Optimal. Leaf size=139 \[ \frac{2}{21} d x^{21/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{17} c x^{17/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^3 x^{9/2}+\frac{2}{13} a c^2 x^{13/2} (3 a d+2 b c)+\frac{2}{25} b d^2 x^{25/2} (2 a d+3 b c)+\frac{2}{29} b^2 d^3 x^{29/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.064872, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {448} \[ \frac{2}{21} d x^{21/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{17} c x^{17/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^3 x^{9/2}+\frac{2}{13} a c^2 x^{13/2} (3 a d+2 b c)+\frac{2}{25} b d^2 x^{25/2} (2 a d+3 b c)+\frac{2}{29} b^2 d^3 x^{29/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 448
Rubi steps
\begin{align*} \int x^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^{7/2}+a c^2 (2 b c+3 a d) x^{11/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{15/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{19/2}+b d^2 (3 b c+2 a d) x^{23/2}+b^2 d^3 x^{27/2}\right ) \, dx\\ &=\frac{2}{9} a^2 c^3 x^{9/2}+\frac{2}{13} a c^2 (2 b c+3 a d) x^{13/2}+\frac{2}{17} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{17/2}+\frac{2}{21} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{21/2}+\frac{2}{25} b d^2 (3 b c+2 a d) x^{25/2}+\frac{2}{29} b^2 d^3 x^{29/2}\\ \end{align*}
Mathematica [A] time = 0.0426096, size = 139, normalized size = 1. \[ \frac{2}{21} d x^{21/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{17} c x^{17/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^3 x^{9/2}+\frac{2}{13} a c^2 x^{13/2} (3 a d+2 b c)+\frac{2}{25} b d^2 x^{25/2} (2 a d+3 b c)+\frac{2}{29} b^2 d^3 x^{29/2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 138, normalized size = 1. \begin{align*}{\frac{696150\,{b}^{2}{d}^{3}{x}^{10}+1615068\,{x}^{8}ab{d}^{3}+2422602\,{x}^{8}{b}^{2}c{d}^{2}+961350\,{x}^{6}{a}^{2}{d}^{3}+5768100\,{x}^{6}abc{d}^{2}+2884050\,{x}^{6}{b}^{2}{c}^{2}d+3562650\,{x}^{4}{a}^{2}c{d}^{2}+7125300\,{x}^{4}ab{c}^{2}d+1187550\,{x}^{4}{b}^{2}{c}^{3}+4658850\,{x}^{2}{a}^{2}{c}^{2}d+3105900\,{x}^{2}ab{c}^{3}+2243150\,{a}^{2}{c}^{3}}{10094175}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03665, size = 171, normalized size = 1.23 \begin{align*} \frac{2}{29} \, b^{2} d^{3} x^{\frac{29}{2}} + \frac{2}{25} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{25}{2}} + \frac{2}{21} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{21}{2}} + \frac{2}{9} \, a^{2} c^{3} x^{\frac{9}{2}} + \frac{2}{17} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.841121, size = 332, normalized size = 2.39 \begin{align*} \frac{2}{10094175} \,{\left (348075 \, b^{2} d^{3} x^{14} + 403767 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{12} + 480675 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{10} + 1121575 \, a^{2} c^{3} x^{4} + 593775 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{8} + 776475 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{6}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 55.1669, size = 192, normalized size = 1.38 \begin{align*} \frac{2 a^{2} c^{3} x^{\frac{9}{2}}}{9} + \frac{6 a^{2} c^{2} d x^{\frac{13}{2}}}{13} + \frac{6 a^{2} c d^{2} x^{\frac{17}{2}}}{17} + \frac{2 a^{2} d^{3} x^{\frac{21}{2}}}{21} + \frac{4 a b c^{3} x^{\frac{13}{2}}}{13} + \frac{12 a b c^{2} d x^{\frac{17}{2}}}{17} + \frac{4 a b c d^{2} x^{\frac{21}{2}}}{7} + \frac{4 a b d^{3} x^{\frac{25}{2}}}{25} + \frac{2 b^{2} c^{3} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c^{2} d x^{\frac{21}{2}}}{7} + \frac{6 b^{2} c d^{2} x^{\frac{25}{2}}}{25} + \frac{2 b^{2} d^{3} x^{\frac{29}{2}}}{29} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17091, size = 182, normalized size = 1.31 \begin{align*} \frac{2}{29} \, b^{2} d^{3} x^{\frac{29}{2}} + \frac{6}{25} \, b^{2} c d^{2} x^{\frac{25}{2}} + \frac{4}{25} \, a b d^{3} x^{\frac{25}{2}} + \frac{2}{7} \, b^{2} c^{2} d x^{\frac{21}{2}} + \frac{4}{7} \, a b c d^{2} x^{\frac{21}{2}} + \frac{2}{21} \, a^{2} d^{3} x^{\frac{21}{2}} + \frac{2}{17} \, b^{2} c^{3} x^{\frac{17}{2}} + \frac{12}{17} \, a b c^{2} d x^{\frac{17}{2}} + \frac{6}{17} \, a^{2} c d^{2} x^{\frac{17}{2}} + \frac{4}{13} \, a b c^{3} x^{\frac{13}{2}} + \frac{6}{13} \, a^{2} c^{2} d x^{\frac{13}{2}} + \frac{2}{9} \, a^{2} c^{3} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]