Optimal. Leaf size=25 \[ \frac{2 (a+b x) \sqrt{c (a+b x)^3}}{5 b} \]
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Rubi [A] time = 0.0112732, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac{2 (a+b x) \sqrt{c (a+b x)^3}}{5 b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \sqrt{c (a+b x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{c x^3} \, dx,x,a+b x\right )}{b}\\ &=\frac{\sqrt{c (a+b x)^3} \operatorname{Subst}\left (\int x^{3/2} \, dx,x,a+b x\right )}{b (a+b x)^{3/2}}\\ &=\frac{2 (a+b x) \sqrt{c (a+b x)^3}}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0072227, size = 25, normalized size = 1. \[ \frac{2 (a+b x) \sqrt{c (a+b x)^3}}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 0.9 \begin{align*}{\frac{2\,bx+2\,a}{5\,b}\sqrt{c \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01085, size = 32, normalized size = 1.28 \begin{align*} \frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.25113, size = 96, normalized size = 3.84 \begin{align*} \frac{2 \, \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}{\left (b x + a\right )}}{5 \, b} \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 \sqrt{c \left (a + b x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13379, size = 89, normalized size = 3.56 \begin{align*} \frac{2 \,{\left (5 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a \mathrm{sgn}\left (b x + a\right ) - \frac{{\left (5 \,{\left (b c x + a c\right )}^{\frac{3}{2}} a c - 3 \,{\left (b c x + a c\right )}^{\frac{5}{2}}\right )} \mathrm{sgn}\left (b x + a\right )}{c}\right )}}{15 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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