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

Optimal. Leaf size=242 \[ \frac{3050}{11} \left (x^4+3 x^2+4\right )^{3/2} x+\frac{1}{33} \left (4516 x^2+18727\right ) \sqrt{x^4+3 x^2+4} x+\frac{51665 \sqrt{x^4+3 x^2+4} x}{33 \left (x^2+2\right )}+\frac{33159 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{11 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{51665 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{33 \sqrt{x^4+3 x^2+4}}+\frac{625}{11} \left (x^4+3 x^2+4\right )^{3/2} x^5+\frac{23500}{99} \left (x^4+3 x^2+4\right )^{3/2} x^3 \]

[Out]

(51665*x*Sqrt[4 + 3*x^2 + x^4])/(33*(2 + x^2)) + (x*(18727 + 4516*x^2)*Sqrt[4 +
3*x^2 + x^4])/33 + (3050*x*(4 + 3*x^2 + x^4)^(3/2))/11 + (23500*x^3*(4 + 3*x^2 +
x^4)^(3/2))/99 + (625*x^5*(4 + 3*x^2 + x^4)^(3/2))/11 - (51665*Sqrt[2]*(2 + x^2
)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(33*S
qrt[4 + 3*x^2 + x^4]) + (33159*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*Ell
ipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(11*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi [A]  time = 0.28012, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{3050}{11} \left (x^4+3 x^2+4\right )^{3/2} x+\frac{1}{33} \left (4516 x^2+18727\right ) \sqrt{x^4+3 x^2+4} x+\frac{51665 \sqrt{x^4+3 x^2+4} x}{33 \left (x^2+2\right )}+\frac{33159 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{11 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{51665 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{33 \sqrt{x^4+3 x^2+4}}+\frac{625}{11} \left (x^4+3 x^2+4\right )^{3/2} x^5+\frac{23500}{99} \left (x^4+3 x^2+4\right )^{3/2} x^3 \]

Antiderivative was successfully verified.

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

[Out]

(51665*x*Sqrt[4 + 3*x^2 + x^4])/(33*(2 + x^2)) + (x*(18727 + 4516*x^2)*Sqrt[4 +
3*x^2 + x^4])/33 + (3050*x*(4 + 3*x^2 + x^4)^(3/2))/11 + (23500*x^3*(4 + 3*x^2 +
x^4)^(3/2))/99 + (625*x^5*(4 + 3*x^2 + x^4)^(3/2))/11 - (51665*Sqrt[2]*(2 + x^2
)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(33*S
qrt[4 + 3*x^2 + x^4]) + (33159*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*Ell
ipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(11*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 56.6838, size = 241, normalized size = 1. \[ \frac{625 x^{5} \left (x^{4} + 3 x^{2} + 4\right )^{\frac{3}{2}}}{11} + \frac{23500 x^{3} \left (x^{4} + 3 x^{2} + 4\right )^{\frac{3}{2}}}{99} + \frac{x \left (\frac{22580 x^{2}}{11} + \frac{93635}{11}\right ) \sqrt{x^{4} + 3 x^{2} + 4}}{15} + \frac{3050 x \left (x^{4} + 3 x^{2} + 4\right )^{\frac{3}{2}}}{11} + \frac{103330 x \sqrt{x^{4} + 3 x^{2} + 4}}{33 \left (2 x^{2} + 4\right )} - \frac{51665 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{33 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{33159 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{22 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625*x**5*(x**4 + 3*x**2 + 4)**(3/2)/11 + 23500*x**3*(x**4 + 3*x**2 + 4)**(3/2)/9
9 + x*(22580*x**2/11 + 93635/11)*sqrt(x**4 + 3*x**2 + 4)/15 + 3050*x*(x**4 + 3*x
**2 + 4)**(3/2)/11 + 103330*x*sqrt(x**4 + 3*x**2 + 4)/(33*(2*x**2 + 4)) - 51665*
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)/(33*sqrt(x**4 + 3*x**2 + 4)) + 33159*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)/(22
*sqrt(x**4 + 3*x**2 + 4))

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Mathematica [C]  time = 0.988836, size = 354, normalized size = 1.46 \[ \frac{3 \sqrt{2} \left (51665 \sqrt{7}-36253 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-154995 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (5625 x^{12}+57250 x^{10}+264075 x^8+712748 x^6+1217475 x^4+1257535 x^2+663924\right )}{396 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(663924 + 1257535*x^2 + 1217475*x^4 + 712748*x^
6 + 264075*x^8 + 57250*x^10 + 5625*x^12) - 154995*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 - Sqr
t[7])/(3*I + Sqrt[7])] + 3*Sqrt[2]*(-36253*I + 51665*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])])/(396*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])

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Maple [C]  time = 0.208, size = 292, normalized size = 1.2 \[{\frac{55327\,x}{33}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{382496}{33\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{1653280}{33\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{189898\,{x}^{3}}{99}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{3650\,{x}^{5}}{3}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{40375\,{x}^{7}}{99}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{625\,{x}^{9}}{11}\sqrt{{x}^{4}+3\,{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

55327/33*x*(x^4+3*x^2+4)^(1/2)+382496/33/(-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)*Ellipt
icF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-1653280/33/(-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)))+189898/99*x^3*(x^4+3*x^2+4)^(1/2)+3650/3*x^5*(x^4+3*x^2+4)^(1/2)+403
75/99*x^7*(x^4+3*x^2+4)^(1/2)+625/11*x^9*(x^4+3*x^2+4)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (625 \, x^{8} + 3500 \, x^{6} + 7350 \, x^{4} + 6860 \, x^{2} + 2401\right )} \sqrt{x^{4} + 3 \, x^{2} + 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((625*x^8 + 3500*x^6 + 7350*x^4 + 6860*x^2 + 2401)*sqrt(x^4 + 3*x^2 + 4)
, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x**2 - x + 2)*(x**2 + x + 2))*(5*x**2 + 7)**4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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