3.1316 \(\int \frac{1}{x^{7/2} \sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2 \sqrt{x^5+1}}{5 x^{5/2}} \]

[Out]

(-2*Sqrt[1 + x^5])/(5*x^(5/2))

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Rubi [A]  time = 0.0029125, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{2 \sqrt{x^5+1}}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*Sqrt[1 + x^5]),x]

[Out]

(-2*Sqrt[1 + x^5])/(5*x^(5/2))

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^{7/2} \sqrt{1+x^5}} \, dx &=-\frac{2 \sqrt{1+x^5}}{5 x^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0033145, size = 18, normalized size = 1. \[ -\frac{2 \sqrt{x^5+1}}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*Sqrt[1 + x^5]),x]

[Out]

(-2*Sqrt[1 + x^5])/(5*x^(5/2))

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Maple [B]  time = 0.003, size = 32, normalized size = 1.8 \begin{align*} -{\frac{ \left ( 2+2\,x \right ) \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) }{5}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(x^5+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.973099, size = 16, normalized size = 0.89 \begin{align*} -\frac{2 \, \sqrt{x^{5} + 1}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

-2/5*sqrt(x^5 + 1)/x^(5/2)

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Fricas [A]  time = 1.77472, size = 38, normalized size = 2.11 \begin{align*} -\frac{2 \, \sqrt{x^{5} + 1}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

-2/5*sqrt(x^5 + 1)/x^(5/2)

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Sympy [A]  time = 22.2405, size = 14, normalized size = 0.78 \begin{align*} - \frac{2 \sqrt{1 + \frac{1}{x^{5}}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(7/2)/(x**5+1)**(1/2),x)

[Out]

-2*sqrt(1 + x**(-5))/5

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Giac [A]  time = 1.17961, size = 15, normalized size = 0.83 \begin{align*} -\frac{2}{5} \, \sqrt{\frac{1}{x^{5}} + 1} + \frac{2}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

-2/5*sqrt(1/x^5 + 1) + 2/5