Optimal. Leaf size=20 \[ \frac {f^{a+b x^n}}{b n \log (f)} \]
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Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2240}
\begin {gather*} \frac {f^{a+b x^n}}{b n \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rubi steps
\begin {align*} \int f^{a+b x^n} x^{-1+n} \, dx &=\frac {f^{a+b x^n}}{b n \log (f)}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {f^{a+b x^n}}{b n \log (f)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 21, normalized size = 1.05
method | result | size |
risch | \(\frac {f^{a +b \,x^{n}}}{b n \ln \left (f \right )}\) | \(21\) |
norman | \(\frac {{\mathrm e}^{\left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right ) \ln \left (f \right )}}{\ln \left (f \right ) b n}\) | \(25\) |
meijerg | \(-\frac {f^{a} \left (-\frac {\left (-1\right )^{-\frac {-1+n}{n}-\frac {1}{n}}}{\Gamma \left (2-\frac {-1+n}{n}-\frac {1}{n}\right )}+\frac {\left (-1\right )^{-\frac {-1+n}{n}-\frac {1}{n}} {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{\Gamma \left (2-\frac {-1+n}{n}-\frac {1}{n}\right )}\right )}{\ln \left (f \right ) b n}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 20, normalized size = 1.00 \begin {gather*} \frac {f^{b x^{n} + a}}{b n \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 24, normalized size = 1.20 \begin {gather*} \frac {e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{b n \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (14) = 28\).
time = 19.91, size = 39, normalized size = 1.95 \begin {gather*} \begin {cases} \log {\left (x \right )} & \text {for}\: b = 0 \wedge f = 1 \wedge n = 0 \\f^{a + b} \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {x^{n}}{n} & \text {for}\: f = 1 \\\frac {f^{a} x^{n}}{n} & \text {for}\: b = 0 \\\frac {f^{a} f^{b x^{n}}}{b n \log {\left (f \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.63, size = 20, normalized size = 1.00 \begin {gather*} \frac {f^{b x^{n} + a}}{b n \log \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.50, size = 20, normalized size = 1.00 \begin {gather*} \frac {f^{a+b\,x^n}}{b\,n\,\ln \left (f\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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