Optimal. Leaf size=80 \[ \frac{a^2 \left (a+b \sqrt{x}\right )^{12}}{2 b^4}-\frac{2 a^3 \left (a+b \sqrt{x}\right )^{11}}{11 b^4}+\frac{\left (a+b \sqrt{x}\right )^{14}}{7 b^4}-\frac{6 a \left (a+b \sqrt{x}\right )^{13}}{13 b^4} \]
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Rubi [A] time = 0.045463, antiderivative size = 80, 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} \[ \frac{a^2 \left (a+b \sqrt{x}\right )^{12}}{2 b^4}-\frac{2 a^3 \left (a+b \sqrt{x}\right )^{11}}{11 b^4}+\frac{\left (a+b \sqrt{x}\right )^{14}}{7 b^4}-\frac{6 a \left (a+b \sqrt{x}\right )^{13}}{13 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^{10} x \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b x)^{10} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{10}}{b^3}+\frac{3 a^2 (a+b x)^{11}}{b^3}-\frac{3 a (a+b x)^{12}}{b^3}+\frac{(a+b x)^{13}}{b^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 a^3 \left (a+b \sqrt{x}\right )^{11}}{11 b^4}+\frac{a^2 \left (a+b \sqrt{x}\right )^{12}}{2 b^4}-\frac{6 a \left (a+b \sqrt{x}\right )^{13}}{13 b^4}+\frac{\left (a+b \sqrt{x}\right )^{14}}{7 b^4}\\ \end{align*}
Mathematica [A] time = 0.03694, size = 50, normalized size = 0.62 \[ -\frac{\left (a+b \sqrt{x}\right )^{11} \left (-11 a^2 b \sqrt{x}+a^3+66 a b^2 x-286 b^3 x^{3/2}\right )}{2002 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 113, normalized size = 1.4 \begin{align*}{\frac{{x}^{7}{b}^{10}}{7}}+{\frac{20\,a{b}^{9}}{13}{x}^{{\frac{13}{2}}}}+{\frac{15\,{x}^{6}{a}^{2}{b}^{8}}{2}}+{\frac{240\,{a}^{3}{b}^{7}}{11}{x}^{{\frac{11}{2}}}}+42\,{x}^{5}{a}^{4}{b}^{6}+56\,{x}^{9/2}{a}^{5}{b}^{5}+{\frac{105\,{x}^{4}{a}^{6}{b}^{4}}{2}}+{\frac{240\,{a}^{7}{b}^{3}}{7}{x}^{{\frac{7}{2}}}}+15\,{x}^{3}{a}^{8}{b}^{2}+4\,{x}^{5/2}{a}^{9}b+{\frac{{x}^{2}{a}^{10}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964791, size = 86, normalized size = 1.08 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{14}}{7 \, b^{4}} - \frac{6 \,{\left (b \sqrt{x} + a\right )}^{13} a}{13 \, b^{4}} + \frac{{\left (b \sqrt{x} + a\right )}^{12} a^{2}}{2 \, b^{4}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a^{3}}{11 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40067, size = 284, normalized size = 3.55 \begin{align*} \frac{1}{7} \, b^{10} x^{7} + \frac{15}{2} \, a^{2} b^{8} x^{6} + 42 \, a^{4} b^{6} x^{5} + \frac{105}{2} \, a^{6} b^{4} x^{4} + 15 \, a^{8} b^{2} x^{3} + \frac{1}{2} \, a^{10} x^{2} + \frac{4}{1001} \,{\left (385 \, a b^{9} x^{6} + 5460 \, a^{3} b^{7} x^{5} + 14014 \, a^{5} b^{5} x^{4} + 8580 \, a^{7} b^{3} x^{3} + 1001 \, a^{9} 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 = 2.51321, size = 136, normalized size = 1.7 \begin{align*} \frac{a^{10} x^{2}}{2} + 4 a^{9} b x^{\frac{5}{2}} + 15 a^{8} b^{2} x^{3} + \frac{240 a^{7} b^{3} x^{\frac{7}{2}}}{7} + \frac{105 a^{6} b^{4} x^{4}}{2} + 56 a^{5} b^{5} x^{\frac{9}{2}} + 42 a^{4} b^{6} x^{5} + \frac{240 a^{3} b^{7} x^{\frac{11}{2}}}{11} + \frac{15 a^{2} b^{8} x^{6}}{2} + \frac{20 a b^{9} x^{\frac{13}{2}}}{13} + \frac{b^{10} x^{7}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11323, size = 151, normalized size = 1.89 \begin{align*} \frac{1}{7} \, b^{10} x^{7} + \frac{20}{13} \, a b^{9} x^{\frac{13}{2}} + \frac{15}{2} \, a^{2} b^{8} x^{6} + \frac{240}{11} \, a^{3} b^{7} x^{\frac{11}{2}} + 42 \, a^{4} b^{6} x^{5} + 56 \, a^{5} b^{5} x^{\frac{9}{2}} + \frac{105}{2} \, a^{6} b^{4} x^{4} + \frac{240}{7} \, a^{7} b^{3} x^{\frac{7}{2}} + 15 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{10} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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