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

Optimal. Leaf size=237 \[ -\frac{325 \sqrt{x^4+3 x^2+2} x}{4704 \left (5 x^2+7\right )}-\frac{25 \sqrt{x^4+3 x^2+2} x}{168 \left (5 x^2+7\right )^2}+\frac{65 \left (x^2+2\right ) x}{4704 \sqrt{x^4+3 x^2+2}}+\frac{631 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{9408 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{65 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2352 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{2525 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{65856 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

[Out]

(65*x*(2 + x^2))/(4704*Sqrt[2 + 3*x^2 + x^4]) - (25*x*Sqrt[2 + 3*x^2 + x^4])/(16
8*(7 + 5*x^2)^2) - (325*x*Sqrt[2 + 3*x^2 + x^4])/(4704*(7 + 5*x^2)) - (65*(1 + x
^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(2352*Sqrt[2]*Sqrt[2 +
3*x^2 + x^4]) + (631*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/
2])/(9408*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) - (2525*(2 + x^2)*EllipticPi[2/7, ArcTa
n[x], 1/2])/(65856*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

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Rubi [A]  time = 0.527152, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{325 \sqrt{x^4+3 x^2+2} x}{4704 \left (5 x^2+7\right )}-\frac{25 \sqrt{x^4+3 x^2+2} x}{168 \left (5 x^2+7\right )^2}+\frac{65 \left (x^2+2\right ) x}{4704 \sqrt{x^4+3 x^2+2}}+\frac{631 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{9408 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{65 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{2352 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{2525 \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{65856 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(65*x*(2 + x^2))/(4704*Sqrt[2 + 3*x^2 + x^4]) - (25*x*Sqrt[2 + 3*x^2 + x^4])/(16
8*(7 + 5*x^2)^2) - (325*x*Sqrt[2 + 3*x^2 + x^4])/(4704*(7 + 5*x^2)) - (65*(1 + x
^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(2352*Sqrt[2]*Sqrt[2 +
3*x^2 + x^4]) + (631*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/
2])/(9408*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) - (2525*(2 + x^2)*EllipticPi[2/7, ArcTa
n[x], 1/2])/(65856*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.317175, size = 186, normalized size = 0.78 \[ \frac{14 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right )^2 F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-455 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right )^2 E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-505 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right )^2 \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-175 x \left (65 x^6+314 x^4+487 x^2+238\right )}{32928 \left (5 x^2+7\right )^2 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-175*x*(238 + 487*x^2 + 314*x^4 + 65*x^6) - (455*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]
*(7 + 5*x^2)^2*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] + (14*I)*Sqrt[1 + x^2]*Sqrt[2
+ x^2]*(7 + 5*x^2)^2*EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - (505*I)*Sqrt[1 + x^2]*
Sqrt[2 + x^2]*(7 + 5*x^2)^2*EllipticPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(32928*(7
+ 5*x^2)^2*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.028, size = 186, normalized size = 0.8 \[ -{\frac{25\,x}{168\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{\frac{325\,x}{23520\,{x}^{2}+32928}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{{\frac{i}{4704}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{65\,i}{9408}}\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{505\,i}{32928}}\sqrt{2}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}\sqrt{2}x,{\frac{10}{7}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25/168*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)^2-325/4704*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+
7)+1/4704*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))-65/9408*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))-505/32928*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. \[ \int \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((125*x^6 + 525*x^4 + 735*x^2 + 343)*sqrt(x^4 + 3*x^2 + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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