Optimal. Leaf size=27 \[ \frac{(a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 b} \]
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Rubi [A] time = 0.0095091, 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} \[ \frac{(a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (c (a+b x)^{3/2}\right )^{2/3} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^{3/2}\right )^{2/3} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c (a+b x)^{3/2}\right )^{2/3} \operatorname{Subst}(\int x \, dx,x,a+b x)}{b (a+b x)}\\ &=\frac{(a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0151122, size = 34, normalized size = 1.26 \[ \frac{x (2 a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, 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 = 1.35201, size = 28, normalized size = 1.04 \begin{align*} \frac{\left ({\left (b x + a\right )}^{\frac{3}{2}} c\right )^{\frac{2}{3}}{\left (b x + a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8888, size = 90, normalized size = 3.33 \begin{align*} \frac{{\left (b x^{2} + 2 \, a x\right )} \left ({\left (b c x + a c\right )} \sqrt{b x + a}\right )^{\frac{2}{3}}}{2 \,{\left (b x + a\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 (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 [A] time = 1.11462, size = 20, normalized size = 0.74 \begin{align*} \frac{{\left (b x + a\right )}^{2} c^{\frac{2}{3}}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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