Optimal. Leaf size=28 \[ -\frac{c \sqrt{\frac{c}{(a+b x)^2}}}{2 b (a+b x)} \]
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Rubi [A] time = 0.0078393, antiderivative size = 28, 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} \[ -\frac{c \sqrt{\frac{c}{(a+b x)^2}}}{2 b (a+b x)} \]
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)^2}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x^2}\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c \sqrt{\frac{c}{(a+b x)^2}} (a+b x)\right ) \operatorname{Subst}\left (\int \frac{1}{x^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac{c \sqrt{\frac{c}{(a+b x)^2}}}{2 b (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0075258, size = 25, normalized size = 0.89 \[ -\frac{(a+b x) \left (\frac{c}{(a+b x)^2}\right )^{3/2}}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 0.8 \begin{align*} -{\frac{bx+a}{2\,b} \left ({\frac{c}{ \left ( bx+a \right ) ^{2}}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21504, size = 36, normalized size = 1.29 \begin{align*} -\frac{c^{\frac{3}{2}}}{2 \,{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24033, size = 76, normalized size = 2.71 \begin{align*} -\frac{c \sqrt{\frac{c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \,{\left (b^{2} x + a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\frac{c}{\left (a + b x\right )^{2}}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10474, size = 28, normalized size = 1. \begin{align*} -\frac{c^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right )}{2 \,{\left (b x + a\right )}^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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