Optimal. Leaf size=210 \[ \frac{7540}{81} \sqrt{3 x^2+5 x+2} \sqrt{x}-\frac{17512 (3 x+2) \sqrt{x}}{243 \sqrt{3 x^2+5 x+2}}-\frac{7540 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}+\frac{17512 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{4 (645 x+536) x^{3/2}}{9 \sqrt{3 x^2+5 x+2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.366505, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{7540}{81} \sqrt{3 x^2+5 x+2} \sqrt{x}-\frac{17512 (3 x+2) \sqrt{x}}{243 \sqrt{3 x^2+5 x+2}}-\frac{7540 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}+\frac{17512 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{4 (645 x+536) x^{3/2}}{9 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In] Int[((2 - 5*x)*x^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 39.298, size = 194, normalized size = 0.92 \[ \frac{2 x^{\frac{7}{2}} \left (95 x + 74\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{4 x^{\frac{3}{2}} \left (1935 x + 1608\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} - \frac{8756 \sqrt{x} \left (6 x + 4\right )}{243 \sqrt{3 x^{2} + 5 x + 2}} + \frac{7540 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}}{81} + \frac{4378 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{243 \sqrt{3 x^{2} + 5 x + 2}} - \frac{1885 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2-5*x)*x**(9/2)/(3*x**2+5*x+2)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.402984, size = 177, normalized size = 0.84 \[ \frac{-5108 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-17512 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (135 x^5-1512 x^4+58590 x^3+155660 x^2+129880 x+35024\right )}{243 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 - 5*x)*x^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.034, size = 320, normalized size = 1.5 \[{\frac{2}{729\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 5472\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-13134\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+9120\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-21890\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+3648\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -8756\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -405\,{x}^{5}+240948\,{x}^{4}+612270\,{x}^{3}+504936\,{x}^{2}+135720\,x \right ){\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2-5*x)*x^(9/2)/(3*x^2+5*x+2)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (5 \, x - 2\right )} x^{\frac{9}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (5 \, x^{5} - 2 \, x^{4}\right )} \sqrt{x}}{{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2-5*x)*x**(9/2)/(3*x**2+5*x+2)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (5 \, x - 2\right )} x^{\frac{9}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="giac")
[Out]