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

Optimal. Leaf size=168 \[ \frac{25}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{21} x \left (114 x^2+407\right ) \sqrt{x^4+3 x^2+2}+\frac{31 x \left (x^2+2\right )}{\sqrt{x^4+3 x^2+2}}+\frac{472 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{21 \sqrt{x^4+3 x^2+2}}-\frac{31 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]

[Out]

(31*x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] + (x*(407 + 114*x^2)*Sqrt[2 + 3*x^2 + x^4
])/21 + (25*x*(2 + 3*x^2 + x^4)^(3/2))/7 - (31*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/
(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/Sqrt[2 + 3*x^2 + x^4] + (472*Sqrt[2]*(1 +
x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(21*Sqrt[2 + 3*x^2 + x
^4])

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Rubi [A]  time = 0.162065, antiderivative size = 168, 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 \[ \frac{25}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{21} x \left (114 x^2+407\right ) \sqrt{x^4+3 x^2+2}+\frac{31 x \left (x^2+2\right )}{\sqrt{x^4+3 x^2+2}}+\frac{472 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{21 \sqrt{x^4+3 x^2+2}}-\frac{31 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(31*x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] + (x*(407 + 114*x^2)*Sqrt[2 + 3*x^2 + x^4
])/21 + (25*x*(2 + 3*x^2 + x^4)^(3/2))/7 - (31*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/
(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/Sqrt[2 + 3*x^2 + x^4] + (472*Sqrt[2]*(1 +
x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(21*Sqrt[2 + 3*x^2 + x
^4])

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Rubi in Sympy [A]  time = 26.8322, size = 162, normalized size = 0.96 \[ \frac{31 x \left (2 x^{2} + 4\right )}{2 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (\frac{570 x^{2}}{7} + \frac{2035}{7}\right ) \sqrt{x^{4} + 3 x^{2} + 2}}{15} + \frac{25 x \left (x^{4} + 3 x^{2} + 2\right )^{\frac{3}{2}}}{7} - \frac{31 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{118 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{21 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

31*x*(2*x**2 + 4)/(2*sqrt(x**4 + 3*x**2 + 2)) + x*(570*x**2/7 + 2035/7)*sqrt(x**
4 + 3*x**2 + 2)/15 + 25*x*(x**4 + 3*x**2 + 2)**(3/2)/7 - 31*sqrt((2*x**2 + 4)/(x
**2 + 1))*(4*x**2 + 4)*elliptic_e(atan(x), 1/2)/(4*sqrt(x**4 + 3*x**2 + 2)) + 11
8*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(21*sqrt(x
**4 + 3*x**2 + 2))

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Mathematica [C]  time = 0.0764958, size = 114, normalized size = 0.68 \[ \frac{75 x^9+564 x^7+1724 x^5+2349 x^3-293 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-651 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+1114 x}{21 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(1114*x + 2349*x^3 + 1724*x^5 + 564*x^7 + 75*x^9 - (651*I)*Sqrt[1 + x^2]*Sqrt[2
+ x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (293*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*
EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(21*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.011, size = 155, normalized size = 0.9 \[{\frac{557\,x}{21}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{472\,i}{21}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{31\,i}{2}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{113\,{x}^{3}}{7}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{25\,{x}^{5}}{7}\sqrt{{x}^{4}+3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

557/21*x*(x^4+3*x^2+2)^(1/2)-472/21*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))+31/2*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))-Elliptic
E(1/2*I*2^(1/2)*x,2^(1/2)))+113/7*x^3*(x^4+3*x^2+2)^(1/2)+25/7*x^5*(x^4+3*x^2+2)
^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x**2 + 1)*(x**2 + 2))*(5*x**2 + 7)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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