Optimal. Leaf size=53 \[ -\frac {4}{567} (3 x+2)^{7/2}+\frac {8}{135} (3 x+2)^{5/2}-\frac {10}{81} (3 x+2)^{3/2}-\frac {4}{81} \sqrt {3 x+2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1593, 772} \[ -\frac {4}{567} (3 x+2)^{7/2}+\frac {8}{135} (3 x+2)^{5/2}-\frac {10}{81} (3 x+2)^{3/2}-\frac {4}{81} \sqrt {3 x+2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 772
Rule 1593
Rubi steps
\begin {align*} \int \frac {x-2 x^3}{\sqrt {2+3 x}} \, dx &=\int \frac {x \left (1-2 x^2\right )}{\sqrt {2+3 x}} \, dx\\ &=\int \left (-\frac {2}{27 \sqrt {2+3 x}}-\frac {5}{9} \sqrt {2+3 x}+\frac {4}{9} (2+3 x)^{3/2}-\frac {2}{27} (2+3 x)^{5/2}\right ) \, dx\\ &=-\frac {4}{81} \sqrt {2+3 x}-\frac {10}{81} (2+3 x)^{3/2}+\frac {8}{135} (2+3 x)^{5/2}-\frac {4}{567} (2+3 x)^{7/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 28, normalized size = 0.53 \[ -\frac {2 \sqrt {3 x+2} \left (270 x^3-216 x^2-123 x+164\right )}{2835} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 24, normalized size = 0.45 \[ -\frac {2}{2835} \, {\left (270 \, x^{3} - 216 \, x^{2} - 123 \, x + 164\right )} \sqrt {3 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 37, normalized size = 0.70 \[ -\frac {4}{567} \, {\left (3 \, x + 2\right )}^{\frac {7}{2}} + \frac {8}{135} \, {\left (3 \, x + 2\right )}^{\frac {5}{2}} - \frac {10}{81} \, {\left (3 \, x + 2\right )}^{\frac {3}{2}} - \frac {4}{81} \, \sqrt {3 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 25, normalized size = 0.47 \[ -\frac {2 \left (270 x^{3}-216 x^{2}-123 x +164\right ) \sqrt {3 x +2}}{2835} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.58, size = 37, normalized size = 0.70 \[ -\frac {4}{567} \, {\left (3 \, x + 2\right )}^{\frac {7}{2}} + \frac {8}{135} \, {\left (3 \, x + 2\right )}^{\frac {5}{2}} - \frac {10}{81} \, {\left (3 \, x + 2\right )}^{\frac {3}{2}} - \frac {4}{81} \, \sqrt {3 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 37, normalized size = 0.70 \[ \frac {8\,{\left (3\,x+2\right )}^{5/2}}{135}-\frac {10\,{\left (3\,x+2\right )}^{3/2}}{81}-\frac {4\,\sqrt {3\,x+2}}{81}-\frac {4\,{\left (3\,x+2\right )}^{7/2}}{567} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 11.73, size = 46, normalized size = 0.87 \[ - \frac {4 \left (3 x + 2\right )^{\frac {7}{2}}}{567} + \frac {8 \left (3 x + 2\right )^{\frac {5}{2}}}{135} - \frac {10 \left (3 x + 2\right )^{\frac {3}{2}}}{81} - \frac {4 \sqrt {3 x + 2}}{81} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________