Optimal. Leaf size=23 \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]
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Rubi [A] time = 0.0076797, antiderivative size = 23, 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} \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \sqrt{\frac{c}{(a+b x)^3}} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{\frac{c}{x^3}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (\sqrt{\frac{c}{(a+b x)^3}} (a+b x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2 \sqrt{\frac{c}{(a+b x)^3}} (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0077133, size = 23, normalized size = 1. \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 1. \begin{align*} -2\,{\frac{bx+a}{b}\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45024, size = 32, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.34563, size = 89, normalized size = 3.87 \begin{align*} -\frac{2 \,{\left (b x + a\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.944818, size = 97, normalized size = 4.22 \begin{align*} \begin{cases} - \frac{2 a \sqrt{c} \sqrt{\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}}}{b} - 2 \sqrt{c} x \sqrt{\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}} & \text{for}\: b \neq 0 \\x \sqrt{\frac{c}{a^{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34431, size = 68, normalized size = 2.96 \begin{align*} -\frac{2 \, c \mathrm{sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \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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