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

Optimal. Leaf size=65 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{10 \sqrt{5}}+\frac{3 x^2+2}{10 x^2 \sqrt{x^4+5}}-\frac{2 \sqrt{x^4+5}}{25 x^2} \]

[Out]

(2 + 3*x^2)/(10*x^2*Sqrt[5 + x^4]) - (2*Sqrt[5 + x^4])/(25*x^2) - (3*ArcTanh[Sqr
t[5 + x^4]/Sqrt[5]])/(10*Sqrt[5])

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Rubi [A]  time = 0.157011, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{10 \sqrt{5}}+\frac{3 x^2+2}{10 x^2 \sqrt{x^4+5}}-\frac{2 \sqrt{x^4+5}}{25 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(2 + 3*x^2)/(10*x^2*Sqrt[5 + x^4]) - (2*Sqrt[5 + x^4])/(25*x^2) - (3*ArcTanh[Sqr
t[5 + x^4]/Sqrt[5]])/(10*Sqrt[5])

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Rubi in Sympy [A]  time = 12.8169, size = 60, normalized size = 0.92 \[ - \frac{3 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )}}{50} + \frac{15 x^{2} + 10}{50 x^{2} \sqrt{x^{4} + 5}} - \frac{2 \sqrt{x^{4} + 5}}{25 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(5)*atanh(sqrt(5)*sqrt(x**4 + 5)/5)/50 + (15*x**2 + 10)/(50*x**2*sqrt(x**
4 + 5)) - 2*sqrt(x**4 + 5)/(25*x**2)

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Mathematica [A]  time = 0.0921717, size = 53, normalized size = 0.82 \[ \frac{1}{50} \left (\frac{-4 x^4+15 x^2-10}{x^2 \sqrt{x^4+5}}-3 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-10 + 15*x^2 - 4*x^4)/(x^2*Sqrt[5 + x^4]) - 3*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/Sq
rt[5]])/50

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Maple [A]  time = 0.019, size = 47, normalized size = 0.7 \[ -{\frac{2\,{x}^{4}+5}{25\,{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3}{10}{\frac{1}{\sqrt{{x}^{4}+5}}}}-{\frac{3\,\sqrt{5}}{50}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25/x^2*(2*x^4+5)/(x^4+5)^(1/2)+3/10/(x^4+5)^(1/2)-3/50*5^(1/2)*arctanh(5^(1/2
)/(x^4+5)^(1/2))

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Maxima [A]  time = 0.779939, size = 92, normalized size = 1.42 \[ -\frac{x^{2}}{25 \, \sqrt{x^{4} + 5}} + \frac{3}{100} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) + \frac{3}{10 \, \sqrt{x^{4} + 5}} - \frac{\sqrt{x^{4} + 5}}{25 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/25*x^2/sqrt(x^4 + 5) + 3/100*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5)
+ sqrt(x^4 + 5))) + 3/10/sqrt(x^4 + 5) - 1/25*sqrt(x^4 + 5)/x^2

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Fricas [A]  time = 0.299276, size = 216, normalized size = 3.32 \[ -\frac{6 \, \sqrt{5} \sqrt{x^{4} + 5} x^{4} + 3 \,{\left (2 \, x^{8} + 10 \, x^{4} -{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5}\right )} \log \left (\frac{5 \, x^{2} + \sqrt{5}{\left (x^{4} + 5\right )} - \sqrt{x^{4} + 5}{\left (\sqrt{5} x^{2} + 5\right )}}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) - \sqrt{5}{\left (6 \, x^{6} + 15 \, x^{2} - 10\right )}}{10 \,{\left (\sqrt{5}{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} - 2 \, \sqrt{5}{\left (x^{8} + 5 \, x^{4}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/10*(6*sqrt(5)*sqrt(x^4 + 5)*x^4 + 3*(2*x^8 + 10*x^4 - (2*x^6 + 5*x^2)*sqrt(x^
4 + 5))*log((5*x^2 + sqrt(5)*(x^4 + 5) - sqrt(x^4 + 5)*(sqrt(5)*x^2 + 5))/(x^4 -
sqrt(x^4 + 5)*x^2)) - sqrt(5)*(6*x^6 + 15*x^2 - 10))/(sqrt(5)*(2*x^6 + 5*x^2)*s
qrt(x^4 + 5) - 2*sqrt(5)*(x^8 + 5*x^4))

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Sympy [A]  time = 27.01, size = 228, normalized size = 3.51 \[ \frac{3 x^{4} \log{\left (x^{4} \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{6 x^{4} \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{3 x^{4} \log{\left (5 \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} + \frac{6 \sqrt{5} \sqrt{x^{4} + 5}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} + \frac{15 \log{\left (x^{4} \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{30 \log{\left (\sqrt{\frac{x^{4}}{5} + 1} + 1 \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{15 \log{\left (5 \right )}}{20 \sqrt{5} x^{4} + 100 \sqrt{5}} - \frac{2}{25 \sqrt{1 + \frac{5}{x^{4}}}} - \frac{1}{5 x^{4} \sqrt{1 + \frac{5}{x^{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**4*log(x**4)/(20*sqrt(5)*x**4 + 100*sqrt(5)) - 6*x**4*log(sqrt(x**4/5 + 1) +
1)/(20*sqrt(5)*x**4 + 100*sqrt(5)) - 3*x**4*log(5)/(20*sqrt(5)*x**4 + 100*sqrt(
5)) + 6*sqrt(5)*sqrt(x**4 + 5)/(20*sqrt(5)*x**4 + 100*sqrt(5)) + 15*log(x**4)/(2
0*sqrt(5)*x**4 + 100*sqrt(5)) - 30*log(sqrt(x**4/5 + 1) + 1)/(20*sqrt(5)*x**4 +
100*sqrt(5)) - 15*log(5)/(20*sqrt(5)*x**4 + 100*sqrt(5)) - 2/(25*sqrt(1 + 5/x**4
)) - 1/(5*x**4*sqrt(1 + 5/x**4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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