3.377 \(\int \frac{\left (7+5 x^2\right )^3}{\left (4+3 x^2+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=181 \[ \frac{4449 \sqrt{x^4+3 x^2+4} x}{28 \left (x^2+2\right )}-\frac{\left (949 x^2+2323\right ) x}{28 \sqrt{x^4+3 x^2+4}}+\frac{973 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{4 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{4449 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

[Out]

-(x*(2323 + 949*x^2))/(28*Sqrt[4 + 3*x^2 + x^4]) + (4449*x*Sqrt[4 + 3*x^2 + x^4]
)/(28*(2 + x^2)) - (4449*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE
[2*ArcTan[x/Sqrt[2]], 1/8])/(14*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) + (973*(2 + x^2)*
Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(4*Sqrt
[2]*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.148654, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{4449 \sqrt{x^4+3 x^2+4} x}{28 \left (x^2+2\right )}-\frac{\left (949 x^2+2323\right ) x}{28 \sqrt{x^4+3 x^2+4}}+\frac{973 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{4 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{4449 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

[In]  Int[(7 + 5*x^2)^3/(4 + 3*x^2 + x^4)^(3/2),x]

[Out]

-(x*(2323 + 949*x^2))/(28*Sqrt[4 + 3*x^2 + x^4]) + (4449*x*Sqrt[4 + 3*x^2 + x^4]
)/(28*(2 + x^2)) - (4449*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE
[2*ArcTan[x/Sqrt[2]], 1/8])/(14*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) + (973*(2 + x^2)*
Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(4*Sqrt
[2]*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 31.3756, size = 178, normalized size = 0.98 \[ - \frac{x \left (25623 x^{2} + 62721\right )}{756 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{4449 x \sqrt{x^{4} + 3 x^{2} + 4}}{14 \left (2 x^{2} + 4\right )} - \frac{4449 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{28 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{973 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{8 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+7)**3/(x**4+3*x**2+4)**(3/2),x)

[Out]

-x*(25623*x**2 + 62721)/(756*sqrt(x**4 + 3*x**2 + 4)) + 4449*x*sqrt(x**4 + 3*x**
2 + 4)/(14*(2*x**2 + 4)) - 4449*sqrt(2)*sqrt((x**4 + 3*x**2 + 4)/(x**2/2 + 1)**2
)*(x**2/2 + 1)*elliptic_e(2*atan(sqrt(2)*x/2), 1/8)/(28*sqrt(x**4 + 3*x**2 + 4))
+ 973*sqrt(2)*sqrt((x**4 + 3*x**2 + 4)/(x**2/2 + 1)**2)*(x**2/2 + 1)*elliptic_f
(2*atan(sqrt(2)*x/2), 1/8)/(8*sqrt(x**4 + 3*x**2 + 4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.690509, size = 328, normalized size = 1.81 \[ \frac{-4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (949 x^2+2323\right )+\sqrt{2} \left (4449 \sqrt{7}+3899 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-4449 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )}{112 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(7 + 5*x^2)^3/(4 + 3*x^2 + x^4)^(3/2),x]

[Out]

(-4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(2323 + 949*x^2) - 4449*Sqrt[2]*(3*I + Sqrt[7]
)*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I
)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3
*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(3899*I + 4449*Sqrt[7])*Sqrt[(-3*I + Sq
rt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqr
t[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*
I + Sqrt[7])])/(112*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.01, size = 301, normalized size = 1.7 \[ -686\,{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}} \left ({\frac{x}{56}}+{\frac{3\,{x}^{3}}{56}} \right ) }+{\frac{4724}{7\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{35592}{7\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-1470\,{\frac{-1/7\,{x}^{3}-3/14\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}-1050\,{\frac{3/14\,{x}^{3}+4/7\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}-250\,{\frac{-1/14\,{x}^{3}-6/7\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+7)^3/(x^4+3*x^2+4)^(3/2),x)

[Out]

-686*(1/56*x+3/56*x^3)/(x^4+3*x^2+4)^(1/2)+4724/7/(-6+2*I*7^(1/2))^(1/2)*(1-(-3/
8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2+4)^(1/
2)*EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-35592/7/(-6
+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x
^2)^(1/2)/(x^4+3*x^2+4)^(1/2)/(I*7^(1/2)+3)*(EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1
/2),1/4*(2+6*I*7^(1/2))^(1/2))-EllipticE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I
*7^(1/2))^(1/2)))-1470*(-1/7*x^3-3/14*x)/(x^4+3*x^2+4)^(1/2)-1050*(3/14*x^3+4/7*
x)/(x^4+3*x^2+4)^(1/2)-250*(-1/14*x^3-6/7*x)/(x^4+3*x^2+4)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)^3/(x^4 + 3*x^2 + 4)^(3/2),x, algorithm="maxima")

[Out]

integrate((5*x^2 + 7)^3/(x^4 + 3*x^2 + 4)^(3/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)^3/(x^4 + 3*x^2 + 4)^(3/2),x, algorithm="fricas")

[Out]

integral((125*x^6 + 525*x^4 + 735*x^2 + 343)/(x^4 + 3*x^2 + 4)^(3/2), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (5 x^{2} + 7\right )^{3}}{\left (\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)**3/(x**4+3*x**2+4)**(3/2),x)

[Out]

Integral((5*x**2 + 7)**3/((x**2 - x + 2)*(x**2 + x + 2))**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{{\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^2 + 7)^3/(x^4 + 3*x^2 + 4)^(3/2),x, algorithm="giac")

[Out]

integrate((5*x^2 + 7)^3/(x^4 + 3*x^2 + 4)^(3/2), x)