Optimal. Leaf size=38 \[ \frac{2 \left (a+b \sqrt{x}\right )^5}{5 b^2}-\frac{a \left (a+b \sqrt{x}\right )^4}{2 b^2} \]
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Rubi [A] time = 0.01534, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac{2 \left (a+b \sqrt{x}\right )^5}{5 b^2}-\frac{a \left (a+b \sqrt{x}\right )^4}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^3 \, dx &=2 \operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a \left (a+b \sqrt{x}\right )^4}{2 b^2}+\frac{2 \left (a+b \sqrt{x}\right )^5}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0157507, size = 28, normalized size = 0.74 \[ -\frac{\left (a-4 b \sqrt{x}\right ) \left (a+b \sqrt{x}\right )^4}{10 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 33, normalized size = 0.9 \begin{align*}{\frac{2\,{b}^{3}}{5}{x}^{{\frac{5}{2}}}}+{\frac{3\,a{b}^{2}{x}^{2}}{2}}+2\,{a}^{2}b{x}^{3/2}+{a}^{3}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976543, size = 43, normalized size = 1.13 \begin{align*} \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + \frac{3}{2} \, a b^{2} x^{2} + 2 \, a^{2} b x^{\frac{3}{2}} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49886, size = 80, normalized size = 2.11 \begin{align*} \frac{3}{2} \, a b^{2} x^{2} + a^{3} x + \frac{2}{5} \,{\left (b^{3} x^{2} + 5 \, a^{2} b x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.65707, size = 39, normalized size = 1.03 \begin{align*} a^{3} x + 2 a^{2} b x^{\frac{3}{2}} + \frac{3 a b^{2} x^{2}}{2} + \frac{2 b^{3} x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50075, size = 43, normalized size = 1.13 \begin{align*} \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + \frac{3}{2} \, a b^{2} x^{2} + 2 \, a^{2} b x^{\frac{3}{2}} + a^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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