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

Optimal. Leaf size=179 \[ \frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{105} x \left (149 x^2+519\right ) \sqrt{x^4+3 x^2+2}+\frac{116 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}+\frac{197 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{x^4+3 x^2+2}}-\frac{116 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]

[Out]

(116*x*(2 + x^2))/(15*Sqrt[2 + 3*x^2 + x^4]) + (x*(519 + 149*x^2)*Sqrt[2 + 3*x^2
+ x^4])/105 + (x*(108 + 35*x^2)*(2 + 3*x^2 + x^4)^(3/2))/63 - (116*Sqrt[2]*(1 +
x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(15*Sqrt[2 + 3*x^2 +
x^4]) + (197*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/
2])/(35*Sqrt[2 + 3*x^2 + x^4])

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Rubi [A]  time = 0.150632, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{105} x \left (149 x^2+519\right ) \sqrt{x^4+3 x^2+2}+\frac{116 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}+\frac{197 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{x^4+3 x^2+2}}-\frac{116 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(116*x*(2 + x^2))/(15*Sqrt[2 + 3*x^2 + x^4]) + (x*(519 + 149*x^2)*Sqrt[2 + 3*x^2
+ x^4])/105 + (x*(108 + 35*x^2)*(2 + 3*x^2 + x^4)^(3/2))/63 - (116*Sqrt[2]*(1 +
x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(15*Sqrt[2 + 3*x^2 +
x^4]) + (197*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/
2])/(35*Sqrt[2 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 23.3048, size = 163, normalized size = 0.91 \[ \frac{58 x \left (2 x^{2} + 4\right )}{15 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (35 x^{2} + 108\right ) \left (x^{4} + 3 x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{x \left (447 x^{2} + 1557\right ) \sqrt{x^{4} + 3 x^{2} + 2}}{315} - \frac{29 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{15 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{197 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{140 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

58*x*(2*x**2 + 4)/(15*sqrt(x**4 + 3*x**2 + 2)) + x*(35*x**2 + 108)*(x**4 + 3*x**
2 + 2)**(3/2)/63 + x*(447*x**2 + 1557)*sqrt(x**4 + 3*x**2 + 2)/315 - 29*sqrt((2*
x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_e(atan(x), 1/2)/(15*sqrt(x**4 + 3*x*
*2 + 2)) + 197*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/
2)/(140*sqrt(x**4 + 3*x**2 + 2))

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Mathematica [C]  time = 0.0748744, size = 119, normalized size = 0.66 \[ \frac{175 x^{11}+1590 x^9+5962 x^7+12018 x^5+12745 x^3-1110 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-2436 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+5274 x}{315 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(5274*x + 12745*x^3 + 12018*x^5 + 5962*x^7 + 1590*x^9 + 175*x^11 - (2436*I)*Sqrt
[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (1110*I)*Sqrt[1 + x
^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(315*Sqrt[2 + 3*x^2 + x^4]
)

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Maple [C]  time = 0.008, size = 172, normalized size = 1. \[{\frac{71\,{x}^{5}}{21}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{2417\,{x}^{3}}{315}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{293\,x}{35}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{197\,i}{35}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{58\,i}{15}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{5\,{x}^{7}}{9}\sqrt{{x}^{4}+3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

71/21*x^5*(x^4+3*x^2+2)^(1/2)+2417/315*x^3*(x^4+3*x^2+2)^(1/2)+293/35*x*(x^4+3*x
^2+2)^(1/2)-197/35*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*E
llipticF(1/2*I*2^(1/2)*x,2^(1/2))+58/15*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))-EllipticE(1/2*I*2^(1/2)*
x,2^(1/2)))+5/9*x^7*(x^4+3*x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (5 \, x^{6} + 22 \, x^{4} + 31 \, x^{2} + 14\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((5*x^6 + 22*x^4 + 31*x^2 + 14)*sqrt(x^4 + 3*x^2 + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(((x**2 + 1)*(x**2 + 2))**(3/2)*(5*x**2 + 7), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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