Optimal. Leaf size=97 \[ \frac{2}{17} x^{17/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} b d x^{21/2} (a d+b c)+\frac{4}{13} a c x^{13/2} (a d+b c)+\frac{2}{25} b^2 d^2 x^{25/2} \]
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Rubi [A] time = 0.0494179, 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} \[ \frac{2}{17} x^{17/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} b d x^{21/2} (a d+b c)+\frac{4}{13} a c x^{13/2} (a d+b c)+\frac{2}{25} b^2 d^2 x^{25/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\int \left (a^2 c^2 x^{7/2}+2 a c (b c+a d) x^{11/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{15/2}+2 b d (b c+a d) x^{19/2}+b^2 d^2 x^{23/2}\right ) \, dx\\ &=\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{13} a c (b c+a d) x^{13/2}+\frac{2}{17} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{17/2}+\frac{4}{21} b d (b c+a d) x^{21/2}+\frac{2}{25} b^2 d^2 x^{25/2}\\ \end{align*}
Mathematica [A] time = 0.0349686, size = 97, normalized size = 1. \[ \frac{2}{17} x^{17/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{9} a^2 c^2 x^{9/2}+\frac{4}{21} b d x^{21/2} (a d+b c)+\frac{4}{13} a c x^{13/2} (a d+b c)+\frac{2}{25} b^2 d^2 x^{25/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 97, normalized size = 1. \begin{align*}{\frac{27846\,{b}^{2}{d}^{2}{x}^{8}+66300\,{x}^{6}ab{d}^{2}+66300\,{x}^{6}{b}^{2}cd+40950\,{x}^{4}{a}^{2}{d}^{2}+163800\,{x}^{4}abcd+40950\,{x}^{4}{b}^{2}{c}^{2}+107100\,{x}^{2}{a}^{2}cd+107100\,a{c}^{2}b{x}^{2}+77350\,{a}^{2}{c}^{2}}{348075}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08637, size = 115, normalized size = 1.19 \begin{align*} \frac{2}{25} \, b^{2} d^{2} x^{\frac{25}{2}} + \frac{4}{21} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{17}{2}} + \frac{2}{9} \, a^{2} c^{2} x^{\frac{9}{2}} + \frac{4}{13} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.758657, size = 227, normalized size = 2.34 \begin{align*} \frac{2}{348075} \,{\left (13923 \, b^{2} d^{2} x^{12} + 33150 \,{\left (b^{2} c d + a b d^{2}\right )} x^{10} + 20475 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{8} + 38675 \, a^{2} c^{2} x^{4} + 53550 \,{\left (a b c^{2} + a^{2} c d\right )} x^{6}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.5392, size = 136, normalized size = 1.4 \begin{align*} \frac{2 a^{2} c^{2} x^{\frac{9}{2}}}{9} + \frac{4 a^{2} c d x^{\frac{13}{2}}}{13} + \frac{2 a^{2} d^{2} x^{\frac{17}{2}}}{17} + \frac{4 a b c^{2} x^{\frac{13}{2}}}{13} + \frac{8 a b c d x^{\frac{17}{2}}}{17} + \frac{4 a b d^{2} x^{\frac{21}{2}}}{21} + \frac{2 b^{2} c^{2} x^{\frac{17}{2}}}{17} + \frac{4 b^{2} c d x^{\frac{21}{2}}}{21} + \frac{2 b^{2} d^{2} x^{\frac{25}{2}}}{25} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14502, size = 127, normalized size = 1.31 \begin{align*} \frac{2}{25} \, b^{2} d^{2} x^{\frac{25}{2}} + \frac{4}{21} \, b^{2} c d x^{\frac{21}{2}} + \frac{4}{21} \, a b d^{2} x^{\frac{21}{2}} + \frac{2}{17} \, b^{2} c^{2} x^{\frac{17}{2}} + \frac{8}{17} \, a b c d x^{\frac{17}{2}} + \frac{2}{17} \, a^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \, a b c^{2} x^{\frac{13}{2}} + \frac{4}{13} \, a^{2} c d x^{\frac{13}{2}} + \frac{2}{9} \, a^{2} c^{2} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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