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

Optimal. Leaf size=170 \[ \frac{20 \sqrt{x^4+3 x^2+4} x}{x^2+2}+\frac{25}{3} \sqrt{x^4+3 x^2+4} x+\frac{167 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{6 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{20 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{x^4+3 x^2+4}} \]

[Out]

(25*x*Sqrt[4 + 3*x^2 + x^4])/3 + (20*x*Sqrt[4 + 3*x^2 + x^4])/(2 + x^2) - (20*Sq
rt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]
], 1/8])/Sqrt[4 + 3*x^2 + x^4] + (167*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)
^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(6*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi [A]  time = 0.146686, antiderivative size = 170, 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 \[ \frac{20 \sqrt{x^4+3 x^2+4} x}{x^2+2}+\frac{25}{3} \sqrt{x^4+3 x^2+4} x+\frac{167 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{6 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{20 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(25*x*Sqrt[4 + 3*x^2 + x^4])/3 + (20*x*Sqrt[4 + 3*x^2 + x^4])/(2 + x^2) - (20*Sq
rt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]
], 1/8])/Sqrt[4 + 3*x^2 + x^4] + (167*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)
^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(6*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 29.766, size = 172, normalized size = 1.01 \[ \frac{25 x \sqrt{x^{4} + 3 x^{2} + 4}}{3} + \frac{40 x \sqrt{x^{4} + 3 x^{2} + 4}}{2 x^{2} + 4} - \frac{20 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{\sqrt{x^{4} + 3 x^{2} + 4}} + \frac{167 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{12 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+7)**2/(x**4+3*x**2+4)**(1/2),x)

[Out]

25*x*sqrt(x**4 + 3*x**2 + 4)/3 + 40*x*sqrt(x**4 + 3*x**2 + 4)/(2*x**2 + 4) - 20*
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)/sqrt(x**4 + 3*x**2 + 4) + 167*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)/(12*sqrt(x
**4 + 3*x**2 + 4))

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Mathematica [C]  time = 0.757333, size = 331, normalized size = 1.95 \[ \frac{\sqrt{2} \left (30 \sqrt{7}+43 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-30 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+50 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (x^4+3 x^2+4\right )}{6 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(50*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(4 + 3*x^2 + x^4) - 30*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]*(43*I + 30*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])
]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + S
qrt[7])])/(6*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])

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Maple [C]  time = 0.01, size = 224, normalized size = 1.3 \[{\frac{188}{3\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-640\,{\frac{\sqrt{1- \left ( -3/8+i/8\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -3/8-i/8\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ( 1/4\,x\sqrt{-6+2\,i\sqrt{7}},1/4\,\sqrt{2+6\,i\sqrt{7}} \right ) -{\it EllipticE} \left ( 1/4\,x\sqrt{-6+2\,i\sqrt{7}},1/4\,\sqrt{2+6\,i\sqrt{7}} \right ) \right ) }{\sqrt{-6+2\,i\sqrt{7}}\sqrt{{x}^{4}+3\,{x}^{2}+4} \left ( i\sqrt{7}+3 \right ) }}+{\frac{25\,x}{3}\sqrt{{x}^{4}+3\,{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+7)^2/(x^4+3*x^2+4)^(1/2),x)

[Out]

188/3/(-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))-640/(-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)*(Ell
ipticF(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)))+25/3*x*(x^4+3*x^2+4)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{25 \, x^{4} + 70 \, x^{2} + 49}{\sqrt{x^{4} + 3 \, x^{2} + 4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^4 + 70*x^2 + 49)/sqrt(x^4 + 3*x^2 + 4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)**2/(x**4+3*x**2+4)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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