Optimal. Leaf size=47 \[ \frac{6}{11} a^2 b x^{11/2}+\frac{a^3 x^5}{5}+\frac{1}{2} a b^2 x^6+\frac{2}{13} b^3 x^{13/2} \]
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Rubi [A] time = 0.0301109, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{6}{11} a^2 b x^{11/2}+\frac{a^3 x^5}{5}+\frac{1}{2} a b^2 x^6+\frac{2}{13} b^3 x^{13/2} \]
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^4 \, dx &=2 \operatorname{Subst}\left (\int x^9 (a+b x)^3 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^3 x^9+3 a^2 b x^{10}+3 a b^2 x^{11}+b^3 x^{12}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^3 x^5}{5}+\frac{6}{11} a^2 b x^{11/2}+\frac{1}{2} a b^2 x^6+\frac{2}{13} b^3 x^{13/2}\\ \end{align*}
Mathematica [A] time = 0.0225418, size = 47, normalized size = 1. \[ \frac{6}{11} a^2 b x^{11/2}+\frac{a^3 x^5}{5}+\frac{1}{2} a b^2 x^6+\frac{2}{13} b^3 x^{13/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{5}}{5}}+{\frac{6\,b{a}^{2}}{11}{x}^{{\frac{11}{2}}}}+{\frac{a{b}^{2}{x}^{6}}{2}}+{\frac{2\,{b}^{3}}{13}{x}^{{\frac{13}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00509, size = 224, normalized size = 4.77 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{13}}{13 \, b^{10}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{12} a}{2 \, b^{10}} + \frac{72 \,{\left (b \sqrt{x} + a\right )}^{11} a^{2}}{11 \, b^{10}} - \frac{84 \,{\left (b \sqrt{x} + a\right )}^{10} a^{3}}{5 \, b^{10}} + \frac{28 \,{\left (b \sqrt{x} + a\right )}^{9} a^{4}}{b^{10}} - \frac{63 \,{\left (b \sqrt{x} + a\right )}^{8} a^{5}}{2 \, b^{10}} + \frac{24 \,{\left (b \sqrt{x} + a\right )}^{7} a^{6}}{b^{10}} - \frac{12 \,{\left (b \sqrt{x} + a\right )}^{6} a^{7}}{b^{10}} + \frac{18 \,{\left (b \sqrt{x} + a\right )}^{5} a^{8}}{5 \, b^{10}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{9}}{2 \, b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47283, size = 99, normalized size = 2.11 \begin{align*} \frac{1}{2} \, a b^{2} x^{6} + \frac{1}{5} \, a^{3} x^{5} + \frac{2}{143} \,{\left (11 \, b^{3} x^{6} + 39 \, a^{2} b x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.78542, size = 42, normalized size = 0.89 \begin{align*} \frac{a^{3} x^{5}}{5} + \frac{6 a^{2} b x^{\frac{11}{2}}}{11} + \frac{a b^{2} x^{6}}{2} + \frac{2 b^{3} x^{\frac{13}{2}}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12035, size = 47, normalized size = 1. \begin{align*} \frac{2}{13} \, b^{3} x^{\frac{13}{2}} + \frac{1}{2} \, a b^{2} x^{6} + \frac{6}{11} \, a^{2} b x^{\frac{11}{2}} + \frac{1}{5} \, a^{3} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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