Optimal. Leaf size=67 \[ -\frac {1}{15} \left (3 x^2+2\right )^{5/2}+\frac {5}{4} x \left (3 x^2+2\right )^{3/2}+\frac {15}{4} x \sqrt {3 x^2+2}+\frac {5}{2} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {641, 195, 215} \begin {gather*} -\frac {1}{15} \left (3 x^2+2\right )^{5/2}+\frac {5}{4} x \left (3 x^2+2\right )^{3/2}+\frac {15}{4} x \sqrt {3 x^2+2}+\frac {5}{2} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 215
Rule 641
Rubi steps
\begin {align*} \int (5-x) \left (2+3 x^2\right )^{3/2} \, dx &=-\frac {1}{15} \left (2+3 x^2\right )^{5/2}+5 \int \left (2+3 x^2\right )^{3/2} \, dx\\ &=\frac {5}{4} x \left (2+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+3 x^2\right )^{5/2}+\frac {15}{2} \int \sqrt {2+3 x^2} \, dx\\ &=\frac {15}{4} x \sqrt {2+3 x^2}+\frac {5}{4} x \left (2+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+3 x^2\right )^{5/2}+\frac {15}{2} \int \frac {1}{\sqrt {2+3 x^2}} \, dx\\ &=\frac {15}{4} x \sqrt {2+3 x^2}+\frac {5}{4} x \left (2+3 x^2\right )^{3/2}-\frac {1}{15} \left (2+3 x^2\right )^{5/2}+\frac {5}{2} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 55, normalized size = 0.82 \begin {gather*} \frac {5}{2} \sqrt {3} \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )-\frac {1}{60} \sqrt {3 x^2+2} \left (36 x^4-225 x^3+48 x^2-375 x+16\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.19, size = 66, normalized size = 0.99 \begin {gather*} \frac {1}{60} \sqrt {3 x^2+2} \left (-36 x^4+225 x^3-48 x^2+375 x-16\right )-\frac {5}{2} \sqrt {3} \log \left (\sqrt {3 x^2+2}-\sqrt {3} x\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 60, normalized size = 0.90 \begin {gather*} -\frac {1}{60} \, {\left (36 \, x^{4} - 225 \, x^{3} + 48 \, x^{2} - 375 \, x + 16\right )} \sqrt {3 \, x^{2} + 2} + \frac {5}{4} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 53, normalized size = 0.79 \begin {gather*} -\frac {1}{60} \, {\left (3 \, {\left ({\left (3 \, {\left (4 \, x - 25\right )} x + 16\right )} x - 125\right )} x + 16\right )} \sqrt {3 \, x^{2} + 2} - \frac {5}{2} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 49, normalized size = 0.73 \begin {gather*} \frac {5 \left (3 x^{2}+2\right )^{\frac {3}{2}} x}{4}+\frac {15 \sqrt {3 x^{2}+2}\, x}{4}+\frac {5 \sqrt {3}\, \arcsinh \left (\frac {\sqrt {6}\, x}{2}\right )}{2}-\frac {\left (3 x^{2}+2\right )^{\frac {5}{2}}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.33, size = 48, normalized size = 0.72 \begin {gather*} -\frac {1}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {5}{4} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {15}{4} \, \sqrt {3 \, x^{2} + 2} x + \frac {5}{2} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 45, normalized size = 0.67 \begin {gather*} \frac {5\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{2}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {9\,x^4}{5}-\frac {45\,x^3}{4}+\frac {12\,x^2}{5}-\frac {75\,x}{4}+\frac {4}{5}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.76, size = 97, normalized size = 1.45 \begin {gather*} - \frac {3 x^{4} \sqrt {3 x^{2} + 2}}{5} + \frac {15 x^{3} \sqrt {3 x^{2} + 2}}{4} - \frac {4 x^{2} \sqrt {3 x^{2} + 2}}{5} + \frac {25 x \sqrt {3 x^{2} + 2}}{4} - \frac {4 \sqrt {3 x^{2} + 2}}{15} + \frac {5 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________