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 = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 15
Rule 30
Rule 247
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.80, size = 17, normalized size = 0.53 \[ \frac {1}{2} \, {\left (b c x^{2} + 2 \, a c x\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 15, normalized size = 0.47 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.91 \[ \frac {\left (b x +2 a \right ) \left (\left (b x +a \right )^{\frac {2}{3}} c \right )^{\frac {3}{2}} x}{2 b x +2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 15, normalized size = 0.47 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (c\,{\left (a+b\,x\right )}^{2/3}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.60, size = 65, normalized size = 2.03 \[ \begin {cases} \frac {2 a^{2} c^{\frac {3}{2}} x}{2 a + 2 b x} + \frac {3 a b c^{\frac {3}{2}} x^{2}}{2 a + 2 b x} + \frac {b^{2} c^{\frac {3}{2}} x^{3}}{2 a + 2 b x} & \text {for}\: a \neq 0 \vee b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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