Optimal. Leaf size=36 \[ \frac{F^{-c} \left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]
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Rubi [A] time = 0.0734713, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2247, 2246, 32} \[ \frac{F^{-c} \left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]
Antiderivative was successfully verified.
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Rule 2247
Rule 2246
Rule 32
Rubi steps
\begin{align*} \int F^{d x} \left (a+b F^{c+d x}\right )^n \, dx &=F^{-c} \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx\\ &=\frac{F^{-c} \operatorname{Subst}\left (\int (a+b x)^n \, dx,x,F^{c+d x}\right )}{d \log (F)}\\ &=\frac{F^{-c} \left (a+b F^{c+d x}\right )^{1+n}}{b d (1+n) \log (F)}\\ \end{align*}
Mathematica [A] time = 0.0428595, size = 35, normalized size = 0.97 \[ \frac{F^{-c} \left (a+b F^{c+d x}\right )^{n+1}}{b d n \log (F)+b d \log (F)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 81, normalized size = 2.3 \begin{align*}{\frac{{{\rm e}^{d\ln \left ( F \right ) x}}{{\rm e}^{n\ln \left ( a+b{{\rm e}^{c\ln \left ( F \right ) }}{{\rm e}^{d\ln \left ( F \right ) x}} \right ) }}}{d\ln \left ( F \right ) \left ( 1+n \right ) }}+{\frac{a{{\rm e}^{n\ln \left ( a+b{{\rm e}^{c\ln \left ( F \right ) }}{{\rm e}^{d\ln \left ( F \right ) x}} \right ) }}}{b{F}^{c}d\ln \left ( F \right ) \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59747, size = 100, normalized size = 2.78 \begin{align*} \frac{{\left (F^{d x + c} b + a\right )}^{n}{\left (\frac{F^{d x + c} b}{F^{c}} + \frac{a}{F^{c}}\right )}}{{\left (b d n + b d\right )} \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26914, size = 82, normalized size = 2.28 \begin{align*} \frac{{\left (F^{d x} F^{c} b + a\right )}^{n} F^{d x} F^{c} b +{\left (F^{d x} F^{c} b + a\right )}^{n} a}{{\left (b n + b\right )} F^{c} d \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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