3.398 \(\int \frac{\left (d+e x^2\right )^3}{\sqrt{2+3 x^2+x^4}} \, dx\)

Optimal. Leaf size=229 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (5 d^3-10 d e^2+8 e^3\right ) F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{3 e \left (x^2+2\right ) x \left (5 d^2-10 d e+6 e^2\right )}{5 \sqrt{x^4+3 x^2+2}}-\frac{3 \sqrt{2} e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (5 d^2-10 d e+6 e^2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{x^4+3 x^2+2}}+\frac{1}{5} e^2 \sqrt{x^4+3 x^2+2} x (5 d-4 e)+\frac{1}{5} e^3 \sqrt{x^4+3 x^2+2} x^3 \]

[Out]

(3*e*(5*d^2 - 10*d*e + 6*e^2)*x*(2 + x^2))/(5*Sqrt[2 + 3*x^2 + x^4]) + ((5*d - 4
*e)*e^2*x*Sqrt[2 + 3*x^2 + x^4])/5 + (e^3*x^3*Sqrt[2 + 3*x^2 + x^4])/5 - (3*Sqrt
[2]*e*(5*d^2 - 10*d*e + 6*e^2)*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[Arc
Tan[x], 1/2])/(5*Sqrt[2 + 3*x^2 + x^4]) + ((5*d^3 - 10*d*e^2 + 8*e^3)*(1 + x^2)*
Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(5*Sqrt[2]*Sqrt[2 + 3*x^2 +
x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.294469, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (5 d^3-10 d e^2+8 e^3\right ) F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{3 e \left (x^2+2\right ) x \left (5 d^2-10 d e+6 e^2\right )}{5 \sqrt{x^4+3 x^2+2}}-\frac{3 \sqrt{2} e \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \left (5 d^2-10 d e+6 e^2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{x^4+3 x^2+2}}+\frac{1}{5} e^2 \sqrt{x^4+3 x^2+2} x (5 d-4 e)+\frac{1}{5} e^3 \sqrt{x^4+3 x^2+2} x^3 \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^2)^3/Sqrt[2 + 3*x^2 + x^4],x]

[Out]

(3*e*(5*d^2 - 10*d*e + 6*e^2)*x*(2 + x^2))/(5*Sqrt[2 + 3*x^2 + x^4]) + ((5*d - 4
*e)*e^2*x*Sqrt[2 + 3*x^2 + x^4])/5 + (e^3*x^3*Sqrt[2 + 3*x^2 + x^4])/5 - (3*Sqrt
[2]*e*(5*d^2 - 10*d*e + 6*e^2)*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[Arc
Tan[x], 1/2])/(5*Sqrt[2 + 3*x^2 + x^4]) + ((5*d^3 - 10*d*e^2 + 8*e^3)*(1 + x^2)*
Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(5*Sqrt[2]*Sqrt[2 + 3*x^2 +
x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 54.6225, size = 211, normalized size = 0.92 \[ \frac{e^{3} x^{3} \sqrt{x^{4} + 3 x^{2} + 2}}{5} + e^{2} x \left (d - \frac{4 e}{5}\right ) \sqrt{x^{4} + 3 x^{2} + 2} + \frac{3 e x \left (2 x^{2} + 4\right ) \left (5 d^{2} - 10 d e + 6 e^{2}\right )}{10 \sqrt{x^{4} + 3 x^{2} + 2}} - \frac{3 e \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) \left (5 d^{2} - 10 d e + 6 e^{2}\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{20 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{\sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) \left (\frac{d^{3}}{8} - \frac{d e^{2}}{4} + \frac{e^{3}}{5}\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d)**3/(x**4+3*x**2+2)**(1/2),x)

