Integrand size = 21, antiderivative size = 31 \[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx=\frac {\left (a+b F^{c+d x}\right )^{1+n}}{b d (1+n) \log (F)} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2278, 32} \[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx=\frac {\left (a+b F^{c+d x}\right )^{n+1}}{b d (n+1) \log (F)} \]
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Rule 32
Rule 2278
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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*}
Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx=\frac {\left (a+b F^{c+d x}\right )^{1+n}}{b d \log (F)+b d n \log (F)} \]
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Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {\left (a +b \,F^{d x +c}\right )^{1+n}}{b d \left (1+n \right ) \ln \left (F \right )}\) | \(32\) |
default | \(\frac {\left (a +b \,F^{d x +c}\right )^{1+n}}{b d \left (1+n \right ) \ln \left (F \right )}\) | \(32\) |
risch | \(\frac {\left (a +b \,F^{d x +c}\right ) \left (a +b \,F^{d x +c}\right )^{n}}{b d \ln \left (F \right ) \left (1+n \right )}\) | \(41\) |
parallelrisch | \(\frac {F^{d x +c} \left (a +b \,F^{d x +c}\right )^{n} b +\left (a +b \,F^{d x +c}\right )^{n} a}{b d \ln \left (F \right ) \left (1+n \right )}\) | \(55\) |
norman | \(\frac {{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )} {\mathrm e}^{n \ln \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )}}{\ln \left (F \right ) d \left (1+n \right )}+\frac {a \,{\mathrm e}^{n \ln \left (a +b \,{\mathrm e}^{\left (d x +c \right ) \ln \left (F \right )}\right )}}{\ln \left (F \right ) d b \left (1+n \right )}\) | \(75\) |
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Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx=\frac {{\left (F^{d x + c} b + a\right )} {\left (F^{d x + c} b + a\right )}^{n}}{{\left (b d n + b d\right )} \log \left (F\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (22) = 44\).
Time = 1.09 (sec) , antiderivative size = 299, normalized size of antiderivative = 9.65 \[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: F = 1 \wedge a = 0 \wedge b = 0 \wedge d = 0 \wedge n = -1 \\x \left (a + b\right )^{n} & \text {for}\: F = 1 \\\frac {0^{n} F^{c + d x}}{d \log {\left (F \right )}} & \text {for}\: a = - F^{c + d x} b \\\frac {F^{c + d x} a^{n}}{d \log {\left (F \right )}} & \text {for}\: b = 0 \\F^{c} x \left (F^{c} b + a\right )^{n} & \text {for}\: d = 0 \\\frac {\log {\left (F^{c + d x} + \frac {a}{b} \right )}}{b d \log {\left (F \right )}} & \text {for}\: n = -1 \\\frac {2 F^{c + d x} a b \left (F^{c + d x} b + a\right )^{n}}{F^{c + d x} b^{2} d n \log {\left (F \right )} + F^{c + d x} b^{2} d \log {\left (F \right )} + a b d n \log {\left (F \right )} + a b d \log {\left (F \right )}} + \frac {F^{2 c + 2 d x} b^{2} \left (F^{c + d x} b + a\right )^{n}}{F^{c + d x} b^{2} d n \log {\left (F \right )} + F^{c + d x} b^{2} d \log {\left (F \right )} + a b d n \log {\left (F \right )} + a b d \log {\left (F \right )}} + \frac {a^{2} \left (F^{c + d x} b + a\right )^{n}}{F^{c + d x} b^{2} d n \log {\left (F \right )} + F^{c + d x} b^{2} d \log {\left (F \right )} + a b d n \log {\left (F \right )} + a b d \log {\left (F \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx=\frac {{\left (F^{d x + c} b + a\right )}^{n + 1}}{b d {\left (n + 1\right )} \log \left (F\right )} \]
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Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx=\frac {{\left (F^{d x + c} b + a\right )}^{n + 1}}{b d {\left (n + 1\right )} \log \left (F\right )} \]
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Time = 0.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int F^{c+d x} \left (a+b F^{c+d x}\right )^n \, dx={\left (a+F^{c+d\,x}\,b\right )}^n\,\left (\frac {F^{c+d\,x}}{d\,\ln \left (F\right )\,\left (n+1\right )}+\frac {a}{b\,d\,\ln \left (F\right )\,\left (n+1\right )}\right ) \]
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