Optimal. Leaf size=95 \[ \frac{2}{3} x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{5 x^{5/2}}+\frac{4}{7} b d x^{7/2} (a d+b c)-\frac{4 a c (a d+b c)}{\sqrt{x}}+\frac{2}{11} b^2 d^2 x^{11/2} \]
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Rubi [A] time = 0.0471844, 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} \[ \frac{2}{3} x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{5 x^{5/2}}+\frac{4}{7} b d x^{7/2} (a d+b c)-\frac{4 a c (a d+b c)}{\sqrt{x}}+\frac{2}{11} b^2 d^2 x^{11/2} \]
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^{7/2}} \, dx &=\int \left (\frac{a^2 c^2}{x^{7/2}}+\frac{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 ) \sqrt{x}+2 b d (b c+a d) x^{5/2}+b^2 d^2 x^{9/2}\right ) \, dx\\ &=-\frac{2 a^2 c^2}{5 x^{5/2}}-\frac{4 a c (b c+a d)}{\sqrt{x}}+\frac{2}{3} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+\frac{4}{7} b d (b c+a d) x^{7/2}+\frac{2}{11} b^2 d^2 x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0373862, size = 83, normalized size = 0.87 \[ \frac{2 \left (385 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-231 a^2 c^2+330 b d x^6 (a d+b c)-2310 a c x^2 (a d+b c)+105 b^2 d^2 x^8\right )}{1155 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 97, normalized size = 1. \begin{align*} -{\frac{-210\,{b}^{2}{d}^{2}{x}^{8}-660\,{x}^{6}ab{d}^{2}-660\,{x}^{6}{b}^{2}cd-770\,{x}^{4}{a}^{2}{d}^{2}-3080\,{x}^{4}abcd-770\,{x}^{4}{b}^{2}{c}^{2}+4620\,{x}^{2}{a}^{2}cd+4620\,a{c}^{2}b{x}^{2}+462\,{a}^{2}{c}^{2}}{1155}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06232, size = 117, normalized size = 1.23 \begin{align*} \frac{2}{11} \, b^{2} d^{2} x^{\frac{11}{2}} + \frac{4}{7} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (a^{2} c^{2} + 10 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.75379, size = 204, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (105 \, b^{2} d^{2} x^{8} + 330 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 385 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 231 \, a^{2} c^{2} - 2310 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{1155 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.94829, size = 133, normalized size = 1.4 \begin{align*} - \frac{2 a^{2} c^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a^{2} c d}{\sqrt{x}} + \frac{2 a^{2} d^{2} x^{\frac{3}{2}}}{3} - \frac{4 a b c^{2}}{\sqrt{x}} + \frac{8 a b c d x^{\frac{3}{2}}}{3} + \frac{4 a b d^{2} x^{\frac{7}{2}}}{7} + \frac{2 b^{2} c^{2} x^{\frac{3}{2}}}{3} + \frac{4 b^{2} c d x^{\frac{7}{2}}}{7} + \frac{2 b^{2} d^{2} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16628, size = 130, normalized size = 1.37 \begin{align*} \frac{2}{11} \, b^{2} d^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b^{2} c d x^{\frac{7}{2}} + \frac{4}{7} \, a b d^{2} x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} c^{2} x^{\frac{3}{2}} + \frac{8}{3} \, a b c d x^{\frac{3}{2}} + \frac{2}{3} \, a^{2} d^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b c^{2} x^{2} + 10 \, a^{2} c d x^{2} + a^{2} c^{2}\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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