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

Optimal. Leaf size=440 \[ \frac{9 \sqrt{x^4+3 x^2+4} x}{1960 \left (x^2+2\right )}+\frac{167 \sqrt{x^4+3 x^2+4} x}{9800 \left (5 x^2+7\right )}+\frac{11 \sqrt{x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )^2}+\frac{1347 \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{7840 \sqrt{385}}-\frac{22 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{13125 \sqrt{x^4+3 x^2+4}}-\frac{817 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{91875 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{6 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{875 \sqrt{x^4+3 x^2+4}}+\frac{111 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{24500 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{7633 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{274400 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

[Out]

(9*x*Sqrt[4 + 3*x^2 + x^4])/(1960*(2 + x^2)) + (11*x*Sqrt[4 + 3*x^2 + x^4])/(175
*(7 + 5*x^2)^2) + (167*x*Sqrt[4 + 3*x^2 + x^4])/(9800*(7 + 5*x^2)) + (1347*ArcTa
n[(2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/(7840*Sqrt[385]) + (111*(2 + x^2)*Sq
rt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(24500*Sq
rt[2]*Sqrt[4 + 3*x^2 + x^4]) - (6*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 +
x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(875*Sqrt[4 + 3*x^2 + x^4]) - (817*
(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8
])/(91875*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) - (22*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2
+ x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(13125*Sqrt[4 + 3*x^2
+ x^4]) + (7633*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280,
2*ArcTan[x/Sqrt[2]], 1/8])/(274400*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 1.0593, antiderivative size = 440, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{9 \sqrt{x^4+3 x^2+4} x}{1960 \left (x^2+2\right )}+\frac{167 \sqrt{x^4+3 x^2+4} x}{9800 \left (5 x^2+7\right )}+\frac{11 \sqrt{x^4+3 x^2+4} x}{175 \left (5 x^2+7\right )^2}+\frac{1347 \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{7840 \sqrt{385}}-\frac{22 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{13125 \sqrt{x^4+3 x^2+4}}-\frac{817 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{91875 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{6 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{875 \sqrt{x^4+3 x^2+4}}+\frac{111 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{24500 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{7633 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{274400 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(9*x*Sqrt[4 + 3*x^2 + x^4])/(1960*(2 + x^2)) + (11*x*Sqrt[4 + 3*x^2 + x^4])/(175
*(7 + 5*x^2)^2) + (167*x*Sqrt[4 + 3*x^2 + x^4])/(9800*(7 + 5*x^2)) + (1347*ArcTa
n[(2*Sqrt[11/35]*x)/Sqrt[4 + 3*x^2 + x^4]])/(7840*Sqrt[385]) + (111*(2 + x^2)*Sq
rt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(24500*Sq
rt[2]*Sqrt[4 + 3*x^2 + x^4]) - (6*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 +
x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(875*Sqrt[4 + 3*x^2 + x^4]) - (817*
(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8
])/(91875*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4]) - (22*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2
+ x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(13125*Sqrt[4 + 3*x^2
+ x^4]) + (7633*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticPi[-9/280,
2*ArcTan[x/Sqrt[2]], 1/8])/(274400*Sqrt[2]*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.884546, size = 309, normalized size = 0.7 \[ \frac{\frac{140 x \left (167 x^2+357\right ) \left (x^4+3 x^2+4\right )}{\left (5 x^2+7\right )^2}-i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left (7 \left (103+45 i \sqrt{7}\right ) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+315 \left (3-i \sqrt{7}\right ) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+2694 \Pi \left (\frac{5}{14} \left (3+i \sqrt{7}\right );i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )\right )}{274400 \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

((140*x*(357 + 167*x^2)*(4 + 3*x^2 + x^4))/(7 + 5*x^2)^2 - I*Sqrt[6 + (2*I)*Sqrt
[7]]*Sqrt[1 - ((2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[1 + ((2*I)*x^2)/(3*I + Sqrt[7])
]*(315*(3 - I*Sqrt[7])*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*
I - Sqrt[7])/(3*I + Sqrt[7])] + 7*(103 + (45*I)*Sqrt[7])*EllipticF[I*ArcSinh[Sqr
t[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 2694*EllipticP
i[(5*(3 + I*Sqrt[7]))/14, I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqr
t[7])/(3*I + Sqrt[7])]))/(274400*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.034, size = 434, normalized size = 1. \[{\frac{11\,x}{175\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{167\,x}{49000\,{x}^{2}+68600}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{17}{350\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{36}{245\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{36}{245\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{1347}{68600\,\sqrt{-3/8+i/8\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticPi} \left ( \sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}x,-{\frac{5}{-{\frac{21}{8}}+{\frac{7\,i}{8}}\sqrt{7}}},{\frac{\sqrt{-{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7}}}{\sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/175*x*(x^4+3*x^2+4)^(1/2)/(5*x^2+7)^2+167/9800*x*(x^4+3*x^2+4)^(1/2)/(5*x^2+7
)+17/350/(-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))-36/245/(-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))+36/245/(
-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))+1347/68600/(-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))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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