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

Optimal. Leaf size=170 \[ \frac{625}{3} \sqrt{x^4+3 x^2+2} x+\frac{637 \left (x^2+2\right ) x}{2 \sqrt{x^4+3 x^2+2}}+\frac{\left (113 x^2+145\right ) x}{2 \sqrt{x^4+3 x^2+2}}+\frac{1067 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+3 x^2+2}}-\frac{637 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+2}} \]

[Out]

(637*x*(2 + x^2))/(2*Sqrt[2 + 3*x^2 + x^4]) + (x*(145 + 113*x^2))/(2*Sqrt[2 + 3*
x^2 + x^4]) + (625*x*Sqrt[2 + 3*x^2 + x^4])/3 - (637*(1 + x^2)*Sqrt[(2 + x^2)/(1
+ x^2)]*EllipticE[ArcTan[x], 1/2])/(Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + (1067*Sqrt
[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(3*Sqrt[2 + 3
*x^2 + x^4])

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Rubi [A]  time = 0.173624, antiderivative size = 170, 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 \[ \frac{625}{3} \sqrt{x^4+3 x^2+2} x+\frac{637 \left (x^2+2\right ) x}{2 \sqrt{x^4+3 x^2+2}}+\frac{\left (113 x^2+145\right ) x}{2 \sqrt{x^4+3 x^2+2}}+\frac{1067 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+3 x^2+2}}-\frac{637 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(637*x*(2 + x^2))/(2*Sqrt[2 + 3*x^2 + x^4]) + (x*(145 + 113*x^2))/(2*Sqrt[2 + 3*
x^2 + x^4]) + (625*x*Sqrt[2 + 3*x^2 + x^4])/3 - (637*(1 + x^2)*Sqrt[(2 + x^2)/(1
+ x^2)]*EllipticE[ArcTan[x], 1/2])/(Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + (1067*Sqrt
[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(3*Sqrt[2 + 3
*x^2 + x^4])

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Rubi in Sympy [A]  time = 16.6691, size = 158, normalized size = 0.93 \[ \frac{637 x \left (2 x^{2} + 4\right )}{4 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (9153 x^{2} + 11745\right )}{162 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{625 x \sqrt{x^{4} + 3 x^{2} + 2}}{3} - \frac{637 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{8 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{1067 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

637*x*(2*x**2 + 4)/(4*sqrt(x**4 + 3*x**2 + 2)) + x*(9153*x**2 + 11745)/(162*sqrt
(x**4 + 3*x**2 + 2)) + 625*x*sqrt(x**4 + 3*x**2 + 2)/3 - 637*sqrt((2*x**2 + 4)/(
x**2 + 1))*(4*x**2 + 4)*elliptic_e(atan(x), 1/2)/(8*sqrt(x**4 + 3*x**2 + 2)) + 1
067*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(12*sqrt
(x**4 + 3*x**2 + 2))

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Mathematica [C]  time = 0.0884903, size = 104, normalized size = 0.61 \[ \frac{1250 x^5+4089 x^3-2357 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-1911 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+2935 x}{6 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2935*x + 4089*x^3 + 1250*x^5 - (1911*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I
*ArcSinh[x/Sqrt[2]], 2] - (2357*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSi
nh[x/Sqrt[2]], 2])/(6*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.014, size = 234, normalized size = 1.4 \[ -4802\,{\frac{-3/4\,{x}^{3}-5/4\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}-{{\frac{1067\,i}{3}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{637\,i}{4}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-13720\,{\frac{{x}^{3}+3/2\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}-14700\,{\frac{-3/2\,{x}^{3}-2\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}-7000\,{\frac{5/2\,{x}^{3}+3\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}-1250\,{\frac{-9/2\,{x}^{3}-5\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}+{\frac{625\,x}{3}\sqrt{{x}^{4}+3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4802*(-3/4*x^3-5/4*x)/(x^4+3*x^2+2)^(1/2)-1067/3*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))+637/4*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)))-13720*(x^3+3/2*x)/(x^4+3*x^2+2)^(1/2)
-14700*(-3/2*x^3-2*x)/(x^4+3*x^2+2)^(1/2)-7000*(5/2*x^3+3*x)/(x^4+3*x^2+2)^(1/2)
-1250*(-9/2*x^3-5*x)/(x^4+3*x^2+2)^(1/2)+625/3*x*(x^4+3*x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (5 x^{2} + 7\right )^{4}}{\left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((5*x**2 + 7)**4/((x**2 + 1)*(x**2 + 2))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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