3.1315 \(\int \frac{1}{x^{17/2} \sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=37 \[ \frac{4 \sqrt{x^5+1}}{15 x^{5/2}}-\frac{2 \sqrt{x^5+1}}{15 x^{15/2}} \]

[Out]

(-2*Sqrt[1 + x^5])/(15*x^(15/2)) + (4*Sqrt[1 + x^5])/(15*x^(5/2))

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Rubi [A]  time = 0.0245888, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{4 \sqrt{x^5+1}}{15 x^{5/2}}-\frac{2 \sqrt{x^5+1}}{15 x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(17/2)*Sqrt[1 + x^5]),x]

[Out]

(-2*Sqrt[1 + x^5])/(15*x^(15/2)) + (4*Sqrt[1 + x^5])/(15*x^(5/2))

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Rubi in Sympy [A]  time = 3.22701, size = 32, normalized size = 0.86 \[ \frac{4 \sqrt{x^{5} + 1}}{15 x^{\frac{5}{2}}} - \frac{2 \sqrt{x^{5} + 1}}{15 x^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(17/2)/(x**5+1)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0150802, size = 25, normalized size = 0.68 \[ \frac{2 \sqrt{x^5+1} \left (2 x^5-1\right )}{15 x^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(17/2)*Sqrt[1 + x^5]),x]

[Out]

(2*Sqrt[1 + x^5]*(-1 + 2*x^5))/(15*x^(15/2))

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Maple [A]  time = 0.007, size = 39, normalized size = 1.1 \[{\frac{ \left ( 2+2\,x \right ) \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) \left ( 2\,{x}^{5}-1 \right ) }{15}{x}^{-{\frac{15}{2}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(17/2)/(x^5+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.57431, size = 34, normalized size = 0.92 \[ \frac{2 \, \sqrt{x^{5} + 1}}{5 \, x^{\frac{5}{2}}} - \frac{2 \,{\left (x^{5} + 1\right )}^{\frac{3}{2}}}{15 \, x^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^5 + 1)*x^(17/2)),x, algorithm="maxima")

[Out]

2/5*sqrt(x^5 + 1)/x^(5/2) - 2/15*(x^5 + 1)^(3/2)/x^(15/2)

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Fricas [A]  time = 0.226632, size = 26, normalized size = 0.7 \[ \frac{2 \,{\left (2 \, x^{5} - 1\right )} \sqrt{x^{5} + 1}}{15 \, x^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^5 + 1)*x^(17/2)),x, algorithm="fricas")

[Out]

2/15*(2*x^5 - 1)*sqrt(x^5 + 1)/x^(15/2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(17/2)/(x**5+1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22572, size = 27, normalized size = 0.73 \[ -\frac{2}{15} \,{\left (\frac{1}{x^{5}} + 1\right )}^{\frac{3}{2}} + \frac{2}{5} \, \sqrt{\frac{1}{x^{5}} + 1} - \frac{4}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^5 + 1)*x^(17/2)),x, algorithm="giac")

[Out]

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