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 = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
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*}
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Mathematica [A] time = 0.03, size = 83, normalized size = 0.87 \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 100, normalized size = 1.05 \begin {gather*} \frac {2 \left (-1155 a^2 c^2+770 a^2 c d x^2+165 a^2 d^2 x^4+770 a b c^2 x^2+660 a b c d x^4+210 a b d^2 x^6+165 b^2 c^2 x^4+210 b^2 c d x^6+77 b^2 d^2 x^8\right )}{1155 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 87, normalized size = 0.92 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 94, normalized size = 0.99 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 97, normalized size = 1.02 \begin {gather*} -\frac {2 \left (-77 b^{2} d^{2} x^{8}-210 a b \,d^{2} x^{6}-210 b^{2} c d \,x^{6}-165 a^{2} d^{2} x^{4}-660 a b c d \,x^{4}-165 b^{2} c^{2} x^{4}-770 a^{2} c d \,x^{2}-770 a b \,c^{2} x^{2}+1155 a^{2} c^{2}\right )}{1155 \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 85, normalized size = 0.89 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 78, normalized size = 0.82 \begin {gather*} x^{7/2}\,\left (\frac {2\,a^2\,d^2}{7}+\frac {8\,a\,b\,c\,d}{7}+\frac {2\,b^2\,c^2}{7}\right )-\frac {2\,a^2\,c^2}{\sqrt {x}}+\frac {2\,b^2\,d^2\,x^{15/2}}{15}+\frac {4\,a\,c\,x^{3/2}\,\left (a\,d+b\,c\right )}{3}+\frac {4\,b\,d\,x^{11/2}\,\left (a\,d+b\,c\right )}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.26, size = 134, normalized size = 1.41 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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