Optimal. Leaf size=27 \[ \frac{a+b x}{2 b \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]
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Rubi [A] time = 0.0101676, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {247, 15, 30} \[ \frac{a+b x}{2 b \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (\frac{c}{x^{3/2}}\right )^{2/3}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}(\int x \, dx,x,a+b x)}{b \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3} (a+b x)}\\ &=\frac{a+b x}{2 b \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0206451, size = 34, normalized size = 1.26 \[ \frac{x (2 a+b x)}{2 (a+b x) \left (\frac{c}{(a+b x)^{3/2}}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 29, normalized size = 1.1 \begin{align*}{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a} \left ({c \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957195, size = 28, normalized size = 1.04 \begin{align*} \frac{b x + a}{2 \, b \left (\frac{c}{{\left (b x + a\right )}^{\frac{3}{2}}}\right )^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.6807, size = 117, normalized size = 4.33 \begin{align*} \frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{b x + a} \left (\frac{\sqrt{b x + a} c}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{\frac{1}{3}}}{2 \, 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}{\left (\frac{c}{\left (a + b x\right )^{\frac{3}{2}}}\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\frac{c}{{\left (b x + a\right )}^{\frac{3}{2}}}\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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