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 = 0.0288909, antiderivative size = 61, normalized size of antiderivative = 1., 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} \[ -\frac{2 a^2 c}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a d+b c)+2 a \sqrt{x} (a d+2 b c)+\frac{2}{9} b^2 d x^{9/2} \]
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*}
Mathematica [A] time = 0.0200688, size = 57, normalized size = 0.93 \[ \frac{-30 a^2 \left (c-3 d x^2\right )+36 a b x^2 \left (5 c+d x^2\right )+2 b^2 x^4 \left (9 c+5 d x^2\right )}{45 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-10\,{b}^{2}d{x}^{6}-36\,{x}^{4}abd-18\,{b}^{2}c{x}^{4}-90\,{x}^{2}{a}^{2}d-180\,abc{x}^{2}+30\,{a}^{2}c}{45}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0184, size = 69, normalized size = 1.13 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.870297, size = 124, normalized size = 2.03 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.97023, size = 76, normalized size = 1.25 \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17291, size = 72, normalized size = 1.18 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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