Optimal. Leaf size=61 \[ -\frac {2 a^2 c}{3 x^{3/2}}+\frac {2}{5} b x^{5/2} (2 a d+b c)+2 a \sqrt {x} (a d+2 b c)+\frac {2}{9} b^2 d x^{9/2} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \begin {gather*} -\frac {2 a^2 c}{3 x^{3/2}}+\frac {2}{5} b x^{5/2} (2 a d+b c)+2 a \sqrt {x} (a d+2 b c)+\frac {2}{9} b^2 d x^{9/2} \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 )}{x^{5/2}} \, dx &=\int \left (\frac {a^2 c}{x^{5/2}}+\frac {a (2 b c+a d)}{\sqrt {x}}+b (b c+2 a d) x^{3/2}+b^2 d x^{7/2}\right ) \, dx\\ &=-\frac {2 a^2 c}{3 x^{3/2}}+2 a (2 b c+a d) \sqrt {x}+\frac {2}{5} b (b c+2 a d) x^{5/2}+\frac {2}{9} b^2 d x^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 0.93 \begin {gather*} \frac {-30 a^2 \left (c-3 d x^2\right )+36 a b x^2 \left (5 c+d x^2\right )+2 b^2 x^4 \left (9 c+5 d x^2\right )}{45 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 59, normalized size = 0.97 \begin {gather*} \frac {2 \left (-15 a^2 c+45 a^2 d x^2+90 a b c x^2+18 a b d x^4+9 b^2 c x^4+5 b^2 d x^6\right )}{45 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 53, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (5 \, b^{2} d x^{6} + 9 \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} - 15 \, a^{2} c + 45 \, {\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{45 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 53, normalized size = 0.87 \begin {gather*} \frac {2}{9} \, b^{2} d x^{\frac {9}{2}} + \frac {2}{5} \, b^{2} c x^{\frac {5}{2}} + \frac {4}{5} \, a b d x^{\frac {5}{2}} + 4 \, a b c \sqrt {x} + 2 \, a^{2} d \sqrt {x} - \frac {2 \, a^{2} c}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.92 \begin {gather*} -\frac {2 \left (-5 b^{2} d \,x^{6}-18 a b d \,x^{4}-9 b^{2} c \,x^{4}-45 a^{2} d \,x^{2}-90 a b c \,x^{2}+15 a^{2} c \right )}{45 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 51, normalized size = 0.84 \begin {gather*} \frac {2}{9} \, b^{2} d x^{\frac {9}{2}} + \frac {2}{5} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {5}{2}} - \frac {2 \, a^{2} c}{3 \, x^{\frac {3}{2}}} + 2 \, {\left (2 \, a b c + a^{2} d\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.84 \begin {gather*} \sqrt {x}\,\left (2\,d\,a^2+4\,b\,c\,a\right )+x^{5/2}\,\left (\frac {2\,c\,b^2}{5}+\frac {4\,a\,d\,b}{5}\right )-\frac {2\,a^2\,c}{3\,x^{3/2}}+\frac {2\,b^2\,d\,x^{9/2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.76, size = 76, normalized size = 1.25 \begin {gather*} - \frac {2 a^{2} c}{3 x^{\frac {3}{2}}} + 2 a^{2} d \sqrt {x} + 4 a b c \sqrt {x} + \frac {4 a b d x^{\frac {5}{2}}}{5} + \frac {2 b^{2} c x^{\frac {5}{2}}}{5} + \frac {2 b^{2} d x^{\frac {9}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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