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

Optimal. Leaf size=177 \[ \frac{1}{3} \left (3 x^2+10\right ) \sqrt{x^4+3 x^2+4} x+\frac{9 \sqrt{x^4+3 x^2+4} x}{x^2+2}+\frac{49 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{9 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{x^4+3 x^2+4}} \]

[Out]

(9*x*Sqrt[4 + 3*x^2 + x^4])/(2 + x^2) + (x*(10 + 3*x^2)*Sqrt[4 + 3*x^2 + x^4])/3
- (9*Sqrt[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] + (49*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2
+ x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(3*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]
)

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Rubi [A]  time = 0.13205, antiderivative size = 177, 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 \[ \frac{1}{3} \left (3 x^2+10\right ) \sqrt{x^4+3 x^2+4} x+\frac{9 \sqrt{x^4+3 x^2+4} x}{x^2+2}+\frac{49 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{9 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(9*x*Sqrt[4 + 3*x^2 + x^4])/(2 + x^2) + (x*(10 + 3*x^2)*Sqrt[4 + 3*x^2 + x^4])/3
- (9*Sqrt[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] + (49*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2
+ x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(3*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]
)

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Rubi in Sympy [A]  time = 26.6132, size = 177, normalized size = 1. \[ \frac{x \left (15 x^{2} + 50\right ) \sqrt{x^{4} + 3 x^{2} + 4}}{15} + \frac{18 x \sqrt{x^{4} + 3 x^{2} + 4}}{2 x^{2} + 4} - \frac{9 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{\sqrt{x^{4} + 3 x^{2} + 4}} + \frac{49 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{6 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(15*x**2 + 50)*sqrt(x**4 + 3*x**2 + 4)/15 + 18*x*sqrt(x**4 + 3*x**2 + 4)/(2*x*
*2 + 4) - 9*sqrt(2)*sqrt((x**4 + 3*x**2 + 4)/(x**2/2 + 1)**2)*(x**2/2 + 1)*ellip
tic_e(2*atan(sqrt(2)*x/2), 1/8)/sqrt(x**4 + 3*x**2 + 4) + 49*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)
/(6*sqrt(x**4 + 3*x**2 + 4))

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Mathematica [C]  time = 0.776209, size = 338, normalized size = 1.91 \[ \frac{\sqrt{2} \left (27 \sqrt{7}-7 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-27 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (3 x^6+19 x^4+42 x^2+40\right )}{12 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(40 + 42*x^2 + 19*x^4 + 3*x^6) - 27*Sqrt[2]*(3*
I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqr
t[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]*(-7*I + 27*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 + Sqrt[7])])/(12*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])

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Maple [C]  time = 0.008, size = 240, normalized size = 1.4 \[{\frac{10\,x}{3}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{176}{3\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-288\,{\frac{\sqrt{1- \left ( -3/8+i/8\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -3/8-i/8\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ( 1/4\,x\sqrt{-6+2\,i\sqrt{7}},1/4\,\sqrt{2+6\,i\sqrt{7}} \right ) -{\it EllipticE} \left ( 1/4\,x\sqrt{-6+2\,i\sqrt{7}},1/4\,\sqrt{2+6\,i\sqrt{7}} \right ) \right ) }{\sqrt{-6+2\,i\sqrt{7}}\sqrt{{x}^{4}+3\,{x}^{2}+4} \left ( i\sqrt{7}+3 \right ) }}+{x}^{3}\sqrt{{x}^{4}+3\,{x}^{2}+4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10/3*x*(x^4+3*x^2+4)^(1/2)+176/3/(-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))-288/(-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)))+x^3
*(x^4+3*x^2+4)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{x^{4} + 3 \, x^{2} + 4}{\left (5 \, x^{2} + 7\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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