Optimal. Leaf size=45 \[ -\frac {f^{a+b x^n}}{b^2 n \log ^2(f)}+\frac {f^{a+b x^n} x^n}{b n \log (f)} \]
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Rubi [A]
time = 0.03, 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 = {2244, 2240}
\begin {gather*} \frac {x^n f^{a+b x^n}}{b n \log (f)}-\frac {f^{a+b x^n}}{b^2 n \log ^2(f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2244
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] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.00, size = 25, normalized size = 0.56 \begin {gather*} -\frac {f^a \Gamma \left (2,-b x^n \log (f)\right )}{b^2 n \log ^2(f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 30, normalized size = 0.67
method | result | size |
risch | \(\frac {\left (b \,x^{n} \ln \left (f \right )-1\right ) f^{a +b \,x^{n}}}{\ln \left (f \right )^{2} b^{2} n}\) | \(30\) |
meijerg | \(\frac {f^{a} \left (1-\frac {\left (2-2 b \,x^{n} \ln \left (f \right )\right ) {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{2}\right )}{\ln \left (f \right )^{2} b^{2} n}\) | \(37\) |
norman | \(\frac {{\mathrm e}^{n \ln \left (x \right )} {\mathrm e}^{\left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right ) \ln \left (f \right )}}{\ln \left (f \right ) b n}-\frac {{\mathrm e}^{\left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right ) \ln \left (f \right )}}{\ln \left (f \right )^{2} b^{2} n}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 34, normalized size = 0.76 \begin {gather*} \frac {{\left (b f^{a} x^{n} \log \left (f\right ) - f^{a}\right )} f^{b x^{n}}}{b^{2} n \log \left (f\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 33, normalized size = 0.73 \begin {gather*} \frac {{\left (b x^{n} \log \left (f\right ) - 1\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{b^{2} n \log \left (f\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 104.09, size = 49, normalized size = 1.09 \begin {gather*} \begin {cases} - \frac {b f^{a} f^{b x^{n}} x^{3 n} \log {\left (f \right )}}{6 n} + \frac {f^{a} f^{b x^{n}} x^{2 n}}{2 n} & \text {for}\: n \neq 0 \\f^{a + b} \log {\left (x \right )} & \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 = -0.02 \begin {gather*} \int f^{a+b\,x^n}\,x^{2\,n-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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