4}{n}}} + \frac {4 \, a {\rm Ei}\left (\frac {4 \, \log \relax (c)}{n} + \frac {4 \, a}{b n} + 4 \, \log \relax (x)\right ) e^{\left (-\frac {4 \, a}{b n}\right )}}{{\left (b^{3} n^{3} \log \relax (x) + b^{3} n^{2} \log \relax (c) + a b^{2} n^{2}\right )} c^{\frac {4}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}\, dx \]
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 \[ -\frac {x^{4}}{b^{2} n \log \relax (c) + b^{2} n \log \left (x^{n}\right ) + a b n} + 4 \, \int \frac {x^{3}}{b^{2} n \log \relax (c) + b^{2} n \log \left (x^{n}\right ) + a b n}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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