Optimal. Leaf size=25 \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]
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Rubi [A] time = 0.0082432, antiderivative size = 25, 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} \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\frac{c}{(a+b x)^3}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{c}{x^3}}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int x^{3/2} \, dx,x,a+b x\right )}{b \sqrt{\frac{c}{(a+b x)^3}} (a+b x)^{3/2}}\\ &=\frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}}\\ \end{align*}
Mathematica [A] time = 0.0152308, size = 25, normalized size = 1. \[ \frac{2 (a+b x)}{5 b \sqrt{\frac{c}{(a+b x)^3}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 0.9 \begin{align*}{\frac{2\,bx+2\,a}{5\,b}{\frac{1}{\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 = 1.5605, size = 36, normalized size = 1.44 \begin{align*} \frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{5 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31795, size = 161, normalized size = 6.44 \begin{align*} \frac{2 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{5 \, 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{\frac{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 [B] time = 1.15508, size = 194, normalized size = 7.76 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{b c x + a c} a^{2} - \frac{10 \,{\left (3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}\right )} a}{c} + \frac{15 \, \sqrt{b c x + a c} a^{2} c^{2} - 10 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c + 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}}{c^{2}}\right )}}{15 \, b 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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