Optimal. Leaf size=32 \[ \frac{c (a+b x)^{5/3} \sqrt{c (a+b x)^{2/3}}}{2 b} \]
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Rubi [A] time = 0.0097607, antiderivative size = 32, 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} \[ \frac{c (a+b x)^{5/3} \sqrt{c (a+b x)^{2/3}}}{2 b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \left (c (a+b x)^{2/3}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (c x^{2/3}\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c \sqrt{c (a+b x)^{2/3}}\right ) \operatorname{Subst}(\int x \, dx,x,a+b x)}{b \sqrt [3]{a+b x}}\\ &=\frac{c \sqrt{c (a+b x)^{2/3}} (a+b x)^{5/3}}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0166755, size = 34, normalized size = 1.06 \[ \frac{x (2 a+b x) \left (c (a+b x)^{2/3}\right )^{3/2}}{2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 29, normalized size = 0.9 \begin{align*}{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a} \left ( c \left ( bx+a \right ) ^{{\frac{2}{3}}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06698, size = 20, normalized size = 0.62 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27893, size = 45, normalized size = 1.41 \begin{align*} \frac{1}{2} \,{\left (b c x^{2} + 2 \, a c x\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.06757, size = 109, normalized size = 3.41 \begin{align*} \begin{cases} \frac{2 a^{3} c^{\frac{3}{2}}}{2 a b + 2 b^{2} x} + \frac{4 a^{2} b c^{\frac{3}{2}} x}{2 a b + 2 b^{2} x} + \frac{3 a b^{2} c^{\frac{3}{2}} x^{2}}{2 a b + 2 b^{2} x} + \frac{b^{3} c^{\frac{3}{2}} x^{3}}{2 a b + 2 b^{2} x} & \text{for}\: b \neq 0 \\x \left (a^{\frac{2}{3}} c\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09666, size = 20, normalized size = 0.62 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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