Optimal. Leaf size=31 \[ \frac {\left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2246, 32} \[ \frac {\left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2246
Rubi steps
\begin {align*} \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^n \, dx,x,F^{c+d x}\right )}{d \log (F)}\\ &=\frac {\left (a+b F^{c+d x}\right )^{1+n}}{b d (1+n) \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 30, normalized size = 0.97 \[ \frac {\left (a+b F^{c+d x}\right )^{n+1}}{b d n \log (F)+b d \log (F)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 39, normalized size = 1.26 \[ \frac {{\left (F^{d x + c} b + a\right )} {\left (F^{d x + c} b + a\right )}^{n}}{{\left (b d n + b d\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 31, normalized size = 1.00 \[ \frac {{\left (F^{d x + c} b + a\right )}^{n + 1}}{b d {\left (n + 1\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 1.03 \[ \frac {\left (b \,F^{d x +c}+a \right )^{n +1}}{\left (n +1\right ) b d \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 31, normalized size = 1.00 \[ \frac {{\left (F^{d x + c} b + a\right )}^{n + 1}}{b d {\left (n + 1\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.43, size = 52, normalized size = 1.68 \[ {\left (a+F^{c+d\,x}\,b\right )}^n\,\left (\frac {F^{c+d\,x}}{d\,\ln \relax (F)\,\left (n+1\right )}+\frac {a}{b\,d\,\ln \relax (F)\,\left (n+1\right )}\right ) \]
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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