Optimal. Leaf size=47 \[ -\frac{71 (1-4 x)}{529 \sqrt{2 x^2-x+3}}-\frac{11 (3 x+5)}{69 \left (2 x^2-x+3\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0223468, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1660, 12, 613} \[ -\frac{71 (1-4 x)}{529 \sqrt{2 x^2-x+3}}-\frac{11 (3 x+5)}{69 \left (2 x^2-x+3\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1660
Rule 12
Rule 613
Rubi steps
\begin{align*} \int \frac{2+3 x+5 x^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac{11 (5+3 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{213}{4 \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{11 (5+3 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{71}{46} \int \frac{1}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{11 (5+3 x)}{69 \left (3-x+2 x^2\right )^{3/2}}-\frac{71 (1-4 x)}{529 \sqrt{3-x+2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.104221, size = 33, normalized size = 0.7 \[ \frac{2 \left (852 x^3-639 x^2+1005 x-952\right )}{1587 \left (2 x^2-x+3\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 30, normalized size = 0.6 \begin{align*}{\frac{1704\,{x}^{3}-1278\,{x}^{2}+2010\,x-1904}{1587} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.07712, size = 80, normalized size = 1.7 \begin{align*} \frac{284 \, x}{529 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{71}{529 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{11 \, x}{23 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{55}{69 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.34092, size = 132, normalized size = 2.81 \begin{align*} \frac{2 \,{\left (852 \, x^{3} - 639 \, x^{2} + 1005 \, x - 952\right )} \sqrt{2 \, x^{2} - x + 3}}{1587 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 x^{2} + 3 x + 2}{\left (2 x^{2} - x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15732, size = 39, normalized size = 0.83 \begin{align*} \frac{2 \,{\left (3 \,{\left (71 \,{\left (4 \, x - 3\right )} x + 335\right )} x - 952\right )}}{1587 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]