Optimal. Leaf size=38 \[ -\frac {f^{a+b x^n} x^{-n}}{n}+\frac {b f^a \text {Ei}\left (b x^n \log (f)\right ) \log (f)}{n} \]
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Rubi [A]
time = 0.03, antiderivative size = 38, 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 = {2246, 2241}
\begin {gather*} \frac {b f^a \log (f) \text {Ei}\left (b x^n \log (f)\right )}{n}-\frac {x^{-n} f^{a+b x^n}}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 2246
Rubi steps
\begin {align*} \int f^{a+b x^n} x^{-1-n} \, dx &=-\frac {f^{a+b x^n} x^{-n}}{n}+(b \log (f)) \int \frac {f^{a+b x^n}}{x} \, dx\\ &=-\frac {f^{a+b x^n} x^{-n}}{n}+\frac {b f^a \text {Ei}\left (b x^n \log (f)\right ) \log (f)}{n}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 0.53 \begin {gather*} \frac {b f^a \Gamma \left (-1,-b x^n \log (f)\right ) \log (f)}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 43, normalized size = 1.13
method | result | size |
risch | \(-\frac {f^{b \,x^{n}} f^{a} x^{-n}}{n}-\frac {\ln \left (f \right ) b \,f^{a} \expIntegral \left (1, -b \,x^{n} \ln \left (f \right )\right )}{n}\) | \(43\) |
meijerg | \(-\frac {f^{a} \ln \left (f \right ) b \left (\frac {\left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} x^{-n} \left (2+2 b \,x^{n} \ln \left (f \right )\right )}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right ) b \ln \left (f \right )}-\frac {2 \left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} x^{-n} {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right ) b \ln \left (f \right )}+\frac {2 \left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} \left (-\gamma -\ln \left (-b \,x^{n} \ln \left (f \right )\right )-\expIntegral \left (1, -b \,x^{n} \ln \left (f \right )\right )\right )}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right )}+\frac {\left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} \left (-\Psi \left (1-\frac {-1-n}{n}-\frac {1}{n}\right )+n \ln \left (x \right )+\ln \left (-b \right )+\ln \left (\ln \left (f \right )\right )\right )}{\Gamma \left (1-\frac {-1-n}{n}-\frac {1}{n}\right )}-x^{\left (\frac {-1-n}{n}+\frac {1}{n}\right ) n} \left (-b \right )^{\frac {-1-n}{n}+\frac {1}{n}} \ln \left (f \right )^{\frac {-1-n}{n}+\frac {1}{n}}\right )}{n}\) | \(324\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 20, normalized size = 0.53 \begin {gather*} \frac {b f^{a} \Gamma \left (-1, -b x^{n} \log \left (f\right )\right ) \log \left (f\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 43, normalized size = 1.13 \begin {gather*} \frac {b f^{a} x^{n} {\rm Ei}\left (b x^{n} \log \left (f\right )\right ) \log \left (f\right ) - e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{n x^{n}} \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} \int f^{a + \frac {b}{x}}\, dx & \text {for}\: n = -1 \\f^{a + b} \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {b f^{a} f^{b x^{n}} n^{2} x^{n} \log {\left (f \right )} \log {\left (x \right )}}{n^{2} x^{n} + n x^{n}} - \frac {b f^{a} f^{b x^{n}} n^{2} x^{n} \log {\left (f \right )}}{n^{2} x^{n} + n x^{n}} + \frac {b f^{a} f^{b x^{n}} n x^{n} \log {\left (f \right )} \log {\left (x \right )}}{n^{2} x^{n} + n x^{n}} - \frac {f^{a} f^{b x^{n}} n}{n^{2} x^{n} + n x^{n}} - \frac {f^{a} f^{b x^{n}}}{n^{2} x^{n} + n x^{n}} & \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 [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {f^{a+b\,x^n}}{x^{n+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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