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

Optimal. Leaf size=104 \[ \frac{13 \sqrt{x^4+5 x^2+3}}{108 x^2}-\frac{\sqrt{x^4+5 x^2+3}}{54 x^4}-\frac{61 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{216 \sqrt{3}}-\frac{\sqrt{x^4+5 x^2+3}}{9 x^6} \]

[Out]

-Sqrt[3 + 5*x^2 + x^4]/(9*x^6) - Sqrt[3 + 5*x^2 + x^4]/(54*x^4) + (13*Sqrt[3 + 5
*x^2 + x^4])/(108*x^2) - (61*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4
])])/(216*Sqrt[3])

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Rubi [A]  time = 0.238328, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{13 \sqrt{x^4+5 x^2+3}}{108 x^2}-\frac{\sqrt{x^4+5 x^2+3}}{54 x^4}-\frac{61 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{216 \sqrt{3}}-\frac{\sqrt{x^4+5 x^2+3}}{9 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[3 + 5*x^2 + x^4]/(9*x^6) - Sqrt[3 + 5*x^2 + x^4]/(54*x^4) + (13*Sqrt[3 + 5
*x^2 + x^4])/(108*x^2) - (61*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4
])])/(216*Sqrt[3])

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Rubi in Sympy [A]  time = 24.956, size = 94, normalized size = 0.9 \[ - \frac{61 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{648} + \frac{13 \sqrt{x^{4} + 5 x^{2} + 3}}{108 x^{2}} - \frac{\sqrt{x^{4} + 5 x^{2} + 3}}{54 x^{4}} - \frac{\sqrt{x^{4} + 5 x^{2} + 3}}{9 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-61*sqrt(3)*atanh(sqrt(3)*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))/648 + 13*sqr
t(x**4 + 5*x**2 + 3)/(108*x**2) - sqrt(x**4 + 5*x**2 + 3)/(54*x**4) - sqrt(x**4
+ 5*x**2 + 3)/(9*x**6)

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Mathematica [A]  time = 0.0944119, size = 83, normalized size = 0.8 \[ \frac{61 \left (\log \left (x^2\right )-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )\right )}{216 \sqrt{3}}+\sqrt{x^4+5 x^2+3} \left (-\frac{1}{9 x^6}-\frac{1}{54 x^4}+\frac{13}{108 x^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1/(9*x^6) - 1/(54*x^4) + 13/(108*x^2))*Sqrt[3 + 5*x^2 + x^4] + (61*(Log[x^2] -
Log[6 + 5*x^2 + 2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]]))/(216*Sqrt[3])

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Maple [A]  time = 0.019, size = 83, normalized size = 0.8 \[ -{\frac{61\,\sqrt{3}}{648}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{1}{9\,{x}^{6}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{1}{54\,{x}^{4}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{13}{108\,{x}^{2}}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-61/648*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)-1/9*(x^4+5*x^
2+3)^(1/2)/x^6-1/54*(x^4+5*x^2+3)^(1/2)/x^4+13/108*(x^4+5*x^2+3)^(1/2)/x^2

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Maxima [A]  time = 0.782997, size = 115, normalized size = 1.11 \[ -\frac{61}{648} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{108 \, x^{2}} - \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}}{54 \, x^{4}} - \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}}{9 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-61/648*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 13/108*sq
rt(x^4 + 5*x^2 + 3)/x^2 - 1/54*sqrt(x^4 + 5*x^2 + 3)/x^4 - 1/9*sqrt(x^4 + 5*x^2
+ 3)/x^6

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Fricas [A]  time = 0.264431, size = 359, normalized size = 3.45 \[ -\frac{2 \, \sqrt{3}{\left (976 \, x^{8} + 3660 \, x^{6} + 41 \, x^{4} - 6874 \, x^{2} - 3660\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 61 \,{\left (32 \, x^{12} + 240 \, x^{10} + 522 \, x^{8} + 305 \, x^{6} - 2 \,{\left (16 \, x^{10} + 80 \, x^{8} + 87 \, x^{6}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )} \log \left (\frac{6 \, x^{2} + \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} + 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - 2 \, \sqrt{3}{\left (976 \, x^{10} + 6100 \, x^{8} + 7605 \, x^{6} - 8754 \, x^{4} - 17244 \, x^{2} - 6264\right )}}{216 \,{\left (2 \, \sqrt{3}{\left (16 \, x^{10} + 80 \, x^{8} + 87 \, x^{6}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (32 \, x^{12} + 240 \, x^{10} + 522 \, x^{8} + 305 \, x^{6}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/216*(2*sqrt(3)*(976*x^8 + 3660*x^6 + 41*x^4 - 6874*x^2 - 3660)*sqrt(x^4 + 5*x
^2 + 3) + 61*(32*x^12 + 240*x^10 + 522*x^8 + 305*x^6 - 2*(16*x^10 + 80*x^8 + 87*
x^6)*sqrt(x^4 + 5*x^2 + 3))*log((6*x^2 + sqrt(3)*(2*x^4 + 5*x^2 + 6) - 2*sqrt(x^
4 + 5*x^2 + 3)*(sqrt(3)*x^2 + 3))/(2*x^4 - 2*sqrt(x^4 + 5*x^2 + 3)*x^2 + 5*x^2))
- 2*sqrt(3)*(976*x^10 + 6100*x^8 + 7605*x^6 - 8754*x^4 - 17244*x^2 - 6264))/(2*
sqrt(3)*(16*x^10 + 80*x^8 + 87*x^6)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(32*x^12 + 2
40*x^10 + 522*x^8 + 305*x^6))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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