Optimal. Leaf size=24 \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0342689, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2667, 32} \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2667
Rule 32
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{2 (a+a \sin (c+d x))^{5/2}}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.0412181, size = 24, normalized size = 1. \[ \frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 21, normalized size = 0.9 \begin{align*}{\frac{2}{5\,da} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.941551, size = 27, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{5 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57513, size = 104, normalized size = 4.33 \begin{align*} -\frac{2 \,{\left (a \cos \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right ) - 2 \, a\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 26.2348, size = 90, normalized size = 3.75 \begin{align*} \begin{cases} \frac{2 a \sqrt{a \sin{\left (c + d x \right )} + a} \sin ^{2}{\left (c + d x \right )}}{5 d} + \frac{4 a \sqrt{a \sin{\left (c + d x \right )} + a} \sin{\left (c + d x \right )}}{5 d} + \frac{2 a \sqrt{a \sin{\left (c + d x \right )} + a}}{5 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{\frac{3}{2}} \cos{\left (c \right )} & \text{otherwise} \end{cases} \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 (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]