3.369 \(\int \frac{\sqrt{\frac{a}{x^7}}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2}{5} x \sqrt{x^5+1} \sqrt{\frac{a}{x^7}} \]

[Out]

(-2*Sqrt[a/x^7]*x*Sqrt[1 + x^5])/5

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Rubi [A]  time = 0.0043128, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 264} \[ -\frac{2}{5} x \sqrt{x^5+1} \sqrt{\frac{a}{x^7}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x^7]/Sqrt[1 + x^5],x]

[Out]

(-2*Sqrt[a/x^7]*x*Sqrt[1 + x^5])/5

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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{\sqrt{\frac{a}{x^7}}}{\sqrt{1+x^5}} \, dx &=\left (\sqrt{\frac{a}{x^7}} x^{7/2}\right ) \int \frac{1}{x^{7/2} \sqrt{1+x^5}} \, dx\\ &=-\frac{2}{5} \sqrt{\frac{a}{x^7}} x \sqrt{1+x^5}\\ \end{align*}

Mathematica [A]  time = 0.0038984, size = 23, normalized size = 1. \[ -\frac{2}{5} x \sqrt{x^5+1} \sqrt{\frac{a}{x^7}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x^7]/Sqrt[1 + x^5],x]

[Out]

(-2*Sqrt[a/x^7]*x*Sqrt[1 + x^5])/5

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Maple [B]  time = 0.004, size = 37, normalized size = 1.6 \begin{align*} -{\frac{2\,x \left ( 1+x \right ) \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) }{5}\sqrt{{\frac{a}{{x}^{7}}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*x*(1+x)*(x^4-x^3+x^2-x+1)*(a/x^7)^(1/2)/(x^5+1)^(1/2)

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Maxima [B]  time = 1.55219, size = 55, normalized size = 2.39 \begin{align*} -\frac{2 \,{\left (\sqrt{a} x^{6} + \sqrt{a} x\right )}}{5 \, \sqrt{x^{4} - x^{3} + x^{2} - x + 1} \sqrt{x + 1} x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(sqrt(a)*x^6 + sqrt(a)*x)/(sqrt(x^4 - x^3 + x^2 - x + 1)*sqrt(x + 1)*x^(7/2))

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Fricas [A]  time = 1.22383, size = 46, normalized size = 2. \begin{align*} -\frac{2}{5} \, \sqrt{x^{5} + 1} x \sqrt{\frac{a}{x^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*sqrt(x^5 + 1)*x*sqrt(a/x^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{7}}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a/x**7)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

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Giac [A]  time = 1.24845, size = 38, normalized size = 1.65 \begin{align*} -\frac{2 \, a^{3}{\left (\frac{\sqrt{a + \frac{a}{x^{5}}}}{a^{2}} - \frac{1}{a^{\frac{3}{2}}}\right )}}{5 \,{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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