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

Optimal. Leaf size=82 \[ -\frac{9}{4} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (4-9 x^2\right ) \sqrt{x^4+5}}{4 x^2}-\frac{\left (9 x^2+4\right ) \left (x^4+5\right )^{3/2}}{12 x^6} \]

[Out]

-((4 - 9*x^2)*Sqrt[5 + x^4])/(4*x^2) - ((4 + 9*x^2)*(5 + x^4)^(3/2))/(12*x^6) +
ArcSinh[x^2/Sqrt[5]] - (9*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/4

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Rubi [A]  time = 0.195773, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{9}{4} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (4-9 x^2\right ) \sqrt{x^4+5}}{4 x^2}-\frac{\left (9 x^2+4\right ) \left (x^4+5\right )^{3/2}}{12 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-((4 - 9*x^2)*Sqrt[5 + x^4])/(4*x^2) - ((4 + 9*x^2)*(5 + x^4)^(3/2))/(12*x^6) +
ArcSinh[x^2/Sqrt[5]] - (9*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/4

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Rubi in Sympy [A]  time = 16.7594, size = 76, normalized size = 0.93 \[ \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} - \frac{9 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )}}{4} - \frac{\left (- 90 x^{2} + 40\right ) \sqrt{x^{4} + 5}}{40 x^{2}} - \frac{\left (45 x^{2} + 20\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(sqrt(5)*x**2/5) - 9*sqrt(5)*atanh(sqrt(5)*sqrt(x**4 + 5)/5)/4 - (-90*x**2
+ 40)*sqrt(x**4 + 5)/(40*x**2) - (45*x**2 + 20)*(x**4 + 5)**(3/2)/(60*x**6)

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Mathematica [A]  time = 0.117886, size = 69, normalized size = 0.84 \[ -\frac{9}{4} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{\sqrt{x^4+5} \left (18 x^6-16 x^4-45 x^2-20\right )}{12 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[5 + x^4]*(-20 - 45*x^2 - 16*x^4 + 18*x^6))/(12*x^6) + ArcSinh[x^2/Sqrt[5]]
- (9*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/4

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Maple [A]  time = 0.024, size = 73, normalized size = 0.9 \[{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) -{\frac{5}{3\,{x}^{6}}\sqrt{{x}^{4}+5}}-{\frac{4}{3\,{x}^{2}}\sqrt{{x}^{4}+5}}+{\frac{3}{2}\sqrt{{x}^{4}+5}}-{\frac{15}{4\,{x}^{4}}\sqrt{{x}^{4}+5}}-{\frac{9\,\sqrt{5}}{4}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsinh(1/5*5^(1/2)*x^2)-5/3*(x^4+5)^(1/2)/x^6-4/3*(x^4+5)^(1/2)/x^2+3/2*(x^4+5)
^(1/2)-15/4*(x^4+5)^(1/2)/x^4-9/4*5^(1/2)*arctanh(5^(1/2)/(x^4+5)^(1/2))

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Maxima [A]  time = 0.780308, size = 151, normalized size = 1.84 \[ \frac{9}{8} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + \frac{3}{2} \, \sqrt{x^{4} + 5} - \frac{\sqrt{x^{4} + 5}}{x^{2}} - \frac{15 \, \sqrt{x^{4} + 5}}{4 \, x^{4}} - \frac{{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{3 \, x^{6}} + \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{1}{2} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9/8*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt(x^4 + 5))) + 3/2*sqrt
(x^4 + 5) - sqrt(x^4 + 5)/x^2 - 15/4*sqrt(x^4 + 5)/x^4 - 1/3*(x^4 + 5)^(3/2)/x^6
+ 1/2*log(sqrt(x^4 + 5)/x^2 + 1) - 1/2*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 0.29371, size = 356, normalized size = 4.34 \[ -\frac{144 \, x^{16} + 720 \, x^{12} - 480 \, x^{10} - 900 \, x^{8} - 2400 \, x^{6} - 4500 \, x^{4} - 2000 \, x^{2} + 12 \,{\left (8 \, x^{14} + 40 \, x^{10} + 25 \, x^{6} - 4 \,{\left (2 \, x^{12} + 5 \, x^{8}\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + 27 \,{\left (4 \, \sqrt{5}{\left (2 \, x^{12} + 5 \, x^{8}\right )} \sqrt{x^{4} + 5} - \sqrt{5}{\left (8 \, x^{14} + 40 \, x^{10} + 25 \, x^{6}\right )}\right )} \log \left (\frac{x^{4} + \sqrt{5} x^{2} - \sqrt{x^{4} + 5}{\left (x^{2} + \sqrt{5}\right )} + 5}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) -{\left (144 \, x^{14} + 360 \, x^{10} - 480 \, x^{8} - 1350 \, x^{6} - 1200 \, x^{4} - 1125 \, x^{2} - 500\right )} \sqrt{x^{4} + 5}}{12 \,{\left (8 \, x^{14} + 40 \, x^{10} + 25 \, x^{6} - 4 \,{\left (2 \, x^{12} + 5 \, x^{8}\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*(144*x^16 + 720*x^12 - 480*x^10 - 900*x^8 - 2400*x^6 - 4500*x^4 - 2000*x^2
+ 12*(8*x^14 + 40*x^10 + 25*x^6 - 4*(2*x^12 + 5*x^8)*sqrt(x^4 + 5))*log(-x^2 +
sqrt(x^4 + 5)) + 27*(4*sqrt(5)*(2*x^12 + 5*x^8)*sqrt(x^4 + 5) - sqrt(5)*(8*x^14
+ 40*x^10 + 25*x^6))*log((x^4 + sqrt(5)*x^2 - sqrt(x^4 + 5)*(x^2 + sqrt(5)) + 5)
/(x^4 - sqrt(x^4 + 5)*x^2)) - (144*x^14 + 360*x^10 - 480*x^8 - 1350*x^6 - 1200*x
^4 - 1125*x^2 - 500)*sqrt(x^4 + 5))/(8*x^14 + 40*x^10 + 25*x^6 - 4*(2*x^12 + 5*x
^8)*sqrt(x^4 + 5))

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Sympy [A]  time = 22.7124, size = 148, normalized size = 1.8 \[ - \frac{x^{2}}{\sqrt{x^{4} + 5}} - \frac{\sqrt{1 + \frac{5}{x^{4}}}}{3} + \frac{3 \sqrt{x^{4} + 5}}{2} + \frac{3 \sqrt{5} \log{\left (x^{4} \right )}}{4} - \frac{3 \sqrt{5} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )}}{2} - \frac{3 \sqrt{5} \operatorname{asinh}{\left (\frac{\sqrt{5}}{x^{2}} \right )}}{4} + \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )} - \frac{15 \sqrt{1 + \frac{5}{x^{4}}}}{4 x^{2}} - \frac{5}{x^{2} \sqrt{x^{4} + 5}} - \frac{5 \sqrt{1 + \frac{5}{x^{4}}}}{3 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2/sqrt(x**4 + 5) - sqrt(1 + 5/x**4)/3 + 3*sqrt(x**4 + 5)/2 + 3*sqrt(5)*log(x
**4)/4 - 3*sqrt(5)*log(sqrt(x**4/5 + 1) + 1)/2 - 3*sqrt(5)*asinh(sqrt(5)/x**2)/4
+ asinh(sqrt(5)*x**2/5) - 15*sqrt(1 + 5/x**4)/(4*x**2) - 5/(x**2*sqrt(x**4 + 5)
) - 5*sqrt(1 + 5/x**4)/(3*x**4)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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