Optimal. Leaf size=63 \[ \frac {2}{5} a^2 c x^{5/2}+\frac {2}{13} b x^{13/2} (2 a d+b c)+\frac {2}{9} a x^{9/2} (a d+2 b c)+\frac {2}{17} b^2 d x^{17/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \begin {gather*} \frac {2}{5} a^2 c x^{5/2}+\frac {2}{13} b x^{13/2} (2 a d+b c)+\frac {2}{9} a x^{9/2} (a d+2 b c)+\frac {2}{17} b^2 d x^{17/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 448
Rubi steps
\begin {align*} \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c x^{3/2}+a (2 b c+a d) x^{7/2}+b (b c+2 a d) x^{11/2}+b^2 d x^{15/2}\right ) \, dx\\ &=\frac {2}{5} a^2 c x^{5/2}+\frac {2}{9} a (2 b c+a d) x^{9/2}+\frac {2}{13} b (b c+2 a d) x^{13/2}+\frac {2}{17} b^2 d x^{17/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 53, normalized size = 0.84 \begin {gather*} \frac {2 x^{5/2} \left (1989 a^2 c+765 b x^4 (2 a d+b c)+1105 a x^2 (a d+2 b c)+585 b^2 d x^6\right )}{9945} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (1989 a^2 c x^{5/2}+1105 a^2 d x^{9/2}+2210 a b c x^{9/2}+1530 a b d x^{13/2}+765 b^2 c x^{13/2}+585 b^2 d x^{17/2}\right )}{9945} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.16, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{9945} \, {\left (585 \, b^{2} d x^{8} + 765 \, {\left (b^{2} c + 2 \, a b d\right )} x^{6} + 1989 \, a^{2} c x^{2} + 1105 \, {\left (2 \, a b c + a^{2} d\right )} x^{4}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{17} \, b^{2} d x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} c x^{\frac {13}{2}} + \frac {4}{13} \, a b d x^{\frac {13}{2}} + \frac {4}{9} \, a b c x^{\frac {9}{2}} + \frac {2}{9} \, a^{2} d x^{\frac {9}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 56, normalized size = 0.89 \begin {gather*} \frac {2 \left (585 b^{2} d \,x^{6}+1530 a b d \,x^{4}+765 b^{2} c \,x^{4}+1105 a^{2} d \,x^{2}+2210 a b c \,x^{2}+1989 a^{2} c \right ) x^{\frac {5}{2}}}{9945} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{17} \, b^{2} d x^{\frac {17}{2}} + \frac {2}{13} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {13}{2}} + \frac {2}{5} \, a^{2} c x^{\frac {5}{2}} + \frac {2}{9} \, {\left (2 \, a b c + a^{2} d\right )} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 51, normalized size = 0.81 \begin {gather*} x^{9/2}\,\left (\frac {2\,d\,a^2}{9}+\frac {4\,b\,c\,a}{9}\right )+x^{13/2}\,\left (\frac {2\,c\,b^2}{13}+\frac {4\,a\,d\,b}{13}\right )+\frac {2\,a^2\,c\,x^{5/2}}{5}+\frac {2\,b^2\,d\,x^{17/2}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.89, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 a^{2} c x^{\frac {5}{2}}}{5} + \frac {2 a^{2} d x^{\frac {9}{2}}}{9} + \frac {4 a b c x^{\frac {9}{2}}}{9} + \frac {4 a b d x^{\frac {13}{2}}}{13} + \frac {2 b^{2} c x^{\frac {13}{2}}}{13} + \frac {2 b^{2} d x^{\frac {17}{2}}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________