Optimal. Leaf size=41 \[ \frac {\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]
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Rubi [A] time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2246, 32} \[ \frac {\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2246
Rubi steps
\begin {align*} \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^p \, dx,x,\left (F^{e (c+d x)}\right )^n\right )}{d e n \log (F)}\\ &=\frac {\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{1+p}}{b d e n (1+p) \log (F)}\\ \end {align*}
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Mathematica [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.43, size = 53, normalized size = 1.29 \[ \frac {{\left (F^{d e n x + c e n} b + a\right )} {\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 43, normalized size = 1.05 \[ \frac {{\left (F^{d n x e + c n e} b + a\right )}^{p + 1} e^{\left (-1\right )}}{b d n {\left (p + 1\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 1.02 \[ \frac {\left (b \left (F^{\left (d x +c \right ) e}\right )^{n}+a \right )^{p +1}}{\left (p +1\right ) b d e n \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 41, normalized size = 1.00 \[ \frac {{\left ({\left (F^{{\left (d x + c\right )} e}\right )}^{n} b + a\right )}^{p + 1}}{b d e n {\left (p + 1\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 74, normalized size = 1.80 \[ \left (\frac {{\left (F^{c\,e+d\,e\,x}\right )}^n}{d\,e\,n\,\ln \relax (F)\,\left (p+1\right )}+\frac {a}{b\,d\,e\,n\,\ln \relax (F)\,\left (p+1\right )}\right )\,{\left (a+b\,{\left (F^{c\,e+d\,e\,x}\right )}^n\right )}^p \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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