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

Optimal. Leaf size=58 \[ -\frac{3 \sqrt{x^4+5}}{4 x^4}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{4 \sqrt{5}}-\frac{\left (x^4+5\right )^{3/2}}{15 x^6} \]

[Out]

(-3*Sqrt[5 + x^4])/(4*x^4) - (5 + x^4)^(3/2)/(15*x^6) - (3*ArcTanh[Sqrt[5 + x^4]
/Sqrt[5]])/(4*Sqrt[5])

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Rubi [A]  time = 0.121051, antiderivative size = 58, 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 \[ -\frac{3 \sqrt{x^4+5}}{4 x^4}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )}{4 \sqrt{5}}-\frac{\left (x^4+5\right )^{3/2}}{15 x^6} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[5 + x^4])/(4*x^4) - (5 + x^4)^(3/2)/(15*x^6) - (3*ArcTanh[Sqrt[5 + x^4]
/Sqrt[5]])/(4*Sqrt[5])

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Rubi in Sympy [A]  time = 11.4744, size = 54, normalized size = 0.93 \[ - \frac{3 \sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{4} + 5}}{5} \right )}}{20} - \frac{3 \sqrt{x^{4} + 5}}{4 x^{4}} - \frac{\left (x^{4} + 5\right )^{\frac{3}{2}}}{15 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3*x**2+2)*(x**4+5)**(1/2)/x**7,x)

[Out]

-3*sqrt(5)*atanh(sqrt(5)*sqrt(x**4 + 5)/5)/20 - 3*sqrt(x**4 + 5)/(4*x**4) - (x**
4 + 5)**(3/2)/(15*x**6)

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Mathematica [A]  time = 0.0549388, size = 54, normalized size = 0.93 \[ \frac{1}{60} \left (-9 \sqrt{5} \tanh ^{-1}\left (\frac{\sqrt{x^4+5}}{\sqrt{5}}\right )-\frac{\sqrt{x^4+5} \left (4 x^4+45 x^2+20\right )}{x^6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-((Sqrt[5 + x^4]*(20 + 45*x^2 + 4*x^4))/x^6) - 9*Sqrt[5]*ArcTanh[Sqrt[5 + x^4]/
Sqrt[5]])/60

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Maple [A]  time = 0.018, size = 52, normalized size = 0.9 \[ -{\frac{1}{15\,{x}^{6}} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}}-{\frac{3}{20\,{x}^{4}} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}}+{\frac{3}{20}\sqrt{{x}^{4}+5}}-{\frac{3\,\sqrt{5}}{20}{\it Artanh} \left ({\sqrt{5}{\frac{1}{\sqrt{{x}^{4}+5}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3*x^2+2)*(x^4+5)^(1/2)/x^7,x)

[Out]

-1/15*(x^4+5)^(3/2)/x^6-3/20/x^4*(x^4+5)^(3/2)+3/20*(x^4+5)^(1/2)-3/20*5^(1/2)*a
rctanh(5^(1/2)/(x^4+5)^(1/2))

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Maxima [A]  time = 0.776061, size = 80, normalized size = 1.38 \[ \frac{3}{40} \, \sqrt{5} \log \left (-\frac{\sqrt{5} - \sqrt{x^{4} + 5}}{\sqrt{5} + \sqrt{x^{4} + 5}}\right ) - \frac{3 \, \sqrt{x^{4} + 5}}{4 \, x^{4}} - \frac{{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{15 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/40*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt(x^4 + 5))) - 3/4*sqr
t(x^4 + 5)/x^4 - 1/15*(x^4 + 5)^(3/2)/x^6

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Fricas [A]  time = 0.266378, size = 263, normalized size = 4.53 \[ \frac{3 \, \sqrt{5}{\left (12 \, x^{8} + 8 \, x^{6} + 45 \, x^{4} + 20 \, x^{2}\right )} \sqrt{x^{4} + 5} - 9 \,{\left (4 \, x^{12} + 15 \, x^{8} -{\left (4 \, x^{10} + 5 \, x^{6}\right )} \sqrt{x^{4} + 5}\right )} \log \left (\frac{5 \, x^{2} + \sqrt{5}{\left (x^{4} + 5\right )} - \sqrt{x^{4} + 5}{\left (\sqrt{5} x^{2} + 5\right )}}{x^{4} - \sqrt{x^{4} + 5} x^{2}}\right ) - \sqrt{5}{\left (36 \, x^{10} + 24 \, x^{8} + 225 \, x^{6} + 120 \, x^{4} + 225 \, x^{2} + 100\right )}}{12 \,{\left (\sqrt{5}{\left (4 \, x^{10} + 5 \, x^{6}\right )} \sqrt{x^{4} + 5} - \sqrt{5}{\left (4 \, x^{12} + 15 \, x^{8}\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*sqrt(5)*(12*x^8 + 8*x^6 + 45*x^4 + 20*x^2)*sqrt(x^4 + 5) - 9*(4*x^12 + 1
5*x^8 - (4*x^10 + 5*x^6)*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)*(36*x^10 + 24*x^8
+ 225*x^6 + 120*x^4 + 225*x^2 + 100))/(sqrt(5)*(4*x^10 + 5*x^6)*sqrt(x^4 + 5) -
sqrt(5)*(4*x^12 + 15*x^8))

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Sympy [A]  time = 11.988, size = 63, normalized size = 1.09 \[ - \frac{\sqrt{1 + \frac{5}{x^{4}}}}{15} - \frac{3 \sqrt{5} \operatorname{asinh}{\left (\frac{\sqrt{5}}{x^{2}} \right )}}{20} - \frac{3 \sqrt{1 + \frac{5}{x^{4}}}}{4 x^{2}} - \frac{\sqrt{1 + \frac{5}{x^{4}}}}{3 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x**2+2)*(x**4+5)**(1/2)/x**7,x)

[Out]

-sqrt(1 + 5/x**4)/15 - 3*sqrt(5)*asinh(sqrt(5)/x**2)/20 - 3*sqrt(1 + 5/x**4)/(4*
x**2) - sqrt(1 + 5/x**4)/(3*x**4)

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GIAC/XCAS [A]  time = 0.269881, size = 84, normalized size = 1.45 \[ -\frac{1}{60} \,{\left (\frac{5 \,{\left (\frac{4}{x^{2}} + 9\right )}}{x^{2}} + 4\right )} \sqrt{\frac{5}{x^{4}} + 1} - \frac{3}{40} \, \sqrt{5}{\rm ln}\left (\sqrt{5} + \sqrt{x^{4} + 5}\right ) + \frac{3}{40} \, \sqrt{5}{\rm ln}\left (-\sqrt{5} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(5*(4/x^2 + 9)/x^2 + 4)*sqrt(5/x^4 + 1) - 3/40*sqrt(5)*ln(sqrt(5) + sqrt(x
^4 + 5)) + 3/40*sqrt(5)*ln(-sqrt(5) + sqrt(x^4 + 5))