Optimal. Leaf size=34 \[ \frac {2 (a+b x)^{5/2}}{5 b^2}-\frac {2 a (a+b x)^{3/2}}{3 b^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2 (a+b x)^{5/2}}{5 b^2}-\frac {2 a (a+b x)^{3/2}}{3 b^2} \end {gather*}
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 = 0.01, size = 24, normalized size = 0.71 \begin {gather*} \frac {2 (a+b x)^{3/2} (3 b x-2 a)}{15 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.01, size = 35, normalized size = 1.03 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (2 a^2-a b x-3 b^2 x^2\right )}{15 b^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.14, size = 30, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt {b x + a}}{15 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.28, size = 66, normalized size = 1.94 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a}{b} + \frac {3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}}{b}\right )}}{15 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 21, normalized size = 0.62 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-3 b x +2 a \right )}{15 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.31, size = 26, normalized size = 0.76 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 25, normalized size = 0.74 \begin {gather*} -\frac {10\,a\,{\left (a+b\,x\right )}^{3/2}-6\,{\left (a+b\,x\right )}^{5/2}}{15\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.39, size = 202, normalized size = 5.94 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________