Optimal. Leaf size=23 \[ -\frac{2 (a+b x)}{b \sqrt{c (a+b x)^3}} \]
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Rubi [A] time = 0.0110628, 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} \[ -\frac{2 (a+b x)}{b \sqrt{c (a+b x)^3}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c (a+b x)^3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{c x^3}} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x)^{3/2} \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,a+b x\right )}{b \sqrt{c (a+b x)^3}}\\ &=-\frac{2 (a+b x)}{b \sqrt{c (a+b x)^3}}\\ \end{align*}
Mathematica [A] time = 0.0089655, size = 23, normalized size = 1. \[ -\frac{2 (a+b x)}{b \sqrt{c (a+b x)^3}} \]
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{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 = 1.01131, size = 36, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.25826, size = 127, normalized size = 5.52 \begin{align*} -\frac{2 \, \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{b^{3} c x^{2} + 2 \, a b^{2} c x + a^{2} b c} \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 \frac{1}{\sqrt{c \left (a + b x\right )^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10042, size = 31, normalized size = 1.35 \begin{align*} -\frac{2}{\sqrt{b c x + a c} b \mathrm{sgn}\left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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