Optimal. Leaf size=44 \[ \frac{6}{5} a^2 b x^{5/2}+\frac{a^3 x^2}{2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]
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Rubi [A] time = 0.0245843, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{6}{5} a^2 b x^{5/2}+\frac{a^3 x^2}{2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^3 x \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b x)^3 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^3 x^3+3 a^2 b x^4+3 a b^2 x^5+b^3 x^6\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^3 x^2}{2}+\frac{6}{5} a^2 b x^{5/2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0157025, size = 44, normalized size = 1. \[ \frac{6}{5} a^2 b x^{5/2}+\frac{a^3 x^2}{2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 35, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{6\,b{a}^{2}}{5}{x}^{{\frac{5}{2}}}}+{x}^{3}a{b}^{2}+{\frac{2\,{b}^{3}}{7}{x}^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.972826, size = 86, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7}}{7 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a}{b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{5} a^{2}}{5 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{3}}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45211, size = 90, normalized size = 2.05 \begin{align*} a b^{2} x^{3} + \frac{1}{2} \, a^{3} x^{2} + \frac{2}{35} \,{\left (5 \, b^{3} x^{3} + 21 \, a^{2} b x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.50303, size = 41, normalized size = 0.93 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + a b^{2} x^{3} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10067, size = 46, normalized size = 1.05 \begin{align*} \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + a b^{2} x^{3} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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