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

Optimal. Leaf size=90 \[ -\frac{\left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{18 x^4}+\frac{13 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{36 \sqrt{3}}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{9 x^6} \]

[Out]

-((6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/(18*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(9*x^6)
+ (13*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(36*Sqrt[3])

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Rubi [A]  time = 0.163155, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{18 x^4}+\frac{13 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{36 \sqrt{3}}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{9 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-((6 + 5*x^2)*Sqrt[3 + 5*x^2 + x^4])/(18*x^4) - (3 + 5*x^2 + x^4)^(3/2)/(9*x^6)
+ (13*ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(36*Sqrt[3])

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Rubi in Sympy [A]  time = 18.302, size = 80, normalized size = 0.89 \[ \frac{13 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{108} - \frac{\left (5 x^{2} + 6\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{18 x^{4}} - \frac{\left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{9 x^{6}} \]

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**7,x)

[Out]

13*sqrt(3)*atanh(sqrt(3)*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))/108 - (5*x**2
+ 6)*sqrt(x**4 + 5*x**2 + 3)/(18*x**4) - (x**4 + 5*x**2 + 3)**(3/2)/(9*x**6)

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Mathematica [A]  time = 0.100609, size = 79, normalized size = 0.88 \[ \frac{1}{108} \left (-13 \sqrt{3} \left (\log \left (x^2\right )-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )\right )-\frac{6 \sqrt{x^4+5 x^2+3} \left (7 x^4+16 x^2+6\right )}{x^6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-6*Sqrt[3 + 5*x^2 + x^4]*(6 + 16*x^2 + 7*x^4))/x^6 - 13*Sqrt[3]*(Log[x^2] - Lo
g[6 + 5*x^2 + 2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]]))/108

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Maple [A]  time = 0.02, size = 118, normalized size = 1.3 \[ -{\frac{1}{9\,{x}^{6}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{9\,{x}^{4}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{5}{54\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{13}{108}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{13\,\sqrt{3}}{108}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{10\,{x}^{2}+25}{108}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

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^7,x)

[Out]

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

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Maxima [A]  time = 0.807117, size = 134, normalized size = 1.49 \[ \frac{13}{108} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{1}{9} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{5 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{18 \, x^{2}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{9 \, x^{4}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{9 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

13/108*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 + 5) + 1/9*sqrt(x
^4 + 5*x^2 + 3) + 5/18*sqrt(x^4 + 5*x^2 + 3)/x^2 - 1/9*(x^4 + 5*x^2 + 3)^(3/2)/x
^4 - 1/9*(x^4 + 5*x^2 + 3)^(3/2)/x^6

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Fricas [A]  time = 0.274619, size = 362, normalized size = 4.02 \[ \frac{2 \, \sqrt{3}{\left (1072 \, x^{8} + 6468 \, x^{6} + 11927 \, x^{4} + 8012 \, x^{2} + 1830\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - 13 \,{\left (32 \, x^{12} + 240 \, x^{10} + 522 \, x^{8} + 305 \, x^{6} - 2 \,{\left (16 \, x^{10} + 80 \, x^{8} + 87 \, x^{6}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}\right )} \log \left (-\frac{6 \, x^{2} - \sqrt{3}{\left (2 \, x^{4} + 5 \, x^{2} + 6\right )} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (\sqrt{3} x^{2} - 3\right )}}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - 2 \, \sqrt{3}{\left (1072 \, x^{10} + 9148 \, x^{8} + 26355 \, x^{6} + 31674 \, x^{4} + 16452 \, x^{2} + 3132\right )}}{36 \,{\left (2 \, \sqrt{3}{\left (16 \, x^{10} + 80 \, x^{8} + 87 \, x^{6}\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (32 \, x^{12} + 240 \, x^{10} + 522 \, x^{8} + 305 \, x^{6}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*(2*sqrt(3)*(1072*x^8 + 6468*x^6 + 11927*x^4 + 8012*x^2 + 1830)*sqrt(x^4 + 5
*x^2 + 3) - 13*(32*x^12 + 240*x^10 + 522*x^8 + 305*x^6 - 2*(16*x^10 + 80*x^8 + 8
7*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)*(1072*x^10 + 9148*x^8 + 26355*x^6 + 31674*x^4 + 16452*x^2 + 3132
))/(2*sqrt(3)*(16*x^10 + 80*x^8 + 87*x^6)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(32*x^
12 + 240*x^10 + 522*x^8 + 305*x^6))

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

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**7,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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