Optimal. Leaf size=95 \[ \frac{2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{3 x^{3/2}}+\frac{4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt{x} (a d+b c)+\frac{2}{13} b^2 d^2 x^{13/2} \]
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Rubi [A] time = 0.0466474, 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} \[ \frac{2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{3 x^{3/2}}+\frac{4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt{x} (a d+b c)+\frac{2}{13} b^2 d^2 x^{13/2} \]
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}{x^{5/2}} \, dx &=\int \left (\frac{a^2 c^2}{x^{5/2}}+\frac{2 a c (b c+a d)}{\sqrt{x}}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+2 b d (b c+a d) x^{7/2}+b^2 d^2 x^{11/2}\right ) \, dx\\ &=-\frac{2 a^2 c^2}{3 x^{3/2}}+4 a c (b c+a d) \sqrt{x}+\frac{2}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5/2}+\frac{4}{9} b d (b c+a d) x^{9/2}+\frac{2}{13} b^2 d^2 x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0318285, size = 83, normalized size = 0.87 \[ \frac{2 \left (117 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-195 a^2 c^2+130 b d x^6 (a d+b c)+1170 a c x^2 (a d+b c)+45 b^2 d^2 x^8\right )}{585 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 97, normalized size = 1. \begin{align*} -{\frac{-90\,{b}^{2}{d}^{2}{x}^{8}-260\,{x}^{6}ab{d}^{2}-260\,{x}^{6}{b}^{2}cd-234\,{x}^{4}{a}^{2}{d}^{2}-936\,{x}^{4}abcd-234\,{x}^{4}{b}^{2}{c}^{2}-2340\,{x}^{2}{a}^{2}cd-2340\,a{c}^{2}b{x}^{2}+390\,{a}^{2}{c}^{2}}{585}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02923, size = 115, normalized size = 1.21 \begin{align*} \frac{2}{13} \, b^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{9}{2}} + \frac{2}{5} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{5}{2}} - \frac{2 \, a^{2} c^{2}}{3 \, x^{\frac{3}{2}}} + 4 \,{\left (a b c^{2} + a^{2} c d\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.802958, size = 201, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (45 \, b^{2} d^{2} x^{8} + 130 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 117 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 195 \, a^{2} c^{2} + 1170 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{585 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.6326, size = 133, normalized size = 1.4 \begin{align*} - \frac{2 a^{2} c^{2}}{3 x^{\frac{3}{2}}} + 4 a^{2} c d \sqrt{x} + \frac{2 a^{2} d^{2} x^{\frac{5}{2}}}{5} + 4 a b c^{2} \sqrt{x} + \frac{8 a b c d x^{\frac{5}{2}}}{5} + \frac{4 a b d^{2} x^{\frac{9}{2}}}{9} + \frac{2 b^{2} c^{2} x^{\frac{5}{2}}}{5} + \frac{4 b^{2} c d x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d^{2} x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15176, size = 127, normalized size = 1.34 \begin{align*} \frac{2}{13} \, b^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \, b^{2} c d x^{\frac{9}{2}} + \frac{4}{9} \, a b d^{2} x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} c^{2} x^{\frac{5}{2}} + \frac{8}{5} \, a b c d x^{\frac{5}{2}} + \frac{2}{5} \, a^{2} d^{2} x^{\frac{5}{2}} + 4 \, a b c^{2} \sqrt{x} + 4 \, a^{2} c d \sqrt{x} - \frac{2 \, a^{2} c^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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