Optimal. Leaf size=61 \[ 2 a^2 c \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a d+b c)+\frac{2}{5} a x^{5/2} (a d+2 b c)+\frac{2}{13} b^2 d x^{13/2} \]
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Rubi [A] time = 0.0290469, 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} \[ 2 a^2 c \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a d+b c)+\frac{2}{5} a x^{5/2} (a d+2 b c)+\frac{2}{13} b^2 d x^{13/2} \]
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 )}{\sqrt{x}} \, dx &=\int \left (\frac{a^2 c}{\sqrt{x}}+a (2 b c+a d) x^{3/2}+b (b c+2 a d) x^{7/2}+b^2 d x^{11/2}\right ) \, dx\\ &=2 a^2 c \sqrt{x}+\frac{2}{5} a (2 b c+a d) x^{5/2}+\frac{2}{9} b (b c+2 a d) x^{9/2}+\frac{2}{13} b^2 d x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0309814, size = 53, normalized size = 0.87 \[ \frac{2}{585} \sqrt{x} \left (585 a^2 c+65 b x^4 (2 a d+b c)+117 a x^2 (a d+2 b c)+45 b^2 d x^6\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{90\,{b}^{2}d{x}^{6}+260\,{x}^{4}abd+130\,{b}^{2}c{x}^{4}+234\,{x}^{2}{a}^{2}d+468\,abc{x}^{2}+1170\,{a}^{2}c}{585}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1155, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{13} \, b^{2} d x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{9}{2}} + 2 \, a^{2} c \sqrt{x} + \frac{2}{5} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.908064, size = 131, normalized size = 2.15 \begin{align*} \frac{2}{585} \,{\left (45 \, b^{2} d x^{6} + 65 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + 585 \, a^{2} c + 117 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.14039, size = 78, normalized size = 1.28 \begin{align*} 2 a^{2} c \sqrt{x} + \frac{2 a^{2} d x^{\frac{5}{2}}}{5} + \frac{4 a b c x^{\frac{5}{2}}}{5} + \frac{4 a b d x^{\frac{9}{2}}}{9} + \frac{2 b^{2} c x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16651, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{13} \, b^{2} d x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} c x^{\frac{9}{2}} + \frac{4}{9} \, a b d x^{\frac{9}{2}} + \frac{4}{5} \, a b c x^{\frac{5}{2}} + \frac{2}{5} \, a^{2} d x^{\frac{5}{2}} + 2 \, a^{2} c \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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