Optimal. Leaf size=21 \[ \frac{c x \sqrt{\frac{c}{a+b x^2}}}{a} \]
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Rubi [A] time = 0.017496, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {6720, 191} \[ \frac{c x \sqrt{\frac{c}{a+b x^2}}}{a} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 191
Rubi steps
\begin{align*} \int \left (\frac{c}{a+b x^2}\right )^{3/2} \, dx &=\left (c \sqrt{\frac{c}{a+b x^2}} \sqrt{a+b x^2}\right ) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx\\ &=\frac{c x \sqrt{\frac{c}{a+b x^2}}}{a}\\ \end{align*}
Mathematica [A] time = 0.0047915, size = 21, normalized size = 1. \[ \frac{c x \sqrt{\frac{c}{a+b x^2}}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 1.2 \begin{align*}{\frac{x \left ( b{x}^{2}+a \right ) }{a} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{c}{b x^{2} + a}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48451, size = 36, normalized size = 1.71 \begin{align*} \frac{c x \sqrt{\frac{c}{b x^{2} + a}}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.46707, size = 66, normalized size = 3.14 \begin{align*} \begin{cases} c^{\frac{3}{2}} x \left (\frac{1}{a + b x^{2}}\right )^{\frac{3}{2}} + \frac{b c^{\frac{3}{2}} x^{3} \left (\frac{1}{a + b x^{2}}\right )^{\frac{3}{2}}}{a} & \text{for}\: a \neq 0 \\- \frac{c^{\frac{3}{2}} x \left (\frac{1}{b}\right )^{\frac{3}{2}} \left (\frac{1}{x^{2}}\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 = 1.2531, size = 38, normalized size = 1.81 \begin{align*} \frac{c^{2} x \mathrm{sgn}\left (b x^{2} + a\right )}{\sqrt{b c x^{2} + a c} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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