3.445 \(\int \frac{1}{x^{10} \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=63 \[ -\frac{5 \sqrt{x^3+1}}{24 x^3}+\frac{5}{24} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{9 x^9}+\frac{5 \sqrt{x^3+1}}{36 x^6} \]

[Out]

-Sqrt[1 + x^3]/(9*x^9) + (5*Sqrt[1 + x^3])/(36*x^6) - (5*Sqrt[1 + x^3])/(24*x^3)
+ (5*ArcTanh[Sqrt[1 + x^3]])/24

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Rubi [A]  time = 0.0615414, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{5 \sqrt{x^3+1}}{24 x^3}+\frac{5}{24} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{9 x^9}+\frac{5 \sqrt{x^3+1}}{36 x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*Sqrt[1 + x^3]),x]

[Out]

-Sqrt[1 + x^3]/(9*x^9) + (5*Sqrt[1 + x^3])/(36*x^6) - (5*Sqrt[1 + x^3])/(24*x^3)
+ (5*ArcTanh[Sqrt[1 + x^3]])/24

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Rubi in Sympy [A]  time = 5.72421, size = 56, normalized size = 0.89 \[ \frac{5 \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{24} - \frac{5 \sqrt{x^{3} + 1}}{24 x^{3}} + \frac{5 \sqrt{x^{3} + 1}}{36 x^{6}} - \frac{\sqrt{x^{3} + 1}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(x**3+1)**(1/2),x)

[Out]

5*atanh(sqrt(x**3 + 1))/24 - 5*sqrt(x**3 + 1)/(24*x**3) + 5*sqrt(x**3 + 1)/(36*x
**6) - sqrt(x**3 + 1)/(9*x**9)

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Mathematica [A]  time = 0.0457752, size = 42, normalized size = 0.67 \[ \frac{1}{72} \left (15 \tanh ^{-1}\left (\sqrt{x^3+1}\right )+\frac{\sqrt{x^3+1} \left (-15 x^6+10 x^3-8\right )}{x^9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*Sqrt[1 + x^3]),x]

[Out]

((Sqrt[1 + x^3]*(-8 + 10*x^3 - 15*x^6))/x^9 + 15*ArcTanh[Sqrt[1 + x^3]])/72

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Maple [A]  time = 0.033, size = 48, normalized size = 0.8 \[{\frac{5}{24}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }-{\frac{1}{9\,{x}^{9}}\sqrt{{x}^{3}+1}}+{\frac{5}{36\,{x}^{6}}\sqrt{{x}^{3}+1}}-{\frac{5}{24\,{x}^{3}}\sqrt{{x}^{3}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(x^3+1)^(1/2),x)

[Out]

5/24*arctanh((x^3+1)^(1/2))-1/9*(x^3+1)^(1/2)/x^9+5/36*(x^3+1)^(1/2)/x^6-5/24*(x
^3+1)^(1/2)/x^3

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Maxima [A]  time = 1.43967, size = 108, normalized size = 1.71 \[ -\frac{15 \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{x^{3} + 1}}{72 \,{\left ({\left (x^{3} + 1\right )}^{3} + 3 \, x^{3} - 3 \,{\left (x^{3} + 1\right )}^{2} + 2\right )}} + \frac{5}{48} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{5}{48} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^10),x, algorithm="maxima")

[Out]

-1/72*(15*(x^3 + 1)^(5/2) - 40*(x^3 + 1)^(3/2) + 33*sqrt(x^3 + 1))/((x^3 + 1)^3
+ 3*x^3 - 3*(x^3 + 1)^2 + 2) + 5/48*log(sqrt(x^3 + 1) + 1) - 5/48*log(sqrt(x^3 +
1) - 1)

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Fricas [A]  time = 0.227765, size = 77, normalized size = 1.22 \[ \frac{15 \, x^{9} \log \left (\sqrt{x^{3} + 1} + 1\right ) - 15 \, x^{9} \log \left (\sqrt{x^{3} + 1} - 1\right ) - 2 \,{\left (15 \, x^{6} - 10 \, x^{3} + 8\right )} \sqrt{x^{3} + 1}}{144 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^10),x, algorithm="fricas")

[Out]

1/144*(15*x^9*log(sqrt(x^3 + 1) + 1) - 15*x^9*log(sqrt(x^3 + 1) - 1) - 2*(15*x^6
- 10*x^3 + 8)*sqrt(x^3 + 1))/x^9

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Sympy [A]  time = 17.4923, size = 85, normalized size = 1.35 \[ \frac{5 \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} - \frac{5}{24 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{5}{72 x^{\frac{9}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{1}{36 x^{\frac{15}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{1}{9 x^{\frac{21}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(x**3+1)**(1/2),x)

[Out]

5*asinh(x**(-3/2))/24 - 5/(24*x**(3/2)*sqrt(1 + x**(-3))) - 5/(72*x**(9/2)*sqrt(
1 + x**(-3))) + 1/(36*x**(15/2)*sqrt(1 + x**(-3))) - 1/(9*x**(21/2)*sqrt(1 + x**
(-3)))

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GIAC/XCAS [A]  time = 0.245071, size = 80, normalized size = 1.27 \[ -\frac{15 \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{x^{3} + 1}}{72 \, x^{9}} + \frac{5}{48} \,{\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) - \frac{5}{48} \,{\rm ln}\left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^10),x, algorithm="giac")

[Out]

-1/72*(15*(x^3 + 1)^(5/2) - 40*(x^3 + 1)^(3/2) + 33*sqrt(x^3 + 1))/x^9 + 5/48*ln
(sqrt(x^3 + 1) + 1) - 5/48*ln(abs(sqrt(x^3 + 1) - 1))