3.1317 \(\int \frac{1}{x^{17/2} \sqrt{1+x^5}} \, dx\)

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 = 0.0067844, 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, Rules used = {271, 264} \[ \frac{4 \sqrt{x^5+1}}{15 x^{5/2}}-\frac{2 \sqrt{x^5+1}}{15 x^{15/2}} \]

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))

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^{17/2} \sqrt{1+x^5}} \, dx &=-\frac{2 \sqrt{1+x^5}}{15 x^{15/2}}-\frac{2}{3} \int \frac{1}{x^{7/2} \sqrt{1+x^5}} \, dx\\ &=-\frac{2 \sqrt{1+x^5}}{15 x^{15/2}}+\frac{4 \sqrt{1+x^5}}{15 x^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0054414, size = 25, normalized size = 0.68 \[ -\frac{2 \left (1-2 x^5\right ) \sqrt{x^5+1}}{15 x^{15/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 39, normalized size = 1.1 \begin{align*}{\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}}}} \end{align*}

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 = 0.975134, size = 34, normalized size = 0.92 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(17/2)/(x^5+1)^(1/2),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 = 1.72838, size = 55, normalized size = 1.49 \begin{align*} \frac{2 \,{\left (2 \, x^{5} - 1\right )} \sqrt{x^{5} + 1}}{15 \, x^{\frac{15}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(17/2)/(x^5+1)^(1/2),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. \begin{align*} \text{Timed out} \end{align*}

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 [A]  time = 1.21011, size = 27, normalized size = 0.73 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(17/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

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