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

Optimal. Leaf size=86 \[ -\frac{3}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (2-3 x^2\right ) \left (x^4+5\right )^{3/2}}{4 x^4}-\frac{3 \left (15-2 x^2\right ) \sqrt{x^4+5}}{4 x^2} \]

[Out]

(-3*(15 - 2*x^2)*Sqrt[5 + x^4])/(4*x^2) - ((2 - 3*x^2)*(5 + x^4)^(3/2))/(4*x^4)
+ (45*ArcSinh[x^2/Sqrt[5]])/4 - (3*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/2

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Rubi [A]  time = 0.198322, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{3}{2} \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+\frac{45}{4} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{\left (2-3 x^2\right ) \left (x^4+5\right )^{3/2}}{4 x^4}-\frac{3 \left (15-2 x^2\right ) \sqrt{x^4+5}}{4 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(15 - 2*x^2)*Sqrt[5 + x^4])/(4*x^2) - ((2 - 3*x^2)*(5 + x^4)^(3/2))/(4*x^4)
+ (45*ArcSinh[x^2/Sqrt[5]])/4 - (3*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/2

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Rubi in Sympy [A]  time = 16.6795, size = 82, normalized size = 0.95 \[ \frac{45 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} - \frac{3 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )}}{2} - \frac{3 \left (- 8 x^{2} + 60\right ) \sqrt{x^{4} + 5}}{16 x^{2}} - \frac{\left (- 6 x^{2} + 4\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{8 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

45*asinh(sqrt(5)*x**2/5)/4 - 3*sqrt(5)*atanh(sqrt(5)*sqrt(x**4 + 5)/5)/2 - 3*(-8
*x**2 + 60)*sqrt(x**4 + 5)/(16*x**2) - (-6*x**2 + 4)*(x**4 + 5)**(3/2)/(8*x**4)

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Mathematica [A]  time = 0.111896, size = 70, normalized size = 0.81 \[ \frac{1}{4} \left (-6 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )+45 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{\sqrt{x^4+5} \left (3 x^6+4 x^4-30 x^2-10\right )}{x^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[5 + x^4]*(-10 - 30*x^2 + 4*x^4 + 3*x^6))/x^4 + 45*ArcSinh[x^2/Sqrt[5]] -
6*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/4

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Maple [A]  time = 0.023, size = 73, normalized size = 0.9 \[ \sqrt{{x}^{4}+5}-{\frac{5}{2\,{x}^{4}}\sqrt{{x}^{4}+5}}-{\frac{3\,\sqrt{5}}{2}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) }+{\frac{3\,{x}^{2}}{4}\sqrt{{x}^{4}+5}}+{\frac{45}{4}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }-{\frac{15}{2\,{x}^{2}}\sqrt{{x}^{4}+5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^4+5)^(1/2)-5/2*(x^4+5)^(1/2)/x^4-3/2*5^(1/2)*arctanh(5^(1/2)/(x^4+5)^(1/2))+3
/4*x^2*(x^4+5)^(1/2)+45/4*arcsinh(1/5*5^(1/2)*x^2)-15/2*(x^4+5)^(1/2)/x^2

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Maxima [A]  time = 0.785619, size = 166, normalized size = 1.93 \[ \frac{3}{4} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + \sqrt{x^{4} + 5} - \frac{15 \, \sqrt{x^{4} + 5}}{2 \, x^{2}} + \frac{15 \, \sqrt{x^{4} + 5}}{4 \, x^{2}{\left (\frac{x^{4} + 5}{x^{4}} - 1\right )}} - \frac{5 \, \sqrt{x^{4} + 5}}{2 \, x^{4}} + \frac{45}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{45}{8} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt(x^4 + 5))) + sqrt(x^4
+ 5) - 15/2*sqrt(x^4 + 5)/x^2 + 15/4*sqrt(x^4 + 5)/(x^2*((x^4 + 5)/x^4 - 1)) -
5/2*sqrt(x^4 + 5)/x^4 + 45/8*log(sqrt(x^4 + 5)/x^2 + 1) - 45/8*log(sqrt(x^4 + 5)
/x^2 - 1)

