Optimal. Leaf size=148 \[ -\frac{b e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )}{2 f}-\frac{b e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.148998, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3317, 3308, 2181} \[ -\frac{b e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )}{2 f}-\frac{b e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3317
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int (c+d x)^m (a+b \sin (e+f x)) \, dx &=\int \left (a (c+d x)^m+b (c+d x)^m \sin (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}+b \int (c+d x)^m \sin (e+f x) \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}+\frac{1}{2} (i b) \int e^{-i (e+f x)} (c+d x)^m \, dx-\frac{1}{2} (i b) \int e^{i (e+f x)} (c+d x)^m \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}-\frac{b e^{i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (-\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{i f (c+d x)}{d}\right )}{2 f}-\frac{b e^{-i \left (e-\frac{c f}{d}\right )} (c+d x)^m \left (\frac{i f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{i f (c+d x)}{d}\right )}{2 f}\\ \end{align*}
Mathematica [A] time = 0.187802, size = 138, normalized size = 0.93 \[ \frac{1}{2} (c+d x)^m \left (-\frac{b e^{i \left (e-\frac{c f}{d}\right )} \left (-\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i f (c+d x)}{d}\right )}{f}-\frac{b e^{-i \left (e-\frac{c f}{d}\right )} \left (\frac{i f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i f (c+d x)}{d}\right )}{f}+\frac{2 a (c+d x)}{d (m+1)}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.073, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+b\sin \left ( fx+e \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.82744, size = 319, normalized size = 2.16 \begin{align*} -\frac{{\left (b d m + b d\right )} e^{\left (-\frac{d m \log \left (\frac{i \, f}{d}\right ) + i \, d e - i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{i \, d f x + i \, c f}{d}\right ) +{\left (b d m + b d\right )} e^{\left (-\frac{d m \log \left (-\frac{i \, f}{d}\right ) - i \, d e + i \, c f}{d}\right )} \Gamma \left (m + 1, \frac{-i \, d f x - i \, c f}{d}\right ) - 2 \,{\left (a d f x + a c f\right )}{\left (d x + c\right )}^{m}}{2 \,{\left (d f m + d f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (e + f x \right )}\right ) \left (c + d x\right )^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right ) + a\right )}{\left (d x + c\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]