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

Optimal. Leaf size=286 \[ -\frac{5 \sqrt{x^4+3 x^2+4} x}{616 \left (x^2+2\right )}+\frac{25 \sqrt{x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}+\frac{37 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{2464}-\frac{\left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{42 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{5 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{308 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{629 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{51744 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

[Out]

(-5*x*Sqrt[4 + 3*x^2 + x^4])/(616*(2 + x^2)) + (25*x*Sqrt[4 + 3*x^2 + x^4])/(616
*(7 + 5*x^2)) + (37*Sqrt[5/77]*ArcTan[(2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/
2464 + (5*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqr
t[2]], 1/8])/(308*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) - ((2 + x^2)*Sqrt[(4 + 3*x^2 +
x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(42*Sqrt[2]*Sqrt[4 + 3*x^
2 + x^4]) + (629*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280
, 2*ArcTan[x/Sqrt[2]], 1/8])/(51744*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi [A]  time = 0.232554, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{5 \sqrt{x^4+3 x^2+4} x}{616 \left (x^2+2\right )}+\frac{25 \sqrt{x^4+3 x^2+4} x}{616 \left (5 x^2+7\right )}+\frac{37 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{2464}-\frac{\left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{42 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{5 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{308 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{629 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{51744 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(-5*x*Sqrt[4 + 3*x^2 + x^4])/(616*(2 + x^2)) + (25*x*Sqrt[4 + 3*x^2 + x^4])/(616
*(7 + 5*x^2)) + (37*Sqrt[5/77]*ArcTan[(2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/
2464 + (5*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqr
t[2]], 1/8])/(308*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) - ((2 + x^2)*Sqrt[(4 + 3*x^2 +
x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(42*Sqrt[2]*Sqrt[4 + 3*x^
2 + x^4]) + (629*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280
, 2*ArcTan[x/Sqrt[2]], 1/8])/(51744*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 32.9841, size = 280, normalized size = 0.98 \[ \frac{25 x \sqrt{x^{4} + 3 x^{2} + 4}}{3080 x^{2} + 4312} - \frac{5 x \sqrt{x^{4} + 3 x^{2} + 4}}{308 \left (2 x^{2} + 4\right )} + \frac{5 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{616 \sqrt{x^{4} + 3 x^{2} + 4}} - \frac{\sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{84 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{629 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) \Pi \left (- \frac{9}{280}; 2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{103488 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{37 \sqrt{385} \operatorname{atan}{\left (\frac{2 \sqrt{385} x}{35 \sqrt{x^{4} + 3 x^{2} + 4}} \right )}}{189728} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25*x*sqrt(x**4 + 3*x**2 + 4)/(3080*x**2 + 4312) - 5*x*sqrt(x**4 + 3*x**2 + 4)/(3
08*(2*x**2 + 4)) + 5*sqrt(2)*sqrt((x**4 + 3*x**2 + 4)/(x**2/2 + 1)**2)*(x**2/2 +
1)*elliptic_e(2*atan(sqrt(2)*x/2), 1/8)/(616*sqrt(x**4 + 3*x**2 + 4)) - sqrt(2)
*sqrt((x**4 + 3*x**2 + 4)/(x**2/2 + 1)**2)*(x**2/2 + 1)*elliptic_f(2*atan(sqrt(2
)*x/2), 1/8)/(84*sqrt(x**4 + 3*x**2 + 4)) + 629*sqrt(2)*sqrt((x**4 + 3*x**2 + 4)
/(x**2/2 + 1)**2)*(x**2/2 + 1)*elliptic_pi(-9/280, 2*atan(sqrt(2)*x/2), 1/8)/(10
3488*sqrt(x**4 + 3*x**2 + 4)) + 37*sqrt(385)*atan(2*sqrt(385)*x/(35*sqrt(x**4 +
3*x**2 + 4)))/189728

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Mathematica [C]  time = 1.2245, size = 481, normalized size = 1.68 \[ \frac{98 i \left (5 x^2+7\right ) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-74 i \left (5 x^2+7\right ) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \Pi \left (\frac{5}{14} \left (3+i \sqrt{7}\right );i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+35 \left (\sqrt{7}+3 i\right ) \left (5 x^2+7\right ) \sqrt{2-\frac{4 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left (E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )\right )+700 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (x^4+3 x^2+4\right )}{17248 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \left (5 x^2+7\right ) \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(700*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(4 + 3*x^2 + x^4) + 35*(3*I + Sqrt[7])*(7 + 5
*x^2)*Sqrt[2 - ((4*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7]
)]*(EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I +
Sqrt[7])] - EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7
])/(3*I + Sqrt[7])]) + (98*I)*(7 + 5*x^2)*Sqrt[2 - ((4*I)*x^2)/(-3*I + Sqrt[7])]
*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + S
qrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - (74*I)*(7 + 5*x^2)*Sqrt[2 - ((4*
I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])]*EllipticPi[(5*(3
+ I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(
3*I + Sqrt[7])])/(17248*Sqrt[(-I)/(-3*I + Sqrt[7])]*(7 + 5*x^2)*Sqrt[4 + 3*x^2 +
x^4])

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Maple [C]  time = 0.031, size = 410, normalized size = 1.4 \[{\frac{25\,x}{3080\,{x}^{2}+4312}\sqrt{{x}^{4}+3\,{x}^{2}+4}}-{\frac{1}{22\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{20}{77\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{20}{77\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{37}{4312\,\sqrt{-3/8+i/8\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticPi} \left ( \sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}x,-{\frac{5}{-{\frac{21}{8}}+{\frac{7\,i}{8}}\sqrt{7}}},{\frac{\sqrt{-{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7}}}{\sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/616*x*(x^4+3*x^2+4)^(1/2)/(5*x^2+7)-1/22/(-6+2*I*7^(1/2))^(1/2)*(1+3/8*x^2-1/
8*I*x^2*7^(1/2))^(1/2)*(1+3/8*x^2+1/8*I*x^2*7^(1/2))^(1/2)/(x^4+3*x^2+4)^(1/2)*E
llipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))+20/77/(-6+2*I*7
^(1/2))^(1/2)*(1+3/8*x^2-1/8*I*x^2*7^(1/2))^(1/2)*(1+3/8*x^2+1/8*I*x^2*7^(1/2))^
(1/2)/(x^4+3*x^2+4)^(1/2)/(I*7^(1/2)+3)*EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1
/4*(2+6*I*7^(1/2))^(1/2))-20/77/(-6+2*I*7^(1/2))^(1/2)*(1+3/8*x^2-1/8*I*x^2*7^(1
/2))^(1/2)*(1+3/8*x^2+1/8*I*x^2*7^(1/2))^(1/2)/(x^4+3*x^2+4)^(1/2)/(I*7^(1/2)+3)
*EllipticE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))+37/4312/(-3/8
+1/8*I*7^(1/2))^(1/2)*(1+3/8*x^2-1/8*I*x^2*7^(1/2))^(1/2)*(1+3/8*x^2+1/8*I*x^2*7
^(1/2))^(1/2)/(x^4+3*x^2+4)^(1/2)*EllipticPi((-3/8+1/8*I*7^(1/2))^(1/2)*x,-5/7/(
-3/8+1/8*I*7^(1/2)),(-3/8-1/8*I*7^(1/2))^(1/2)/(-3/8+1/8*I*7^(1/2))^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt{x^{4} + 3 \, x^{2} + 4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((25*x^4 + 70*x^2 + 49)*sqrt(x^4 + 3*x^2 + 4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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