Integrand size = 21, antiderivative size = 77 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx=\frac {a (c+d x)^2}{2 d}-\frac {b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)}{f g n \log (F)} \]
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Time = 0.05 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2214, 2207, 2225} \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx=\frac {a (c+d x)^2}{2 d}+\frac {b (c+d x) \left (F^{e g+f g x}\right )^n}{f g n \log (F)}-\frac {b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)} \]
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Rule 2207
Rule 2214
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \int \left (a (c+d x)+b \left (F^{e g+f g x}\right )^n (c+d x)\right ) \, dx \\ & = \frac {a (c+d x)^2}{2 d}+b \int \left (F^{e g+f g x}\right )^n (c+d x) \, dx \\ & = \frac {a (c+d x)^2}{2 d}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)}{f g n \log (F)}-\frac {(b d) \int \left (F^{e g+f g x}\right )^n \, dx}{f g n \log (F)} \\ & = \frac {a (c+d x)^2}{2 d}-\frac {b d \left (F^{e g+f g x}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{e g+f g x}\right )^n (c+d x)}{f g n \log (F)} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.95 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx=\frac {1}{2} a x (2 c+d x)-\frac {b d \left (F^{g (e+f x)}\right )^n}{f^2 g^2 n^2 \log ^2(F)}+\frac {b \left (F^{g (e+f x)}\right )^n (c+d x)}{f g n \log (F)} \]
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Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.09
method | result | size |
norman | \(c a x +\frac {b \left (\ln \left (F \right ) c f g n -d \right ) {\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{n^{2} g^{2} f^{2} \ln \left (F \right )^{2}}+\frac {b d x \,{\mathrm e}^{n \ln \left ({\mathrm e}^{g \left (f x +e \right ) \ln \left (F \right )}\right )}}{n g f \ln \left (F \right )}+\frac {a d \,x^{2}}{2}\) | \(84\) |
parallelrisch | \(\frac {a d \,x^{2} n^{2} g^{2} f^{2} \ln \left (F \right )^{2}+2 c a x \,n^{2} g^{2} f^{2} \ln \left (F \right )^{2}+2 \left (F^{g \left (f x +e \right )}\right )^{n} \ln \left (F \right ) b d f g n x +2 \left (F^{g \left (f x +e \right )}\right )^{n} \ln \left (F \right ) b c f g n -2 \left (F^{g \left (f x +e \right )}\right )^{n} b d}{2 n^{2} g^{2} f^{2} \ln \left (F \right )^{2}}\) | \(110\) |
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Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx=\frac {{\left (a d f^{2} g^{2} n^{2} x^{2} + 2 \, a c f^{2} g^{2} n^{2} x\right )} \log \left (F\right )^{2} - 2 \, {\left (b d - {\left (b d f g n x + b c f g n\right )} \log \left (F\right )\right )} F^{f g n x + e g n}}{2 \, f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (68) = 136\).
Time = 0.42 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.87 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx=\begin {cases} \left (a + b\right ) \left (c x + \frac {d x^{2}}{2}\right ) & \text {for}\: F = 1 \wedge f = 0 \wedge g = 0 \wedge n = 0 \\\left (a + b \left (F^{e g}\right )^{n}\right ) \left (c x + \frac {d x^{2}}{2}\right ) & \text {for}\: f = 0 \\\left (a + b\right ) \left (c x + \frac {d x^{2}}{2}\right ) & \text {for}\: F = 1 \vee g = 0 \vee n = 0 \\a c x + \frac {a d x^{2}}{2} + \frac {b c \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} + \frac {b d x \left (F^{e g + f g x}\right )^{n}}{f g n \log {\left (F \right )}} - \frac {b d \left (F^{e g + f g x}\right )^{n}}{f^{2} g^{2} n^{2} \log {\left (F \right )}^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx=\frac {1}{2} \, a d x^{2} + a c x + \frac {F^{f g n x + e g n} b c}{f g n \log \left (F\right )} + \frac {{\left (F^{e g n} f g n x \log \left (F\right ) - F^{e g n}\right )} F^{f g n x} b d}{f^{2} g^{2} n^{2} \log \left (F\right )^{2}} \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 1101, normalized size of antiderivative = 14.30 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int \left (a+b \left (F^{g (e+f x)}\right )^n\right ) (c+d x) \, dx=a\,c\,x-\left (\frac {b\,\left (d-c\,f\,g\,n\,\ln \left (F\right )\right )}{f^2\,g^2\,n^2\,{\ln \left (F\right )}^2}-\frac {b\,d\,x}{f\,g\,n\,\ln \left (F\right )}\right )\,{\left (F^{f\,g\,x}\,F^{e\,g}\right )}^n+\frac {a\,d\,x^2}{2} \]
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