Optimal. Leaf size=145 \[ -\frac{x \left (x^2+2\right )}{2 \sqrt{x^4+3 x^2+2}}+\frac{x \left (x^2+5\right )}{2 \sqrt{x^4+3 x^2+2}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.107024, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{x \left (x^2+2\right )}{2 \sqrt{x^4+3 x^2+2}}+\frac{x \left (x^2+5\right )}{2 \sqrt{x^4+3 x^2+2}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In] Int[(7 + 5*x^2)/(2 + 3*x^2 + x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.2562, size = 133, normalized size = 0.92 \[ \frac{x \left (x^{2} + 5\right )}{2 \sqrt{x^{4} + 3 x^{2} + 2}} - \frac{x \left (2 x^{2} + 4\right )}{4 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{\sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{8 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{\sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{8 \sqrt{x^{4} + 3 x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**2+7)/(x**4+3*x**2+2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0709652, size = 97, normalized size = 0.67 \[ \frac{x^3-3 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+5 x}{2 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(7 + 5*x^2)/(2 + 3*x^2 + x^4)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.008, size = 150, normalized size = 1. \[ -14\,{\frac{-3/4\,{x}^{3}-5/4\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}-{{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{i}{4}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-10\,{\frac{{x}^{3}+3/2\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^2+7)/(x^4+3*x^2+2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{5 \, x^{2} + 7}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 7)/(x^4 + 3*x^2 + 2)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{5 \, x^{2} + 7}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 7)/(x^4 + 3*x^2 + 2)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{5 x^{2} + 7}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**2+7)/(x**4+3*x**2+2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{5 \, x^{2} + 7}{{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^2 + 7)/(x^4 + 3*x^2 + 2)^(3/2),x, algorithm="giac")
[Out]