Optimal. Leaf size=34 \[ \frac{2 (a+b x)^{5/2}}{5 b^2}-\frac{2 a (a+b x)^{3/2}}{3 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0086508, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2 (a+b x)^{5/2}}{5 b^2}-\frac{2 a (a+b x)^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int x \sqrt{a+b x} \, dx &=\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx\\ &=-\frac{2 a (a+b x)^{3/2}}{3 b^2}+\frac{2 (a+b x)^{5/2}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0113039, size = 24, normalized size = 0.71 \[ \frac{2 (a+b x)^{3/2} (3 b x-2 a)}{15 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-6\,bx+4\,a}{15\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.954078, size = 35, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, b^{2}} - \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}} a}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6055, size = 70, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.04818, size = 202, normalized size = 5.94 \begin{align*} - \frac{4 a^{\frac{9}{2}} \sqrt{1 + \frac{b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac{4 a^{\frac{9}{2}}}{15 a^{2} b^{2} + 15 a b^{3} x} - \frac{2 a^{\frac{7}{2}} b x \sqrt{1 + \frac{b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac{4 a^{\frac{7}{2}} b x}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac{8 a^{\frac{5}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac{6 a^{\frac{3}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.05599, size = 34, normalized size = 1. \begin{align*} \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )}}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]