Optimal. Leaf size=95 \[ \frac{2}{7} x^{7/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{\sqrt{x}}+\frac{4}{11} b d x^{11/2} (a d+b c)+\frac{4}{3} a c x^{3/2} (a d+b c)+\frac{2}{15} b^2 d^2 x^{15/2} \]
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Rubi [A] time = 0.0475267, 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} \[ \frac{2}{7} x^{7/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{\sqrt{x}}+\frac{4}{11} b d x^{11/2} (a d+b c)+\frac{4}{3} a c x^{3/2} (a d+b c)+\frac{2}{15} b^2 d^2 x^{15/2} \]
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^{3/2}} \, dx &=\int \left (\frac{a^2 c^2}{x^{3/2}}+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^{5/2}+2 b d (b c+a d) x^{9/2}+b^2 d^2 x^{13/2}\right ) \, dx\\ &=-\frac{2 a^2 c^2}{\sqrt{x}}+\frac{4}{3} a c (b c+a d) x^{3/2}+\frac{2}{7} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{7/2}+\frac{4}{11} b d (b c+a d) x^{11/2}+\frac{2}{15} b^2 d^2 x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0316485, size = 83, normalized size = 0.87 \[ \frac{2 \left (165 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-1155 a^2 c^2+210 b d x^6 (a d+b c)+770 a c x^2 (a d+b c)+77 b^2 d^2 x^8\right )}{1155 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 97, normalized size = 1. \begin{align*} -{\frac{-154\,{b}^{2}{d}^{2}{x}^{8}-420\,{x}^{6}ab{d}^{2}-420\,{x}^{6}{b}^{2}cd-330\,{x}^{4}{a}^{2}{d}^{2}-1320\,{x}^{4}abcd-330\,{x}^{4}{b}^{2}{c}^{2}-1540\,{x}^{2}{a}^{2}cd-1540\,a{c}^{2}b{x}^{2}+2310\,{a}^{2}{c}^{2}}{1155}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0307, size = 115, normalized size = 1.21 \begin{align*} \frac{2}{15} \, b^{2} d^{2} x^{\frac{15}{2}} + \frac{4}{11} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{7}{2}} - \frac{2 \, a^{2} c^{2}}{\sqrt{x}} + \frac{4}{3} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.721341, size = 203, normalized size = 2.14 \begin{align*} \frac{2 \,{\left (77 \, b^{2} d^{2} x^{8} + 210 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 165 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 1155 \, a^{2} c^{2} + 770 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{1155 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.54516, size = 134, normalized size = 1.41 \begin{align*} - \frac{2 a^{2} c^{2}}{\sqrt{x}} + \frac{4 a^{2} c d x^{\frac{3}{2}}}{3} + \frac{2 a^{2} d^{2} x^{\frac{7}{2}}}{7} + \frac{4 a b c^{2} x^{\frac{3}{2}}}{3} + \frac{8 a b c d x^{\frac{7}{2}}}{7} + \frac{4 a b d^{2} x^{\frac{11}{2}}}{11} + \frac{2 b^{2} c^{2} x^{\frac{7}{2}}}{7} + \frac{4 b^{2} c d x^{\frac{11}{2}}}{11} + \frac{2 b^{2} d^{2} x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.62242, size = 127, normalized size = 1.34 \begin{align*} \frac{2}{15} \, b^{2} d^{2} x^{\frac{15}{2}} + \frac{4}{11} \, b^{2} c d x^{\frac{11}{2}} + \frac{4}{11} \, a b d^{2} x^{\frac{11}{2}} + \frac{2}{7} \, b^{2} c^{2} x^{\frac{7}{2}} + \frac{8}{7} \, a b c d x^{\frac{7}{2}} + \frac{2}{7} \, a^{2} d^{2} x^{\frac{7}{2}} + \frac{4}{3} \, a b c^{2} x^{\frac{3}{2}} + \frac{4}{3} \, a^{2} c d x^{\frac{3}{2}} - \frac{2 \, a^{2} c^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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