Optimal. Leaf size=50 \[ -\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n} \]
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Rubi [A]
time = 0.00, antiderivative size = 50, 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 = {2239}
\begin {gather*} -\frac {F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-b \log (F) (c+d x)^n\right )}{d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^n} \, dx &=-\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} -\frac {F^a (c+d x) \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-1/n}}{d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{a +b \left (d x +c \right )^{n}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 + \frac {b}{c}} x & \text {for}\: d = 0 \wedge n = -1 \\F^{a + b c^{n}} x & \text {for}\: d = 0 \\\int F^{a + \frac {b}{c + d x}}\, dx & \text {for}\: n = -1 \\\frac {F^{a} F^{b \left (c + d x\right )^{n}} b c n \left (c + d x\right )^{n} \log {\left (F \right )}}{d n + d} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} b d n x \left (c + d x\right )^{n} \log {\left (F \right )}}{d n + d} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} c n}{d n + d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} c}{d n + d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} d n x}{d n + d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} d x}{d n + d} & \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\,{\left (c+d\,x\right )}^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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