Optimal. Leaf size=71 \[ \frac {b^2 f^a \log ^2(f) \text {Ei}\left (b x^n \log (f)\right )}{2 n}-\frac {x^{-2 n} f^{a+b x^n}}{2 n}-\frac {b \log (f) x^{-n} f^{a+b x^n}}{2 n} \]
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Rubi [A] time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2215, 2210} \[ \frac {b^2 f^a \log ^2(f) \text {Ei}\left (b x^n \log (f)\right )}{2 n}-\frac {x^{-2 n} f^{a+b x^n}}{2 n}-\frac {b \log (f) x^{-n} f^{a+b x^n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2215
Rubi steps
\begin {align*} \int f^{a+b x^n} x^{-1-2 n} \, dx &=-\frac {f^{a+b x^n} x^{-2 n}}{2 n}+\frac {1}{2} (b \log (f)) \int f^{a+b x^n} x^{-1-n} \, dx\\ &=-\frac {f^{a+b x^n} x^{-2 n}}{2 n}-\frac {b f^{a+b x^n} x^{-n} \log (f)}{2 n}+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+b x^n}}{x} \, dx\\ &=-\frac {f^{a+b x^n} x^{-2 n}}{2 n}-\frac {b f^{a+b x^n} x^{-n} \log (f)}{2 n}+\frac {b^2 f^a \text {Ei}\left (b x^n \log (f)\right ) \log ^2(f)}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 25, normalized size = 0.35 \[ -\frac {b^2 f^a \log ^2(f) \Gamma \left (-2,-b x^n \log (f)\right )}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 61, normalized size = 0.86 \[ \frac {b^{2} f^{a} x^{2 \, n} {\rm Ei}\left (b x^{n} \log \relax (f)\right ) \log \relax (f)^{2} - {\left (b x^{n} \log \relax (f) + 1\right )} e^{\left (b x^{n} \log \relax (f) + a \log \relax (f)\right )}}{2 \, n x^{2 \, n}} \]
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 \[ \int f^{b x^{n} + a} x^{-2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 70, normalized size = 0.99 \[ -\frac {b^{2} f^{a} \Ei \left (1, -b \,x^{n} \ln \relax (f )\right ) \ln \relax (f )^{2}}{2 n}-\frac {b \,f^{a} f^{b \,x^{n}} x^{-n} \ln \relax (f )}{2 n}-\frac {f^{a} f^{b \,x^{n}} x^{-2 n}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 25, normalized size = 0.35 \[ -\frac {b^{2} f^{a} \Gamma \left (-2, -b x^{n} \log \relax (f)\right ) \log \relax (f)^{2}}{n} \]
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 \[ \int \frac {f^{a+b\,x^n}}{x^{2\,n+1}} \,d x \]
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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