Optimal. Leaf size=80 \[ \frac{6 a^2 \left (a+b \sqrt{x}\right )^{17}}{17 b^4}-\frac{a^3 \left (a+b \sqrt{x}\right )^{16}}{8 b^4}+\frac{2 \left (a+b \sqrt{x}\right )^{19}}{19 b^4}-\frac{a \left (a+b \sqrt{x}\right )^{18}}{3 b^4} \]
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Rubi [A] time = 0.0574581, 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} \[ \frac{6 a^2 \left (a+b \sqrt{x}\right )^{17}}{17 b^4}-\frac{a^3 \left (a+b \sqrt{x}\right )^{16}}{8 b^4}+\frac{2 \left (a+b \sqrt{x}\right )^{19}}{19 b^4}-\frac{a \left (a+b \sqrt{x}\right )^{18}}{3 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^{15} x \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b x)^{15} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{15}}{b^3}+\frac{3 a^2 (a+b x)^{16}}{b^3}-\frac{3 a (a+b x)^{17}}{b^3}+\frac{(a+b x)^{18}}{b^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^3 \left (a+b \sqrt{x}\right )^{16}}{8 b^4}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^{17}}{17 b^4}-\frac{a \left (a+b \sqrt{x}\right )^{18}}{3 b^4}+\frac{2 \left (a+b \sqrt{x}\right )^{19}}{19 b^4}\\ \end{align*}
Mathematica [A] time = 0.0520202, size = 50, normalized size = 0.62 \[ -\frac{\left (a+b \sqrt{x}\right )^{16} \left (-16 a^2 b \sqrt{x}+a^3+136 a b^2 x-816 b^3 x^{3/2}\right )}{7752 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 168, normalized size = 2.1 \begin{align*}{\frac{2\,{b}^{15}}{19}{x}^{{\frac{19}{2}}}}+{\frac{5\,{x}^{9}a{b}^{14}}{3}}+{\frac{210\,{a}^{2}{b}^{13}}{17}{x}^{{\frac{17}{2}}}}+{\frac{455\,{x}^{8}{a}^{3}{b}^{12}}{8}}+182\,{x}^{15/2}{a}^{4}{b}^{11}+429\,{x}^{7}{a}^{5}{b}^{10}+770\,{x}^{13/2}{a}^{6}{b}^{9}+{\frac{2145\,{x}^{6}{a}^{7}{b}^{8}}{2}}+1170\,{x}^{11/2}{a}^{8}{b}^{7}+1001\,{x}^{5}{a}^{9}{b}^{6}+{\frac{2002\,{a}^{10}{b}^{5}}{3}{x}^{{\frac{9}{2}}}}+{\frac{1365\,{x}^{4}{a}^{11}{b}^{4}}{4}}+130\,{x}^{7/2}{a}^{12}{b}^{3}+35\,{x}^{3}{a}^{13}{b}^{2}+6\,{x}^{5/2}{a}^{14}b+{\frac{{x}^{2}{a}^{15}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962299, size = 86, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{19}}{19 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{18} a}{3 \, b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{17} a^{2}}{17 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a^{3}}{8 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.40549, size = 439, normalized size = 5.49 \begin{align*} \frac{5}{3} \, a b^{14} x^{9} + \frac{455}{8} \, a^{3} b^{12} x^{8} + 429 \, a^{5} b^{10} x^{7} + \frac{2145}{2} \, a^{7} b^{8} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac{1365}{4} \, a^{11} b^{4} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac{1}{2} \, a^{15} x^{2} + \frac{2}{969} \,{\left (51 \, b^{15} x^{9} + 5985 \, a^{2} b^{13} x^{8} + 88179 \, a^{4} b^{11} x^{7} + 373065 \, a^{6} b^{9} x^{6} + 566865 \, a^{8} b^{7} x^{5} + 323323 \, a^{10} b^{5} x^{4} + 62985 \, a^{12} b^{3} x^{3} + 2907 \, a^{14} b x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.22273, size = 204, normalized size = 2.55 \begin{align*} \frac{a^{15} x^{2}}{2} + 6 a^{14} b x^{\frac{5}{2}} + 35 a^{13} b^{2} x^{3} + 130 a^{12} b^{3} x^{\frac{7}{2}} + \frac{1365 a^{11} b^{4} x^{4}}{4} + \frac{2002 a^{10} b^{5} x^{\frac{9}{2}}}{3} + 1001 a^{9} b^{6} x^{5} + 1170 a^{8} b^{7} x^{\frac{11}{2}} + \frac{2145 a^{7} b^{8} x^{6}}{2} + 770 a^{6} b^{9} x^{\frac{13}{2}} + 429 a^{5} b^{10} x^{7} + 182 a^{4} b^{11} x^{\frac{15}{2}} + \frac{455 a^{3} b^{12} x^{8}}{8} + \frac{210 a^{2} b^{13} x^{\frac{17}{2}}}{17} + \frac{5 a b^{14} x^{9}}{3} + \frac{2 b^{15} x^{\frac{19}{2}}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10811, size = 225, normalized size = 2.81 \begin{align*} \frac{2}{19} \, b^{15} x^{\frac{19}{2}} + \frac{5}{3} \, a b^{14} x^{9} + \frac{210}{17} \, a^{2} b^{13} x^{\frac{17}{2}} + \frac{455}{8} \, a^{3} b^{12} x^{8} + 182 \, a^{4} b^{11} x^{\frac{15}{2}} + 429 \, a^{5} b^{10} x^{7} + 770 \, a^{6} b^{9} x^{\frac{13}{2}} + \frac{2145}{2} \, a^{7} b^{8} x^{6} + 1170 \, a^{8} b^{7} x^{\frac{11}{2}} + 1001 \, a^{9} b^{6} x^{5} + \frac{2002}{3} \, a^{10} b^{5} x^{\frac{9}{2}} + \frac{1365}{4} \, a^{11} b^{4} x^{4} + 130 \, a^{12} b^{3} x^{\frac{7}{2}} + 35 \, a^{13} b^{2} x^{3} + 6 \, a^{14} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{15} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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