Optimal. Leaf size=38 \[ \frac{2 \left (a+b \sqrt{x}\right )^{17}}{17 b^2}-\frac{a \left (a+b \sqrt{x}\right )^{16}}{8 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.013963, 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} \[ \frac{2 \left (a+b \sqrt{x}\right )^{17}}{17 b^2}-\frac{a \left (a+b \sqrt{x}\right )^{16}}{8 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^{15} \, dx &=2 \operatorname{Subst}\left (\int x (a+b x)^{15} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{15}}{b}+\frac{(a+b x)^{16}}{b}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a \left (a+b \sqrt{x}\right )^{16}}{8 b^2}+\frac{2 \left (a+b \sqrt{x}\right )^{17}}{17 b^2}\\ \end{align*}
Mathematica [A] time = 0.0406283, size = 28, normalized size = 0.74 \[ -\frac{\left (a-16 b \sqrt{x}\right ) \left (a+b \sqrt{x}\right )^{16}}{136 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.003, size = 165, normalized size = 4.3 \begin{align*}{\frac{2\,{b}^{15}}{17}{x}^{{\frac{17}{2}}}}+{\frac{15\,{x}^{8}a{b}^{14}}{8}}+14\,{x}^{15/2}{a}^{2}{b}^{13}+65\,{x}^{7}{a}^{3}{b}^{12}+210\,{x}^{13/2}{a}^{4}{b}^{11}+{\frac{1001\,{x}^{6}{a}^{5}{b}^{10}}{2}}+910\,{x}^{11/2}{a}^{6}{b}^{9}+1287\,{x}^{5}{a}^{7}{b}^{8}+1430\,{x}^{9/2}{a}^{8}{b}^{7}+{\frac{5005\,{x}^{4}{a}^{9}{b}^{6}}{4}}+858\,{x}^{7/2}{a}^{10}{b}^{5}+455\,{x}^{3}{a}^{11}{b}^{4}+182\,{x}^{5/2}{a}^{12}{b}^{3}+{\frac{105\,{x}^{2}{a}^{13}{b}^{2}}{2}}+10\,{x}^{3/2}{a}^{14}b+x{a}^{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.961453, size = 41, normalized size = 1.08 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{17}}{17 \, b^{2}} - \frac{{\left (b \sqrt{x} + a\right )}^{16} a}{8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.25577, size = 410, normalized size = 10.79 \begin{align*} \frac{15}{8} \, a b^{14} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac{1001}{2} \, a^{5} b^{10} x^{6} + 1287 \, a^{7} b^{8} x^{5} + \frac{5005}{4} \, a^{9} b^{6} x^{4} + 455 \, a^{11} b^{4} x^{3} + \frac{105}{2} \, a^{13} b^{2} x^{2} + a^{15} x + \frac{2}{17} \,{\left (b^{15} x^{8} + 119 \, a^{2} b^{13} x^{7} + 1785 \, a^{4} b^{11} x^{6} + 7735 \, a^{6} b^{9} x^{5} + 12155 \, a^{8} b^{7} x^{4} + 7293 \, a^{10} b^{5} x^{3} + 1547 \, a^{12} b^{3} x^{2} + 85 \, a^{14} b x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 4.50528, size = 197, normalized size = 5.18 \begin{align*} a^{15} x + 10 a^{14} b x^{\frac{3}{2}} + \frac{105 a^{13} b^{2} x^{2}}{2} + 182 a^{12} b^{3} x^{\frac{5}{2}} + 455 a^{11} b^{4} x^{3} + 858 a^{10} b^{5} x^{\frac{7}{2}} + \frac{5005 a^{9} b^{6} x^{4}}{4} + 1430 a^{8} b^{7} x^{\frac{9}{2}} + 1287 a^{7} b^{8} x^{5} + 910 a^{6} b^{9} x^{\frac{11}{2}} + \frac{1001 a^{5} b^{10} x^{6}}{2} + 210 a^{4} b^{11} x^{\frac{13}{2}} + 65 a^{3} b^{12} x^{7} + 14 a^{2} b^{13} x^{\frac{15}{2}} + \frac{15 a b^{14} x^{8}}{8} + \frac{2 b^{15} x^{\frac{17}{2}}}{17} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.09959, size = 221, normalized size = 5.82 \begin{align*} \frac{2}{17} \, b^{15} x^{\frac{17}{2}} + \frac{15}{8} \, a b^{14} x^{8} + 14 \, a^{2} b^{13} x^{\frac{15}{2}} + 65 \, a^{3} b^{12} x^{7} + 210 \, a^{4} b^{11} x^{\frac{13}{2}} + \frac{1001}{2} \, a^{5} b^{10} x^{6} + 910 \, a^{6} b^{9} x^{\frac{11}{2}} + 1287 \, a^{7} b^{8} x^{5} + 1430 \, a^{8} b^{7} x^{\frac{9}{2}} + \frac{5005}{4} \, a^{9} b^{6} x^{4} + 858 \, a^{10} b^{5} x^{\frac{7}{2}} + 455 \, a^{11} b^{4} x^{3} + 182 \, a^{12} b^{3} x^{\frac{5}{2}} + \frac{105}{2} \, a^{13} b^{2} x^{2} + 10 \, a^{14} b x^{\frac{3}{2}} + a^{15} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]