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

Optimal. Leaf size=200 \[ \frac{14523 \sqrt{x^4+3 x^2+4} x}{28 \left (x^2+2\right )}+\frac{625}{3} \sqrt{x^4+3 x^2+4} x+\frac{\left (2719-4023 x^2\right ) x}{28 \sqrt{x^4+3 x^2+4}}+\frac{4243 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{12 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{14523 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

[Out]

(x*(2719 - 4023*x^2))/(28*Sqrt[4 + 3*x^2 + x^4]) + (625*x*Sqrt[4 + 3*x^2 + x^4])
/3 + (14523*x*Sqrt[4 + 3*x^2 + x^4])/(28*(2 + x^2)) - (14523*(2 + x^2)*Sqrt[(4 +
3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(14*Sqrt[2]*Sqrt
[4 + 3*x^2 + x^4]) + (4243*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*Ellipti
cF[2*ArcTan[x/Sqrt[2]], 1/8])/(12*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi [A]  time = 0.193057, antiderivative size = 200, 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 \[ \frac{14523 \sqrt{x^4+3 x^2+4} x}{28 \left (x^2+2\right )}+\frac{625}{3} \sqrt{x^4+3 x^2+4} x+\frac{\left (2719-4023 x^2\right ) x}{28 \sqrt{x^4+3 x^2+4}}+\frac{4243 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{12 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{14523 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{14 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2719 - 4023*x^2))/(28*Sqrt[4 + 3*x^2 + x^4]) + (625*x*Sqrt[4 + 3*x^2 + x^4])
/3 + (14523*x*Sqrt[4 + 3*x^2 + x^4])/(28*(2 + x^2)) - (14523*(2 + x^2)*Sqrt[(4 +
3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(14*Sqrt[2]*Sqrt
[4 + 3*x^2 + x^4]) + (4243*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*Ellipti
cF[2*ArcTan[x/Sqrt[2]], 1/8])/(12*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 41.7213, size = 197, normalized size = 0.98 \[ \frac{x \left (- 325863 x^{2} + 220239\right )}{2268 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{625 x \sqrt{x^{4} + 3 x^{2} + 4}}{3} + \frac{14523 x \sqrt{x^{4} + 3 x^{2} + 4}}{14 \left (2 x^{2} + 4\right )} - \frac{14523 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{28 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{4243 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{24 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-325863*x**2 + 220239)/(2268*sqrt(x**4 + 3*x**2 + 4)) + 625*x*sqrt(x**4 + 3*x
**2 + 4)/3 + 14523*x*sqrt(x**4 + 3*x**2 + 4)/(14*(2*x**2 + 4)) - 14523*sqrt(2)*s
qrt((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)/(28*sqrt(x**4 + 3*x**2 + 4)) + 4243*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)/(24*sqrt(x**4
+ 3*x**2 + 4))

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Mathematica [C]  time = 0.797547, size = 333, normalized size = 1.66 \[ \frac{\sqrt{2} \left (43569 \sqrt{7}+186179 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-43569 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (17500 x^4+40431 x^2+78157\right )}{336 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(78157 + 40431*x^2 + 17500*x^4) - 43569*Sqrt[2]
*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I +
Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + S
qrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(186179*I + 43569*Sqrt[7
])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*
I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (
3*I - Sqrt[7])/(3*I + Sqrt[7])])/(336*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2
+ x^4])

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Maple [C]  time = 0.014, size = 339, normalized size = 1.7 \[ -4802\,{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}} \left ({\frac{x}{56}}+{\frac{3\,{x}^{3}}{56}} \right ) }-{\frac{27736}{21\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{116184}{7\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-13720\,{\frac{-1/7\,{x}^{3}-3/14\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}-14700\,{\frac{3/14\,{x}^{3}+4/7\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}-7000\,{\frac{-1/14\,{x}^{3}-6/7\,x}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}-1250\,{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}} \left ( -{\frac{9\,{x}^{3}}{14}}+2/7\,x \right ) }+{\frac{625\,x}{3}\sqrt{{x}^{4}+3\,{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4802*(1/56*x+3/56*x^3)/(x^4+3*x^2+4)^(1/2)-27736/21/(-6+2*I*7^(1/2))^(1/2)*(1-(
-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2+4)^
(1/2)*EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-116184/7
/(-6+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2
))*x^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))-EllipticE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2
+6*I*7^(1/2))^(1/2)))-13720*(-1/7*x^3-3/14*x)/(x^4+3*x^2+4)^(1/2)-14700*(3/14*x^
3+4/7*x)/(x^4+3*x^2+4)^(1/2)-7000*(-1/14*x^3-6/7*x)/(x^4+3*x^2+4)^(1/2)-1250*(-9
/14*x^3+2/7*x)/(x^4+3*x^2+4)^(1/2)+625/3*x*(x^4+3*x^2+4)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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