3.318 \(\int \frac{1}{\left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=235 \[ \frac{625 \sqrt{x^4+3 x^2+2} x}{504 \left (5 x^2+7\right )}-\frac{31 \left (x^2+2\right ) x}{56 \sqrt{x^4+3 x^2+2}}+\frac{\left (11 x^2+20\right ) x}{36 \sqrt{x^4+3 x^2+2}}-\frac{463 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{336 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{31 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{375 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{784 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

[Out]

(-31*x*(2 + x^2))/(56*Sqrt[2 + 3*x^2 + x^4]) + (x*(20 + 11*x^2))/(36*Sqrt[2 + 3*
x^2 + x^4]) + (625*x*Sqrt[2 + 3*x^2 + x^4])/(504*(7 + 5*x^2)) + (31*(1 + x^2)*Sq
rt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(28*Sqrt[2]*Sqrt[2 + 3*x^2 +
x^4]) - (463*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(336
*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + (375*(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1/2]
)/(784*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

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Rubi [A]  time = 0.806312, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{625 \sqrt{x^4+3 x^2+2} x}{504 \left (5 x^2+7\right )}-\frac{31 \left (x^2+2\right ) x}{56 \sqrt{x^4+3 x^2+2}}+\frac{\left (11 x^2+20\right ) x}{36 \sqrt{x^4+3 x^2+2}}-\frac{463 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{336 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{31 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{375 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{784 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-31*x*(2 + x^2))/(56*Sqrt[2 + 3*x^2 + x^4]) + (x*(20 + 11*x^2))/(36*Sqrt[2 + 3*
x^2 + x^4]) + (625*x*Sqrt[2 + 3*x^2 + x^4])/(504*(7 + 5*x^2)) + (31*(1 + x^2)*Sq
rt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(28*Sqrt[2]*Sqrt[2 + 3*x^2 +
x^4]) - (463*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(336
*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) + (375*(2 + x^2)*EllipticPi[2/7, ArcTan[x], 1/2]
)/(784*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 129.892, size = 330, normalized size = 1.4 \[ \frac{625 x \sqrt{x^{4} + 3 x^{2} + 2}}{12 \left (210 x^{2} + 294\right )} - \frac{125 x \sqrt{x^{4} + 3 x^{2} + 2}}{504 \left (x^{2} + 1\right )} + \frac{4 x \sqrt{x^{4} + 3 x^{2} + 2}}{27 \left (\frac{x^{2}}{2} + 1\right ) \left (x^{2} + 1\right )} + \frac{305 \sqrt{x^{4} + 3 x^{2} + 2} E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{1512 \sqrt{\frac{\frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} - \frac{7667 \sqrt{x^{4} + 3 x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{6048 \sqrt{\frac{\frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} + \frac{6875 \sqrt{x^{4} + 3 x^{2} + 2} \Pi \left (\frac{2}{7}; \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{14112 \sqrt{\frac{\frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} + \frac{19 \sqrt{2} \sqrt{x^{4} + 3 x^{2} + 2} E\left (\operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -1\right )}{108 \sqrt{\frac{x^{2} + 1}{\frac{x^{2}}{2} + 1}} \left (\frac{x^{2}}{2} + 1\right )} + \frac{125 \sqrt{2} \sqrt{x^{4} + 3 x^{2} + 2} \Pi \left (- \frac{3}{7}; \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -1\right )}{378 \sqrt{\frac{x^{2} + 1}{\frac{x^{2}}{2} + 1}} \left (\frac{x^{2}}{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625*x*sqrt(x**4 + 3*x**2 + 2)/(12*(210*x**2 + 294)) - 125*x*sqrt(x**4 + 3*x**2 +
2)/(504*(x**2 + 1)) + 4*x*sqrt(x**4 + 3*x**2 + 2)/(27*(x**2/2 + 1)*(x**2 + 1))
+ 305*sqrt(x**4 + 3*x**2 + 2)*elliptic_e(atan(x), 1/2)/(1512*sqrt((x**2/2 + 1)/(
x**2 + 1))*(x**2 + 1)) - 7667*sqrt(x**4 + 3*x**2 + 2)*elliptic_f(atan(x), 1/2)/(
6048*sqrt((x**2/2 + 1)/(x**2 + 1))*(x**2 + 1)) + 6875*sqrt(x**4 + 3*x**2 + 2)*el
liptic_pi(2/7, atan(x), 1/2)/(14112*sqrt((x**2/2 + 1)/(x**2 + 1))*(x**2 + 1)) +
19*sqrt(2)*sqrt(x**4 + 3*x**2 + 2)*elliptic_e(atan(sqrt(2)*x/2), -1)/(108*sqrt((
x**2 + 1)/(x**2/2 + 1))*(x**2/2 + 1)) + 125*sqrt(2)*sqrt(x**4 + 3*x**2 + 2)*elli
ptic_pi(-3/7, atan(sqrt(2)*x/2), -1)/(378*sqrt((x**2 + 1)/(x**2/2 + 1))*(x**2/2
+ 1))

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Mathematica [C]  time = 0.224743, size = 208, normalized size = 0.89 \[ \frac{3255 x^5+10157 x^3+182 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right ) F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+651 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right ) E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+1125 i \sqrt{x^2+1} \sqrt{x^2+2} x^2 \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+1575 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+7490 x}{1176 \left (5 x^2+7\right ) \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(7490*x + 10157*x^3 + 3255*x^5 + (651*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)
*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + (182*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5
*x^2)*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] + (1575*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*
EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2] + (1125*I)*x^2*Sqrt[1 + x^2]*Sqrt[2 +
x^2]*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(1176*(7 + 5*x^2)*Sqrt[2 + 3*x^2
+ x^4])

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Maple [C]  time = 0.03, size = 185, normalized size = 0.8 \[{\frac{625\,x}{2520\,{x}^{2}+3528}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-2\,{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}} \left ( -{\frac{11\,{x}^{3}}{72}}-{\frac{5\,x}{18}} \right ) }+{{\frac{13\,i}{168}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{31\,i}{112}}\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{75\,i}{392}}\sqrt{2}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}\sqrt{2}x,{\frac{10}{7}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625/504*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)-2*(-11/72*x^3-5/18*x)/(x^4+3*x^2+2)^(1/2
)+13/168*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))+31/112*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^
2+2)^(1/2)*EllipticE(1/2*I*2^(1/2)*x,2^(1/2))+75/392*I*2^(1/2)*(1+1/2*x^2)^(1/2)
*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticPi(1/2*I*2^(1/2)*x,10/7,2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7)^2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (25 \, x^{8} + 145 \, x^{6} + 309 \, x^{4} + 287 \, x^{2} + 98\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7)^2),x, algorithm="fricas")

[Out]

integral(1/((25*x^8 + 145*x^6 + 309*x^4 + 287*x^2 + 98)*sqrt(x^4 + 3*x^2 + 2)),
x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7)^2),x, algorithm="giac")

[Out]

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