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 = 0.01, antiderivative size = 27, 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} \[ \frac {(a+b x) \left (c (a+b x)^{3/2}\right )^{2/3}}{2 b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \left (c (a+b x)^{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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.86, size = 37, normalized size = 1.37 \[ \frac {{\left (b x^{2} + 2 \, a x\right )} \left ({\left (b c x + a c\right )} \sqrt {b x + a}\right )^{\frac {2}{3}}}{2 \, {\left (b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 15, normalized size = 0.56 \[ \frac {{\left (b x + a\right )}^{2} c^{\frac {2}{3}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.07 \[ \frac {\left (b x +2 a \right ) \left (\left (b x +a \right )^{\frac {3}{2}} c \right )^{\frac {2}{3}} x}{2 b x +2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 21, normalized size = 0.78 \[ \frac {\left ({\left (b x + a\right )}^{\frac {3}{2}} c\right )^{\frac {2}{3}} {\left (b x + a\right )}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (c\,{\left (a+b\,x\right )}^{3/2}\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (a + b x\right )^{\frac {3}{2}}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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