Optimal. Leaf size=45 \[ \frac {x^n f^{a+b x^n}}{b n \log (f)}-\frac {f^{a+b x^n}}{b^2 n \log ^2(f)} \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2213, 2209} \[ \frac {x^n f^{a+b x^n}}{b n \log (f)}-\frac {f^{a+b x^n}}{b^2 n \log ^2(f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2213
Rubi steps
\begin {align*} \int f^{a+b x^n} x^{-1+2 n} \, dx &=\frac {f^{a+b x^n} x^n}{b n \log (f)}-\frac {\int f^{a+b x^n} x^{-1+n} \, dx}{b \log (f)}\\ &=-\frac {f^{a+b x^n}}{b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^n}{b n \log (f)}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 25, normalized size = 0.56 \[ -\frac {f^a \Gamma \left (2,-b x^n \log (f)\right )}{b^2 n \log ^2(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 33, normalized size = 0.73 \[ \frac {{\left (b x^{n} \log \relax (f) - 1\right )} e^{\left (b x^{n} \log \relax (f) + a \log \relax (f)\right )}}{b^{2} n \log \relax (f)^{2}} \]
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 = 0.05, size = 56, normalized size = 1.24 \[ \frac {{\mathrm e}^{n \ln \relax (x )} {\mathrm e}^{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right ) \ln \relax (f )}}{b n \ln \relax (f )}-\frac {{\mathrm e}^{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right ) \ln \relax (f )}}{b^{2} n \ln \relax (f )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 34, normalized size = 0.76 \[ \frac {{\left (b f^{a} x^{n} \log \relax (f) - f^{a}\right )} f^{b x^{n}}}{b^{2} n \log \relax (f)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int f^{a+b\,x^n}\,x^{2\,n-1} \,d x \]
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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