Optimal. Leaf size=71 \[ \frac {2 f^{a+b x^n}}{b^3 n \log ^3(f)}-\frac {2 x^n f^{a+b x^n}}{b^2 n \log ^2(f)}+\frac {x^{2 n} f^{a+b x^n}}{b n \log (f)} \]
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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 = {2213, 2209} \[ -\frac {2 x^n f^{a+b x^n}}{b^2 n \log ^2(f)}+\frac {2 f^{a+b x^n}}{b^3 n \log ^3(f)}+\frac {x^{2 n} f^{a+b x^n}}{b n \log (f)} \]
Antiderivative was successfully verified.
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Rule 2209
Rule 2213
Rubi steps
\begin {align*} \int f^{a+b x^n} x^{-1+3 n} \, dx &=\frac {f^{a+b x^n} x^{2 n}}{b n \log (f)}-\frac {2 \int f^{a+b x^n} x^{-1+2 n} \, dx}{b \log (f)}\\ &=-\frac {2 f^{a+b x^n} x^n}{b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^{2 n}}{b n \log (f)}+\frac {2 \int f^{a+b x^n} x^{-1+n} \, dx}{b^2 \log ^2(f)}\\ &=\frac {2 f^{a+b x^n}}{b^3 n \log ^3(f)}-\frac {2 f^{a+b x^n} x^n}{b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^{2 n}}{b n \log (f)}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 24, normalized size = 0.34 \[ \frac {f^a \Gamma \left (3,-b x^n \log (f)\right )}{b^3 n \log ^3(f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 47, normalized size = 0.66 \[ \frac {{\left (b^{2} x^{2 \, n} \log \relax (f)^{2} - 2 \, b x^{n} \log \relax (f) + 2\right )} e^{\left (b x^{n} \log \relax (f) + a \log \relax (f)\right )}}{b^{3} n \log \relax (f)^{3}} \]
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 \[ \int f^{b x^{n} + a} x^{3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 44, normalized size = 0.62 \[ \frac {\left (b^{2} x^{2 n} \ln \relax (f )^{2}-2 b \,x^{n} \ln \relax (f )+2\right ) f^{b \,x^{n}+a}}{b^{3} n \ln \relax (f )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 51, normalized size = 0.72 \[ \frac {{\left (b^{2} f^{a} x^{2 \, n} \log \relax (f)^{2} - 2 \, b f^{a} x^{n} \log \relax (f) + 2 \, f^{a}\right )} f^{b x^{n}}}{b^{3} n \log \relax (f)^{3}} \]
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 \[ \int f^{a+b\,x^n}\,x^{3\,n-1} \,d x \]
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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