3.150 \(\int \frac{\left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4}}{x^{11}} \, dx\)

Optimal. Leaf size=132 \[ -\frac{161 \left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{5184 x^4}+\frac{2093 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{10368 \sqrt{3}}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}+\frac{173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6} \]

[Out]

(-161*(6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/(5184*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(1
5*x^10) - (3 + 5*x^2 + x^4)^(3/2)/(36*x^8) + (173*(3 + 5*x^2 + x^4)^(3/2))/(3240
*x^6) + (2093*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(10368*Sqr
t[3])

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Rubi [A]  time = 0.268982, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{161 \left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{5184 x^4}+\frac{2093 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{10368 \sqrt{3}}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}+\frac{173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^11,x]

[Out]

(-161*(6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/(5184*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(1
5*x^10) - (3 + 5*x^2 + x^4)^(3/2)/(36*x^8) + (173*(3 + 5*x^2 + x^4)^(3/2))/(3240
*x^6) + (2093*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(10368*Sqr
t[3])

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Rubi in Sympy [A]  time = 27.3705, size = 121, normalized size = 0.92 \[ \frac{2093 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{31104} - \frac{161 \left (5 x^{2} + 6\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{5184 x^{4}} + \frac{173 \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{3240 x^{6}} - \frac{\left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{36 x^{8}} - \frac{\left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{15 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)*(x**4+5*x**2+3)**(1/2)/x**11,x)

[Out]

2093*sqrt(3)*atanh(sqrt(3)*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))/31104 - 161
*(5*x**2 + 6)*sqrt(x**4 + 5*x**2 + 3)/(5184*x**4) + 173*(x**4 + 5*x**2 + 3)**(3/
2)/(3240*x**6) - (x**4 + 5*x**2 + 3)**(3/2)/(36*x**8) - (x**4 + 5*x**2 + 3)**(3/
2)/(15*x**10)

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Mathematica [A]  time = 0.124435, size = 89, normalized size = 0.67 \[ \frac{-10465 \sqrt{3} \left (\log \left (x^2\right )-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )\right )-\frac{6 \sqrt{x^4+5 x^2+3} \left (2641 x^8-1370 x^6+1176 x^4+10800 x^2+5184\right )}{x^{10}}}{155520} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x^2)*Sqrt[3 + 5*x^2 + x^4])/x^11,x]

[Out]

((-6*Sqrt[3 + 5*x^2 + x^4]*(5184 + 10800*x^2 + 1176*x^4 - 1370*x^6 + 2641*x^8))/
x^10 - 10465*Sqrt[3]*(Log[x^2] - Log[6 + 5*x^2 + 2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]
]))/155520

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Maple [A]  time = 0.025, size = 152, normalized size = 1.2 \[ -{\frac{1}{15\,{x}^{10}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{36\,{x}^{8}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{173}{3240\,{x}^{6}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{161}{2592\,{x}^{4}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{805}{15552\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{2093}{31104}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{2093\,\sqrt{3}}{31104}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{1610\,{x}^{2}+4025}{31104}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)*(x^4+5*x^2+3)^(1/2)/x^11,x)

[Out]

-1/15*(x^4+5*x^2+3)^(3/2)/x^10-1/36*(x^4+5*x^2+3)^(3/2)/x^8+173/3240*(x^4+5*x^2+
3)^(3/2)/x^6-161/2592/x^4*(x^4+5*x^2+3)^(3/2)+805/15552/x^2*(x^4+5*x^2+3)^(3/2)-
2093/31104*(x^4+5*x^2+3)^(1/2)+2093/31104*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x
^2+3)^(1/2))*3^(1/2)-805/31104*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.815431, size = 180, normalized size = 1.36 \[ \frac{2093}{31104} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{161}{2592} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{805 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{5184 \, x^{2}} - \frac{161 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{2592 \, x^{4}} + \frac{173 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{3240 \, x^{6}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{36 \, x^{8}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{15 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)/x^11,x, algorithm="maxima")

[Out]

2093/31104*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 161/25
92*sqrt(x^4 + 5*x^2 + 3) + 805/5184*sqrt(x^4 + 5*x^2 + 3)/x^2 - 161/2592*(x^4 +
5*x^2 + 3)^(3/2)/x^4 + 173/3240*(x^4 + 5*x^2 + 3)^(3/2)/x^6 - 1/36*(x^4 + 5*x^2
+ 3)^(3/2)/x^8 - 1/15*(x^4 + 5*x^2 + 3)^(3/2)/x^10

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Fricas [A]  time = 0.275158, size = 470, normalized size = 3.56 \[ \frac{10 \, \sqrt{3}{\left (535808 \, x^{16} + 4688320 \, x^{14} + 14822768 \, x^{12} + 30236712 \, x^{10} + 80707685 \, x^{8} + 178917094 \, x^{6} + 207290040 \, x^{4} + 111922992 \, x^{2} + 22524480\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 10465 \,{\left (512 \, x^{20} + 6400 \, x^{18} + 29920 \, x^{16} + 64400 \, x^{14} + 62690 \, x^{12} + 21725 \, x^{10} - 2 \,{\left (256 \, x^{18} + 2560 \, x^{16} + 8976 \, x^{14} + 12880 \, x^{12} + 6269 \, x^{10}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )} \log \left (-\frac{6 \, x^{2} - \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} - 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - 2 \, \sqrt{3}{\left (2679040 \, x^{18} + 30139200 \, x^{16} + 128364400 \, x^{14} + 309259160 \, x^{12} + 725547665 \, x^{10} + 1784368010 \, x^{8} + 2899650660 \, x^{6} + 2567795760 \, x^{4} + 1131835680 \, x^{2} + 194990976\right )}}{51840 \,{\left (2 \, \sqrt{3}{\left (256 \, x^{18} + 2560 \, x^{16} + 8976 \, x^{14} + 12880 \, x^{12} + 6269 \, x^{10}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (512 \, x^{20} + 6400 \, x^{18} + 29920 \, x^{16} + 64400 \, x^{14} + 62690 \, x^{12} + 21725 \, x^{10}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)/x^11,x, algorithm="fricas")

[Out]

1/51840*(10*sqrt(3)*(535808*x^16 + 4688320*x^14 + 14822768*x^12 + 30236712*x^10
+ 80707685*x^8 + 178917094*x^6 + 207290040*x^4 + 111922992*x^2 + 22524480)*sqrt(
x^4 + 5*x^2 + 3) - 10465*(512*x^20 + 6400*x^18 + 29920*x^16 + 64400*x^14 + 62690
*x^12 + 21725*x^10 - 2*(256*x^18 + 2560*x^16 + 8976*x^14 + 12880*x^12 + 6269*x^1
0)*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)*(2679040*x^18 + 30139200*x^16 + 128364400*x^14 + 309259160*x^12 + 72
5547665*x^10 + 1784368010*x^8 + 2899650660*x^6 + 2567795760*x^4 + 1131835680*x^2
+ 194990976))/(2*sqrt(3)*(256*x^18 + 2560*x^16 + 8976*x^14 + 12880*x^12 + 6269*
x^10)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(512*x^20 + 6400*x^18 + 29920*x^16 + 64400
*x^14 + 62690*x^12 + 21725*x^10))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{x^{11}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)*(x**4+5*x**2+3)**(1/2)/x**11,x)

[Out]

Integral((3*x**2 + 2)*sqrt(x**4 + 5*x**2 + 3)/x**11, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)/x^11,x, algorithm="giac")

[Out]

integrate(sqrt(x^4 + 5*x^2 + 3)*(3*x^2 + 2)/x^11, x)