Optimal. Leaf size=97 \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]
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Rubi [A] time = 0.0497726, antiderivative size = 97, 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} \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\int \left (a^2 c^2 x^{3/2}+2 a c (b c+a d) x^{7/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{11/2}+2 b d (b c+a d) x^{15/2}+b^2 d^2 x^{19/2}\right ) \, dx\\ &=\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{9} a c (b c+a d) x^{9/2}+\frac{2}{13} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{13/2}+\frac{4}{17} b d (b c+a d) x^{17/2}+\frac{2}{21} b^2 d^2 x^{21/2}\\ \end{align*}
Mathematica [A] time = 0.0316799, size = 97, normalized size = 1. \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 97, normalized size = 1. \begin{align*}{\frac{6630\,{b}^{2}{d}^{2}{x}^{8}+16380\,{x}^{6}ab{d}^{2}+16380\,{x}^{6}{b}^{2}cd+10710\,{x}^{4}{a}^{2}{d}^{2}+42840\,{x}^{4}abcd+10710\,{x}^{4}{b}^{2}{c}^{2}+30940\,{x}^{2}{a}^{2}cd+30940\,a{c}^{2}b{x}^{2}+27846\,{a}^{2}{c}^{2}}{69615}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10538, size = 115, normalized size = 1.19 \begin{align*} \frac{2}{21} \, b^{2} d^{2} x^{\frac{21}{2}} + \frac{4}{17} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{5} \, a^{2} c^{2} x^{\frac{5}{2}} + \frac{4}{9} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.884202, size = 220, normalized size = 2.27 \begin{align*} \frac{2}{69615} \,{\left (3315 \, b^{2} d^{2} x^{10} + 8190 \,{\left (b^{2} c d + a b d^{2}\right )} x^{8} + 5355 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + 13923 \, a^{2} c^{2} x^{2} + 15470 \,{\left (a b c^{2} + a^{2} c d\right )} x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.3792, size = 136, normalized size = 1.4 \begin{align*} \frac{2 a^{2} c^{2} x^{\frac{5}{2}}}{5} + \frac{4 a^{2} c d x^{\frac{9}{2}}}{9} + \frac{2 a^{2} d^{2} x^{\frac{13}{2}}}{13} + \frac{4 a b c^{2} x^{\frac{9}{2}}}{9} + \frac{8 a b c d x^{\frac{13}{2}}}{13} + \frac{4 a b d^{2} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c^{2} x^{\frac{13}{2}}}{13} + \frac{4 b^{2} c d x^{\frac{17}{2}}}{17} + \frac{2 b^{2} d^{2} x^{\frac{21}{2}}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.59996, size = 127, normalized size = 1.31 \begin{align*} \frac{2}{21} \, b^{2} d^{2} x^{\frac{21}{2}} + \frac{4}{17} \, b^{2} c d x^{\frac{17}{2}} + \frac{4}{17} \, a b d^{2} x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} c^{2} x^{\frac{13}{2}} + \frac{8}{13} \, a b c d x^{\frac{13}{2}} + \frac{2}{13} \, a^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \, a b c^{2} x^{\frac{9}{2}} + \frac{4}{9} \, a^{2} c d x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} c^{2} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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