Optimal. Leaf size=31 \[ \frac {1}{48} \left (4 x^2+9\right )^{3/2}-\frac {9}{16} \sqrt {4 x^2+9} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {1}{48} \left (4 x^2+9\right )^{3/2}-\frac {9}{16} \sqrt {4 x^2+9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {9+4 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {9+4 x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {9}{4 \sqrt {9+4 x}}+\frac {1}{4} \sqrt {9+4 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {9}{16} \sqrt {9+4 x^2}+\frac {1}{48} \left (9+4 x^2\right )^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 22, normalized size = 0.71 \[ \frac {1}{24} \left (2 x^2-9\right ) \sqrt {4 x^2+9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 18, normalized size = 0.58 \[ \frac {1}{24} \, \sqrt {4 \, x^{2} + 9} {\left (2 \, x^{2} - 9\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.13, size = 23, normalized size = 0.74 \[ \frac {1}{48} \, {\left (4 \, x^{2} + 9\right )}^{\frac {3}{2}} - \frac {9}{16} \, \sqrt {4 \, x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 19, normalized size = 0.61 \[ \frac {\sqrt {4 x^{2}+9}\, \left (2 x^{2}-9\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.91, size = 26, normalized size = 0.84 \[ \frac {1}{12} \, \sqrt {4 \, x^{2} + 9} x^{2} - \frac {3}{8} \, \sqrt {4 \, x^{2} + 9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.02, size = 15, normalized size = 0.48 \[ \sqrt {x^2+\frac {9}{4}}\,\left (\frac {x^2}{6}-\frac {3}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.38, size = 27, normalized size = 0.87 \[ \frac {x^{2} \sqrt {4 x^{2} + 9}}{12} - \frac {3 \sqrt {4 x^{2} + 9}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________