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Fricas [A]  time = 0.298746, size = 370, normalized size = 4.3 \[ -\frac{24 \, x^{16} + 32 \, x^{14} + 180 \, x^{12} + 160 \, x^{10} - 300 \, x^{8} - 200 \, x^{6} - 2250 \, x^{4} - 1000 \, x^{2} + 45 \,{\left (8 \, x^{12} + 40 \, x^{8} + 25 \, x^{4} - 4 \,{\left (2 \, x^{10} + 5 \, x^{6}\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + 6 \,{\left (4 \, \sqrt{5}{\left (2 \, x^{10} + 5 \, x^{6}\right )} \sqrt{x^{4} + 5} - \sqrt{5}{\left (8 \, x^{12} + 40 \, x^{8} + 25 \, x^{4}\right )}\right )} \log \left (\frac{x^{4} + \sqrt{5} x^{2} - \sqrt{x^{4} + 5}{\left (x^{2} + \sqrt{5}\right )} + 5}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) -{\left (24 \, x^{14} + 32 \, x^{12} + 120 \, x^{10} + 80 \, x^{8} - 525 \, x^{6} - 300 \, x^{4} - 750 \, x^{2} - 250\right )} \sqrt{x^{4} + 5}}{4 \,{\left (8 \, x^{12} + 40 \, x^{8} + 25 \, x^{4} - 4 \,{\left (2 \, x^{10} + 5 \, x^{6}\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(24*x^16 + 32*x^14 + 180*x^12 + 160*x^10 - 300*x^8 - 200*x^6 - 2250*x^4 - 1
000*x^2 + 45*(8*x^12 + 40*x^8 + 25*x^4 - 4*(2*x^10 + 5*x^6)*sqrt(x^4 + 5))*log(-
x^2 + sqrt(x^4 + 5)) + 6*(4*sqrt(5)*(2*x^10 + 5*x^6)*sqrt(x^4 + 5) - sqrt(5)*(8*
x^12 + 40*x^8 + 25*x^4))*log((x^4 + sqrt(5)*x^2 - sqrt(x^4 + 5)*(x^2 + sqrt(5))
+ 5)/(x^4 - sqrt(x^4 + 5)*x^2)) - (24*x^14 + 32*x^12 + 120*x^10 + 80*x^8 - 525*x
^6 - 300*x^4 - 750*x^2 - 250)*sqrt(x^4 + 5))/(8*x^12 + 40*x^8 + 25*x^4 - 4*(2*x^
10 + 5*x^6)*sqrt(x^4 + 5))

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Sympy [A]  time = 23.105, size = 133, normalized size = 1.55 \[ \frac{3 x^{6}}{4 \sqrt{x^{4} + 5}} - \frac{15 x^{2}}{4 \sqrt{x^{4} + 5}} + \sqrt{x^{4} + 5} + \frac{\sqrt{5} \log{\left (x^{4} \right )}}{2} - \sqrt{5} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )} - \frac{\sqrt{5} \operatorname{asinh}{\left (\frac{\sqrt{5}}{x^{2}} \right )}}{2} + \frac{45 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{4} - \frac{5 \sqrt{1 + \frac{5}{x^{4}}}}{2 x^{2}} - \frac{75}{2 x^{2} \sqrt{x^{4} + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**6/(4*sqrt(x**4 + 5)) - 15*x**2/(4*sqrt(x**4 + 5)) + sqrt(x**4 + 5) + sqrt(5
)*log(x**4)/2 - sqrt(5)*log(sqrt(x**4/5 + 1) + 1) - sqrt(5)*asinh(sqrt(5)/x**2)/
2 + 45*asinh(sqrt(5)*x**2/5)/4 - 5*sqrt(1 + 5/x**4)/(2*x**2) - 75/(2*x**2*sqrt(x
**4 + 5))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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