[Out]

e**3*x**3*sqrt(x**4 + 3*x**2 + 2)/5 + e**2*x*(d - 4*e/5)*sqrt(x**4 + 3*x**2 + 2)
+ 3*e*x*(2*x**2 + 4)*(5*d**2 - 10*d*e + 6*e**2)/(10*sqrt(x**4 + 3*x**2 + 2)) -
3*e*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*(5*d**2 - 10*d*e + 6*e**2)*ellipt
ic_e(atan(x), 1/2)/(20*sqrt(x**4 + 3*x**2 + 2)) + sqrt((2*x**2 + 4)/(x**2 + 1))*
(4*x**2 + 4)*(d**3/8 - d*e**2/4 + e**3/5)*elliptic_f(atan(x), 1/2)/sqrt(x**4 + 3
*x**2 + 2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.304077, size = 154, normalized size = 0.67 \[ \frac{-3 i e \sqrt{x^2+1} \sqrt{x^2+2} \left (5 d^2-10 d e+6 e^2\right ) E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-5 i \sqrt{x^2+1} \sqrt{x^2+2} \left (d^3-3 d^2 e+4 d e^2-2 e^3\right ) F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+e^2 x \left (x^4+3 x^2+2\right ) \left (5 d+e \left (x^2-4\right )\right )}{5 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^2)^3/Sqrt[2 + 3*x^2 + x^4],x]

[Out]

(e^2*x*(2 + 3*x^2 + x^4)*(5*d + e*(-4 + x^2)) - (3*I)*e*(5*d^2 - 10*d*e + 6*e^2)
*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (5*I)*(d^3 - 3
*d^2*e + 4*d*e^2 - 2*e^3)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt
[2]], 2])/(5*Sqrt[2 + 3*x^2 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.014, size = 380, normalized size = 1.7 \[{-{\frac{i}{2}}{d}^{3}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{e}^{3} \left ({\frac{{x}^{3}}{5}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{\frac{4\,x}{5}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{4\,i}{5}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{9\,i}{5}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \right ) +{{\frac{3\,i}{2}}{d}^{2}e\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+3\,{e}^{2}d \left ( 1/3\,x\sqrt{{x}^{4}+3\,{x}^{2}+2}+{\frac{i/3\sqrt{2}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ( i/2\sqrt{2}x,\sqrt{2} \right ) }{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}-{\frac{i\sqrt{2}\sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ( i/2\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ( i/2\sqrt{2}x,\sqrt{2} \right ) \right ) }{\sqrt{{x}^{4}+3\,{x}^{2}+2}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^3/(x^4+3*x^2+2)^(1/2),x)

[Out]

-1/2*I*d^3*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticF(1
/2*I*2^(1/2)*x,2^(1/2))+e^3*(1/5*x^3*(x^4+3*x^2+2)^(1/2)-4/5*x*(x^4+3*x^2+2)^(1/
2)-4/5*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticF(1/2
*I*2^(1/2)*x,2^(1/2))+9/5*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^
(1/2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(1/2*I*2^(1/2)*x,2^(1/2))))+3
/2*I*d^2*e*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*(EllipticF(
1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(1/2*I*2^(1/2)*x,2^(1/2)))+3*e^2*d*(1/3*x*(x^4
+3*x^2+2)^(1/2)+1/3*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*
EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+
3*x^2+2)^(1/2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(1/2*I*2^(1/2)*x,2^(
1/2))))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{3}}{\sqrt{x^{4} + 3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^3/sqrt(x^4 + 3*x^2 + 2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^3/sqrt(x^4 + 3*x^2 + 2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}{\sqrt{x^{4} + 3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^3/sqrt(x^4 + 3*x^2 + 2),x, algorithm="fricas")

[Out]

integral((e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3)/sqrt(x^4 + 3*x^2 + 2), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x^{2}\right )^{3}}{\sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d)**3/(x**4+3*x**2+2)**(1/2),x)

[Out]

Integral((d + e*x**2)**3/sqrt((x**2 + 1)*(x**2 + 2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x^{2} + d\right )}^{3}}{\sqrt{x^{4} + 3 \, x^{2} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d)^3/sqrt(x^4 + 3*x^2 + 2),x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^3/sqrt(x^4 + 3*x^2 + 2), x)