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

Optimal. Leaf size=312 \[ -\frac{139 \sqrt{x^4+3 x^2+4} x}{86240 \left (x^2+2\right )}+\frac{139 \sqrt{x^4+3 x^2+4} x}{17248 \left (5 x^2+7\right )}+\frac{\sqrt{x^4+3 x^2+4} x}{28 \left (5 x^2+7\right )^2}+\frac{14999 \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{344960 \sqrt{385}}-\frac{23 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2940 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{139 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{43120 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{254983 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{36220800 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

[Out]

(-139*x*Sqrt[4 + 3*x^2 + x^4])/(86240*(2 + x^2)) + (x*Sqrt[4 + 3*x^2 + x^4])/(28
*(7 + 5*x^2)^2) + (139*x*Sqrt[4 + 3*x^2 + x^4])/(17248*(7 + 5*x^2)) + (14999*Arc
Tan[(2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/(344960*Sqrt[385]) + (139*(2 + x^2
)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(4312
0*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) - (23*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2
)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(2940*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) +
(254983*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280, 2*ArcT
an[x/Sqrt[2]], 1/8])/(36220800*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi [A]  time = 0.849995, antiderivative size = 312, 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 \[ -\frac{139 \sqrt{x^4+3 x^2+4} x}{86240 \left (x^2+2\right )}+\frac{139 \sqrt{x^4+3 x^2+4} x}{17248 \left (5 x^2+7\right )}+\frac{\sqrt{x^4+3 x^2+4} x}{28 \left (5 x^2+7\right )^2}+\frac{14999 \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{344960 \sqrt{385}}-\frac{23 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2940 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{139 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{43120 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{254983 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{36220800 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(-139*x*Sqrt[4 + 3*x^2 + x^4])/(86240*(2 + x^2)) + (x*Sqrt[4 + 3*x^2 + x^4])/(28
*(7 + 5*x^2)^2) + (139*x*Sqrt[4 + 3*x^2 + x^4])/(17248*(7 + 5*x^2)) + (14999*Arc
Tan[(2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/(344960*Sqrt[385]) + (139*(2 + x^2
)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(4312
0*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) - (23*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2
)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(2940*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) +
(254983*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280, 2*ArcT
an[x/Sqrt[2]], 1/8])/(36220800*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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Mathematica [C]  time = 0.88365, size = 308, normalized size = 0.99 \[ \frac{\frac{700 x \left (695 x^2+1589\right ) \left (x^4+3 x^2+4\right )}{\left (5 x^2+7\right )^2}+i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left (\left (-9597+4865 i \sqrt{7}\right ) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+4865 \left (3-i \sqrt{7}\right ) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-29998 \Pi \left (\frac{5}{14} \left (3+i \sqrt{7}\right );i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )\right )}{12073600 \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

((700*x*(1589 + 695*x^2)*(4 + 3*x^2 + x^4))/(7 + 5*x^2)^2 + I*Sqrt[6 + (2*I)*Sqr
t[7]]*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7]
)]*(4865*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (
3*I - Sqrt[7])/(3*I + Sqrt[7])] + (-9597 + (4865*I)*Sqrt[7])*EllipticF[I*ArcSinh
[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] - 29998*Elli
pticPi[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I
- Sqrt[7])/(3*I + Sqrt[7])]))/(12073600*Sqrt[4 + 3*x^2 + x^4])

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Maple [C]  time = 0.034, size = 434, normalized size = 1.4 \[{\frac{x}{28\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{139\,x}{86240\,{x}^{2}+120736}\sqrt{{x}^{4}+3\,{x}^{2}+4}}-{\frac{51}{15400\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{139}{2695\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{139}{2695\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{14999}{3018400\,\sqrt{-3/8+i/8\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticPi} \left ( \sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}x,-{\frac{5}{-{\frac{21}{8}}+{\frac{7\,i}{8}}\sqrt{7}}},{\frac{\sqrt{-{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7}}}{\sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/28*x*(x^4+3*x^2+4)^(1/2)/(5*x^2+7)^2+139/17248*x*(x^4+3*x^2+4)^(1/2)/(5*x^2+7)
-51/15400/(-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)*EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1
/2),1/4*(2+6*I*7^(1/2))^(1/2))+139/2695/(-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))-139/2
695/(-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))+14999/3018400/(-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. \[ \int \frac{\sqrt{x^{4} + 3 \, x^{2} + 4}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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