Integrand size = 29, antiderivative size = 41 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx=\frac {\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{1+p}}{b d e n (1+p) \log (F)} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2278, 32} \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx=\frac {\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{p+1}}{b d e n (p+1) \log (F)} \]
[In]
[Out]
Rule 32
Rule 2278
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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*}
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx=\frac {\left (a+b \left (F^{e (c+d x)}\right )^n\right )^{1+p}}{b d e n (1+p) \log (F)} \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {{\left (a +b \left (F^{e \left (d x +c \right )}\right )^{n}\right )}^{p +1}}{b d e n \left (p +1\right ) \ln \left (F \right )}\) | \(42\) |
default | \(\frac {{\left (a +b \left (F^{e \left (d x +c \right )}\right )^{n}\right )}^{p +1}}{b d e n \left (p +1\right ) \ln \left (F \right )}\) | \(42\) |
risch | \(\frac {\left (a +b \left (F^{e \left (d x +c \right )}\right )^{n}\right ) {\left (a +b \left (F^{e \left (d x +c \right )}\right )^{n}\right )}^{p}}{b \left (p +1\right ) \ln \left (F \right ) e d n}\) | \(55\) |
parallelrisch | \(\frac {\left (F^{e \left (d x +c \right )}\right )^{n} {\left (a +b \left (F^{e \left (d x +c \right )}\right )^{n}\right )}^{p} b +{\left (a +b \left (F^{e \left (d x +c \right )}\right )^{n}\right )}^{p} a}{b \left (p +1\right ) \ln \left (F \right ) e d n}\) | \(73\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.29 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx=\frac {{\left (F^{d e n x + c e n} b + a\right )} {\left (F^{d e n x + c e n} b + a\right )}^{p}}{{\left (b d e n p + b d e n\right )} \log \left (F\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (29) = 58\).
Time = 12.54 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.20 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx=\begin {cases} x \left (a + b \left (F^{c e}\right )^{n}\right )^{p} \left (F^{c e}\right )^{n} & \text {for}\: d = 0 \\x \left (a + b\right )^{p} & \text {for}\: e = 0 \vee n = 0 \vee \log {\left (F \right )} = 0 \\\frac {\begin {cases} a^{p} \left (F^{e \left (c + d x\right )}\right )^{n} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b \left (F^{e \left (c + d x\right )}\right )^{n}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b \left (F^{e \left (c + d x\right )}\right )^{n} \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}}{d e n \log {\left (F \right )}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx=\frac {{\left (F^{{\left (d x + c\right )} e n} b + a\right )}^{p + 1}}{b d e n {\left (p + 1\right )} \log \left (F\right )} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.02 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx=\frac {{\left (F^{d e n x + c e n} b + a\right )}^{p + 1}}{b d e n {\left (p + 1\right )} \log \left (F\right )} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.80 \[ \int \left (F^{e (c+d x)}\right )^n \left (a+b \left (F^{e (c+d x)}\right )^n\right )^p \, dx=\left (\frac {{\left (F^{c\,e+d\,e\,x}\right )}^n}{d\,e\,n\,\ln \left (F\right )\,\left (p+1\right )}+\frac {a}{b\,d\,e\,n\,\ln \left (F\right )\,\left (p+1\right )}\right )\,{\left (a+b\,{\left (F^{c\,e+d\,e\,x}\right )}^n\right )}^p \]
[In]
[Out]