Optimal. Leaf size=19 \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]
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Rubi [A] time = 0.0060305, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (\frac{c}{a+b x}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x}\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c \sqrt{\frac{c}{a+b x}} \sqrt{a+b x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2 c \sqrt{\frac{c}{a+b x}}}{b}\\ \end{align*}
Mathematica [A] time = 0.0061148, size = 19, normalized size = 1. \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 1.2 \begin{align*} -2\,{\frac{bx+a}{b} \left ({\frac{c}{bx+a}} \right ) ^{3/2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956174, size = 23, normalized size = 1.21 \begin{align*} -\frac{2 \, c \sqrt{\frac{c}{b x + a}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33943, size = 35, normalized size = 1.84 \begin{align*} -\frac{2 \, c \sqrt{\frac{c}{b x + a}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.842649, size = 48, normalized size = 2.53 \begin{align*} \begin{cases} - \frac{2 a c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}}{b} - 2 c^{\frac{3}{2}} x \left (\frac{1}{a + b x}\right )^{\frac{3}{2}} & \text{for}\: b \neq 0 \\x \left (\frac{c}{a}\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09958, size = 32, normalized size = 1.68 \begin{align*} -\frac{2 \, c^{2} \mathrm{sgn}\left (b x + a\right )}{\sqrt{b c x + a c} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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