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 = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 32
Rule 2246
Rule 2247
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.43, size = 50, normalized size = 1.39 \[ \frac {{\left (F^{d x + c} b + a\right )}^{n} {\left (\frac {F^{d x + c} b}{F^{c}} + \frac {a}{F^{c}}\right )}}{{\left (b d n + b d\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 36, normalized size = 1.00 \[ \frac {{\left (F^{d x + c} b + a\right )}^{n + 1}}{F^{c} b d {\left (n + 1\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 81, normalized size = 2.25 \[ \frac {a \,F^{-c} {\mathrm e}^{n \ln \left (b \,{\mathrm e}^{c \ln \relax (F )} {\mathrm e}^{d x \ln \relax (F )}+a \right )}}{\left (n +1\right ) b d \ln \relax (F )}+\frac {{\mathrm e}^{n \ln \left (b \,{\mathrm e}^{c \ln \relax (F )} {\mathrm e}^{d x \ln \relax (F )}+a \right )} {\mathrm e}^{d x \ln \relax (F )}}{\left (n +1\right ) d \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 36, normalized size = 1.00 \[ \frac {{\left (F^{d x + c} b + a\right )}^{n + 1}}{F^{c} b d {\left (n + 1\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 55, normalized size = 1.53 \[ {\left (a+F^{c+d\,x}\,b\right )}^n\,\left (\frac {F^{d\,x}}{d\,\ln \relax (F)\,\left (n+1\right )}+\frac {a}{F^c\,b\,d\,\ln \relax (F)\,\left (n+1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 145.67, size = 141, normalized size = 3.92 \[ \begin {cases} \frac {x}{a} & \text {for}\: F = 1 \wedge b = 0 \wedge d = 0 \wedge n = -1 \\x \left (a + b\right )^{n} & \text {for}\: F = 1 \\\frac {F^{d x} a^{n}}{d \log {\relax (F )}} & \text {for}\: b = 0 \\x \left (F^{c} b + a\right )^{n} & \text {for}\: d = 0 \\\frac {F^{- c} \log {\left (F^{c} F^{d x} + \frac {a}{b} \right )}}{b d \log {\relax (F )}} & \text {for}\: n = -1 \\\frac {F^{c} F^{d x} b \left (F^{c} F^{d x} b + a\right )^{n}}{F^{c} b d n \log {\relax (F )} + F^{c} b d \log {\relax (F )}} + \frac {a \left (F^{c} F^{d x} b + a\right )^{n}}{F^{c} b d n \log {\relax (F )} + F^{c} b d \log {\relax (F )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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