Optimal. Leaf size=34 \[ \frac{(a+b x)^{5/3}}{2 b c \sqrt{\frac{c}{(a+b x)^{2/3}}}} \]
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Rubi [A] time = 0.0114691, antiderivative size = 34, 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} \[ \frac{(a+b x)^{5/3}}{2 b c \sqrt{\frac{c}{(a+b x)^{2/3}}}} \]
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)^{2/3}}\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (\frac{c}{x^{2/3}}\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}(\int x \, dx,x,a+b x)}{b c \sqrt{\frac{c}{(a+b x)^{2/3}}} \sqrt [3]{a+b x}}\\ &=\frac{(a+b x)^{5/3}}{2 b c \sqrt{\frac{c}{(a+b x)^{2/3}}}}\\ \end{align*}
Mathematica [A] time = 0.0210627, size = 34, normalized size = 1. \[ \frac{x (2 a+b x)}{2 (a+b x) \left (\frac{c}{(a+b x)^{2/3}}\right )^{3/2}} \]
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 = 0.954062, size = 20, normalized size = 0.59 \begin{align*} \frac{b x^{2} + 2 \, a x}{2 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50003, size = 39, normalized size = 1.15 \begin{align*} \frac{b x^{2} + 2 \, a x}{2 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.60487, size = 76, normalized size = 2.24 \begin{align*} \begin{cases} \frac{2 a x}{\frac{2 a c^{\frac{3}{2}}}{a + b x} + \frac{2 b c^{\frac{3}{2}} x}{a + b x}} + \frac{b x^{2}}{\frac{2 a c^{\frac{3}{2}}}{a + b x} + \frac{2 b c^{\frac{3}{2}} x}{a + b x}} & \text{for}\: a \neq 0 \vee b \neq 0 \\\frac{x}{\left (\tilde{\infty } 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 = 1.13091, size = 20, normalized size = 0.59 \begin{align*} \frac{b x^{2} + 2 \, a x}{2 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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