Optimal. Leaf size=50 \[ -\frac {F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right )}{d n} \]
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Rubi [A] time = 0.01, 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 = {2208} \[ -\frac {F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-b \log (F) (c+d x)^n\right )}{d n} \]
Antiderivative was successfully verified.
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Rule 2208
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 \[ -\frac {F^a (c+d x) \left (-b \log (F) (c+d x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-b (c+d x)^n \log (F)\right )}{d n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{{\left (d x + c\right )}^{n} b + a}, x\right ) \]
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 \[ \int F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int F^{b \left (d x +c \right )^{n}+a}\, dx \]
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 \[ \int F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]
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 \[ \int F^{a+b\,{\left (c+d\,x\right )}^n} \,d x \]
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 \[ \begin {cases} F^{a + \frac {b}{c}} x & \text {for}\: d = 0 \wedge n = -1 \\F^{a + b c^{n}} x & \text {for}\: d = 0 \\\int F^{a + \frac {b}{c + d x}}\, dx & \text {for}\: n = -1 \\\frac {F^{a} F^{b \left (c + d x\right )^{n}} b c n \left (c + d x\right )^{n} \log {\relax (F )}}{d n + d} + \frac {2 F^{a} F^{b \left (c + d x\right )^{n}} b c \left (c + d x\right )^{n} \log {\relax (F )}}{d n + d} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} b d n x \left (c + d x\right )^{n} \log {\relax (F )}}{d n + d} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} c n}{d n + d} - \frac {F^{a} F^{b \left (c + d x\right )^{n}} c}{d n + d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} d n x}{d n + d} + \frac {F^{a} F^{b \left (c + d x\right )^{n}} d x}{d n + d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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