Optimal. Leaf size=46 \[ -\frac {f^a x^{1+m} \Gamma \left (\frac {1+m}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-\frac {1+m}{n}}}{n} \]
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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2250}
\begin {gather*} -\frac {f^a x^{m+1} \left (-b \log (f) x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-b \log (f) x^n\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rubi steps
\begin {align*} \int f^{a+b x^n} x^m \, dx &=-\frac {f^a x^{1+m} \Gamma \left (\frac {1+m}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-\frac {1+m}{n}}}{n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 46, normalized size = 1.00 \begin {gather*} -\frac {f^a x^{1+m} \Gamma \left (\frac {1+m}{n},-b x^n \log (f)\right ) \left (-b x^n \log (f)\right )^{-\frac {1+m}{n}}}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.04, size = 280, normalized size = 6.09
method | result | size |
meijerg | \(\frac {f^{a} \left (-b \right )^{-\frac {m}{n}-\frac {1}{n}} \ln \left (f \right )^{-\frac {m}{n}-\frac {1}{n}} \left (\frac {n \,x^{1+m} \left (-b \right )^{\frac {m}{n}+\frac {1}{n}} \ln \left (f \right )^{\frac {m}{n}+\frac {1}{n}} \left (\ln \left (f \right ) x^{n} b n +m +n +1\right ) L_{-\frac {1+m}{n}}^{\left (\frac {1+m +n}{n}\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (-\frac {1+m}{n}+1\right ) \Gamma \left (\frac {1+m +n}{n}+1\right )}{\left (1+m \right ) \left (1+m +n \right ) \Gamma \left (-\frac {1+m}{n}+\frac {1+m +n}{n}+1\right )}-\frac {n^{2} x^{1+m +n} \left (-b \right )^{\frac {m}{n}+\frac {1}{n}} \ln \left (f \right )^{1+\frac {m}{n}+\frac {1}{n}} b L_{-\frac {1+m}{n}}^{\left (\frac {1+m +n}{n}+1\right )}\left (b \,x^{n} \ln \left (f \right )\right ) \Gamma \left (-\frac {1+m}{n}+1\right ) \Gamma \left (\frac {1+m +n}{n}+1\right )}{\left (1+m \right ) \left (1+m +n \right ) \Gamma \left (-\frac {1+m}{n}+\frac {1+m +n}{n}+1\right )}\right )}{n}\) | \(280\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.07, size = 47, normalized size = 1.02 \begin {gather*} -\frac {f^{a} x^{m + 1} \Gamma \left (\frac {m + 1}{n}, -b x^{n} \log \left (f\right )\right )}{\left (-b x^{n} \log \left (f\right )\right )^{\frac {m + 1}{n}} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \begin {cases} f^{a + b} \log {\left (x \right )} & \text {for}\: m = -1 \wedge n = 0 \\\int \frac {f^{a + b x^{n}}}{x}\, dx & \text {for}\: m = -1 \\\int f^{a + b x^{- m - 1}} x^{m}\, dx & \text {for}\: n = - m - 1 \\- \frac {b f^{a} f^{b x^{n}} n x x^{m} x^{n} \log {\left (f \right )}}{m^{2} + m n + 2 m + n + 1} + \frac {f^{a} f^{b x^{n}} m x x^{m}}{m^{2} + m n + 2 m + n + 1} + \frac {f^{a} f^{b x^{n}} n x x^{m}}{m^{2} + m n + 2 m + n + 1} + \frac {f^{a} f^{b x^{n}} x x^{m}}{m^{2} + m n + 2 m + n + 1} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.76, size = 79, normalized size = 1.72 \begin {gather*} \frac {f^a\,f^{b\,x^n}\,x^{m+1}\,{\mathrm {e}}^{-\frac {b\,x^n\,\ln \left (f\right )}{2}}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left (b\,x^n\,\ln \left (f\right )\right )}{\left (m+1\right )\,{\left (b\,x^n\,\ln \left (f\right )\right )}^{\frac {m+n+1}{2\,n}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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