Optimal. Leaf size=95 \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]
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Rubi [A] time = 0.0459258, antiderivative size = 95, 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} \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{\sqrt{x}} \, dx &=\int \left (\frac{a^2 c^2}{\sqrt{x}}+2 a c (b c+a d) x^{3/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{7/2}+2 b d (b c+a d) x^{11/2}+b^2 d^2 x^{15/2}\right ) \, dx\\ &=2 a^2 c^2 \sqrt{x}+\frac{4}{5} a c (b c+a d) x^{5/2}+\frac{2}{9} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{9/2}+\frac{4}{13} b d (b c+a d) x^{13/2}+\frac{2}{17} b^2 d^2 x^{17/2}\\ \end{align*}
Mathematica [A] time = 0.0313195, size = 95, normalized size = 1. \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 97, normalized size = 1. \begin{align*}{\frac{1170\,{b}^{2}{d}^{2}{x}^{8}+3060\,{x}^{6}ab{d}^{2}+3060\,{x}^{6}{b}^{2}cd+2210\,{x}^{4}{a}^{2}{d}^{2}+8840\,{x}^{4}abcd+2210\,{x}^{4}{b}^{2}{c}^{2}+7956\,{x}^{2}{a}^{2}cd+7956\,a{c}^{2}b{x}^{2}+19890\,{a}^{2}{c}^{2}}{9945}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02062, size = 115, normalized size = 1.21 \begin{align*} \frac{2}{17} \, b^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{9}{2}} + 2 \, a^{2} c^{2} \sqrt{x} + \frac{4}{5} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.70194, size = 208, normalized size = 2.19 \begin{align*} \frac{2}{9945} \,{\left (585 \, b^{2} d^{2} x^{8} + 1530 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 1105 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + 9945 \, a^{2} c^{2} + 3978 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.98539, size = 134, normalized size = 1.41 \begin{align*} 2 a^{2} c^{2} \sqrt{x} + \frac{4 a^{2} c d x^{\frac{5}{2}}}{5} + \frac{2 a^{2} d^{2} x^{\frac{9}{2}}}{9} + \frac{4 a b c^{2} x^{\frac{5}{2}}}{5} + \frac{8 a b c d x^{\frac{9}{2}}}{9} + \frac{4 a b d^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{2} c^{2} x^{\frac{9}{2}}}{9} + \frac{4 b^{2} c d x^{\frac{13}{2}}}{13} + \frac{2 b^{2} d^{2} x^{\frac{17}{2}}}{17} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17831, size = 127, normalized size = 1.34 \begin{align*} \frac{2}{17} \, b^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b^{2} c d x^{\frac{13}{2}} + \frac{4}{13} \, a b d^{2} x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} c^{2} x^{\frac{9}{2}} + \frac{8}{9} \, a b c d x^{\frac{9}{2}} + \frac{2}{9} \, a^{2} d^{2} x^{\frac{9}{2}} + \frac{4}{5} \, a b c^{2} x^{\frac{5}{2}} + \frac{4}{5} \, a^{2} c d x^{\frac{5}{2}} + 2 \, a^{2} c^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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