Optimal. Leaf size=97 \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]
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Rubi [A] time = 0.0459392, 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} \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\int \left (a^2 c^2 x^{5/2}+2 a c (b c+a d) x^{9/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{13/2}+2 b d (b c+a d) x^{17/2}+b^2 d^2 x^{21/2}\right ) \, dx\\ &=\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{11} a c (b c+a d) x^{11/2}+\frac{2}{15} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{15/2}+\frac{4}{19} b d (b c+a d) x^{19/2}+\frac{2}{23} b^2 d^2 x^{23/2}\\ \end{align*}
Mathematica [A] time = 0.0330486, size = 97, normalized size = 1. \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 97, normalized size = 1. \begin{align*}{\frac{43890\,{b}^{2}{d}^{2}{x}^{8}+106260\,{x}^{6}ab{d}^{2}+106260\,{x}^{6}{b}^{2}cd+67298\,{x}^{4}{a}^{2}{d}^{2}+269192\,{x}^{4}abcd+67298\,{x}^{4}{b}^{2}{c}^{2}+183540\,{x}^{2}{a}^{2}cd+183540\,a{c}^{2}b{x}^{2}+144210\,{a}^{2}{c}^{2}}{504735}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04838, size = 115, normalized size = 1.19 \begin{align*} \frac{2}{23} \, b^{2} d^{2} x^{\frac{23}{2}} + \frac{4}{19} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, a^{2} c^{2} x^{\frac{7}{2}} + \frac{4}{11} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.74764, size = 225, normalized size = 2.32 \begin{align*} \frac{2}{504735} \,{\left (21945 \, b^{2} d^{2} x^{11} + 53130 \,{\left (b^{2} c d + a b d^{2}\right )} x^{9} + 33649 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{7} + 72105 \, a^{2} c^{2} x^{3} + 91770 \,{\left (a b c^{2} + a^{2} c d\right )} x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.5765, size = 136, normalized size = 1.4 \begin{align*} \frac{2 a^{2} c^{2} x^{\frac{7}{2}}}{7} + \frac{4 a^{2} c d x^{\frac{11}{2}}}{11} + \frac{2 a^{2} d^{2} x^{\frac{15}{2}}}{15} + \frac{4 a b c^{2} x^{\frac{11}{2}}}{11} + \frac{8 a b c d x^{\frac{15}{2}}}{15} + \frac{4 a b d^{2} x^{\frac{19}{2}}}{19} + \frac{2 b^{2} c^{2} x^{\frac{15}{2}}}{15} + \frac{4 b^{2} c d x^{\frac{19}{2}}}{19} + \frac{2 b^{2} d^{2} x^{\frac{23}{2}}}{23} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14413, size = 127, normalized size = 1.31 \begin{align*} \frac{2}{23} \, b^{2} d^{2} x^{\frac{23}{2}} + \frac{4}{19} \, b^{2} c d x^{\frac{19}{2}} + \frac{4}{19} \, a b d^{2} x^{\frac{19}{2}} + \frac{2}{15} \, b^{2} c^{2} x^{\frac{15}{2}} + \frac{8}{15} \, a b c d x^{\frac{15}{2}} + \frac{2}{15} \, a^{2} d^{2} x^{\frac{15}{2}} + \frac{4}{11} \, a b c^{2} x^{\frac{11}{2}} + \frac{4}{11} \, a^{2} c d x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} c^{2} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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