Optimal. Leaf size=21 \[ -\frac{c \sqrt{\frac{c}{a+b x^2}}}{b} \]
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Rubi [A] time = 0.0175141, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1591, 15, 30} \[ -\frac{c \sqrt{\frac{c}{a+b x^2}}}{b} \]
Antiderivative was successfully verified.
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Rule 1591
Rule 15
Rule 30
Rubi steps
\begin{align*} \int x \left (\frac{c}{a+b x^2}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x}\right )^{3/2} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac{\left (c \sqrt{\frac{c}{a+b x^2}} \sqrt{a+b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac{c \sqrt{\frac{c}{a+b x^2}}}{b}\\ \end{align*}
Mathematica [A] time = 0.0048062, size = 21, normalized size = 1. \[ -\frac{c \sqrt{\frac{c}{a+b x^2}}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 1.2 \begin{align*} -{\frac{b{x}^{2}+a}{b} \left ({\frac{c}{b{x}^{2}+a}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967719, size = 26, normalized size = 1.24 \begin{align*} -\frac{c \sqrt{\frac{c}{b x^{2} + a}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49485, size = 35, normalized size = 1.67 \begin{align*} -\frac{c \sqrt{\frac{c}{b x^{2} + a}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.4509, size = 53, normalized size = 2.52 \begin{align*} \begin{cases} - \frac{a c^{\frac{3}{2}} \left (\frac{1}{a + b x^{2}}\right )^{\frac{3}{2}}}{b} - c^{\frac{3}{2}} x^{2} \left (\frac{1}{a + b x^{2}}\right )^{\frac{3}{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \left (\frac{c}{a}\right )^{\frac{3}{2}}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19169, size = 38, normalized size = 1.81 \begin{align*} -\frac{c^{2} \mathrm{sgn}\left (b x^{2} + a\right )}{\sqrt{b c x^{2} + a c} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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