Optimal. Leaf size=162 \[ \frac{21 a^2 \left (a+b \sqrt{x}\right )^{16}}{8 b^8}-\frac{14 a^3 \left (a+b \sqrt{x}\right )^{15}}{3 b^8}+\frac{5 a^4 \left (a+b \sqrt{x}\right )^{14}}{b^8}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{13}}{13 b^8}+\frac{7 a^6 \left (a+b \sqrt{x}\right )^{12}}{6 b^8}-\frac{2 a^7 \left (a+b \sqrt{x}\right )^{11}}{11 b^8}+\frac{\left (a+b \sqrt{x}\right )^{18}}{9 b^8}-\frac{14 a \left (a+b \sqrt{x}\right )^{17}}{17 b^8} \]
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Rubi [A] time = 0.0804328, antiderivative size = 162, 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} \[ \frac{21 a^2 \left (a+b \sqrt{x}\right )^{16}}{8 b^8}-\frac{14 a^3 \left (a+b \sqrt{x}\right )^{15}}{3 b^8}+\frac{5 a^4 \left (a+b \sqrt{x}\right )^{14}}{b^8}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{13}}{13 b^8}+\frac{7 a^6 \left (a+b \sqrt{x}\right )^{12}}{6 b^8}-\frac{2 a^7 \left (a+b \sqrt{x}\right )^{11}}{11 b^8}+\frac{\left (a+b \sqrt{x}\right )^{18}}{9 b^8}-\frac{14 a \left (a+b \sqrt{x}\right )^{17}}{17 b^8} \]
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^3 \, dx &=2 \operatorname{Subst}\left (\int x^7 (a+b x)^{10} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^7 (a+b x)^{10}}{b^7}+\frac{7 a^6 (a+b x)^{11}}{b^7}-\frac{21 a^5 (a+b x)^{12}}{b^7}+\frac{35 a^4 (a+b x)^{13}}{b^7}-\frac{35 a^3 (a+b x)^{14}}{b^7}+\frac{21 a^2 (a+b x)^{15}}{b^7}-\frac{7 a (a+b x)^{16}}{b^7}+\frac{(a+b x)^{17}}{b^7}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 a^7 \left (a+b \sqrt{x}\right )^{11}}{11 b^8}+\frac{7 a^6 \left (a+b \sqrt{x}\right )^{12}}{6 b^8}-\frac{42 a^5 \left (a+b \sqrt{x}\right )^{13}}{13 b^8}+\frac{5 a^4 \left (a+b \sqrt{x}\right )^{14}}{b^8}-\frac{14 a^3 \left (a+b \sqrt{x}\right )^{15}}{3 b^8}+\frac{21 a^2 \left (a+b \sqrt{x}\right )^{16}}{8 b^8}-\frac{14 a \left (a+b \sqrt{x}\right )^{17}}{17 b^8}+\frac{\left (a+b \sqrt{x}\right )^{18}}{9 b^8}\\ \end{align*}
Mathematica [A] time = 0.0514132, size = 136, normalized size = 0.84 \[ 9 a^8 b^2 x^5+\frac{240}{11} a^7 b^3 x^{11/2}+35 a^6 b^4 x^6+\frac{504}{13} a^5 b^5 x^{13/2}+30 a^4 b^6 x^7+16 a^3 b^7 x^{15/2}+\frac{45}{8} a^2 b^8 x^8+\frac{20}{9} a^9 b x^{9/2}+\frac{a^{10} x^4}{4}+\frac{20}{17} a b^9 x^{17/2}+\frac{b^{10} x^9}{9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 113, normalized size = 0.7 \begin{align*}{\frac{{x}^{9}{b}^{10}}{9}}+{\frac{20\,a{b}^{9}}{17}{x}^{{\frac{17}{2}}}}+{\frac{45\,{x}^{8}{a}^{2}{b}^{8}}{8}}+16\,{x}^{15/2}{a}^{3}{b}^{7}+30\,{a}^{4}{b}^{6}{x}^{7}+{\frac{504\,{a}^{5}{b}^{5}}{13}{x}^{{\frac{13}{2}}}}+35\,{x}^{6}{a}^{6}{b}^{4}+{\frac{240\,{a}^{7}{b}^{3}}{11}{x}^{{\frac{11}{2}}}}+9\,{x}^{5}{a}^{8}{b}^{2}+{\frac{20\,{a}^{9}b}{9}{x}^{{\frac{9}{2}}}}+{\frac{{a}^{10}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966325, size = 178, normalized size = 1.1 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{18}}{9 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{17} a}{17 \, b^{8}} + \frac{21 \,{\left (b \sqrt{x} + a\right )}^{16} a^{2}}{8 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{15} a^{3}}{3 \, b^{8}} + \frac{5 \,{\left (b \sqrt{x} + a\right )}^{14} a^{4}}{b^{8}} - \frac{42 \,{\left (b \sqrt{x} + a\right )}^{13} a^{5}}{13 \, b^{8}} + \frac{7 \,{\left (b \sqrt{x} + a\right )}^{12} a^{6}}{6 \, b^{8}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11} a^{7}}{11 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55635, size = 288, normalized size = 1.78 \begin{align*} \frac{1}{9} \, b^{10} x^{9} + \frac{45}{8} \, a^{2} b^{8} x^{8} + 30 \, a^{4} b^{6} x^{7} + 35 \, a^{6} b^{4} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac{1}{4} \, a^{10} x^{4} + \frac{4}{21879} \,{\left (6435 \, a b^{9} x^{8} + 87516 \, a^{3} b^{7} x^{7} + 212058 \, a^{5} b^{5} x^{6} + 119340 \, a^{7} b^{3} x^{5} + 12155 \, a^{9} b x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.22995, size = 136, normalized size = 0.84 \begin{align*} \frac{a^{10} x^{4}}{4} + \frac{20 a^{9} b x^{\frac{9}{2}}}{9} + 9 a^{8} b^{2} x^{5} + \frac{240 a^{7} b^{3} x^{\frac{11}{2}}}{11} + 35 a^{6} b^{4} x^{6} + \frac{504 a^{5} b^{5} x^{\frac{13}{2}}}{13} + 30 a^{4} b^{6} x^{7} + 16 a^{3} b^{7} x^{\frac{15}{2}} + \frac{45 a^{2} b^{8} x^{8}}{8} + \frac{20 a b^{9} x^{\frac{17}{2}}}{17} + \frac{b^{10} x^{9}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12067, size = 151, normalized size = 0.93 \begin{align*} \frac{1}{9} \, b^{10} x^{9} + \frac{20}{17} \, a b^{9} x^{\frac{17}{2}} + \frac{45}{8} \, a^{2} b^{8} x^{8} + 16 \, a^{3} b^{7} x^{\frac{15}{2}} + 30 \, a^{4} b^{6} x^{7} + \frac{504}{13} \, a^{5} b^{5} x^{\frac{13}{2}} + 35 \, a^{6} b^{4} x^{6} + \frac{240}{11} \, a^{7} b^{3} x^{\frac{11}{2}} + 9 \, a^{8} b^{2} x^{5} + \frac{20}{9} \, a^{9} b x^{\frac{9}{2}} + \frac{1}{4} \, a^{10} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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