3.370 \(\int \frac{\sqrt{\frac{a}{x^{17}}}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=49 \[ \frac{4}{15} x^6 \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}}-\frac{2}{15} x \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}} \]

[Out]

(-2*Sqrt[a/x^17]*x*Sqrt[1 + x^5])/15 + (4*Sqrt[a/x^17]*x^6*Sqrt[1 + x^5])/15

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Rubi [A]  time = 0.0093764, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {15, 271, 264} \[ \frac{4}{15} x^6 \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}}-\frac{2}{15} x \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x^17]/Sqrt[1 + x^5],x]

[Out]

(-2*Sqrt[a/x^17]*x*Sqrt[1 + x^5])/15 + (4*Sqrt[a/x^17]*x^6*Sqrt[1 + x^5])/15

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 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{\sqrt{\frac{a}{x^{17}}}}{\sqrt{1+x^5}} \, dx &=\left (\sqrt{\frac{a}{x^{17}}} x^{17/2}\right ) \int \frac{1}{x^{17/2} \sqrt{1+x^5}} \, dx\\ &=-\frac{2}{15} \sqrt{\frac{a}{x^{17}}} x \sqrt{1+x^5}-\frac{1}{3} \left (2 \sqrt{\frac{a}{x^{17}}} x^{17/2}\right ) \int \frac{1}{x^{7/2} \sqrt{1+x^5}} \, dx\\ &=-\frac{2}{15} \sqrt{\frac{a}{x^{17}}} x \sqrt{1+x^5}+\frac{4}{15} \sqrt{\frac{a}{x^{17}}} x^6 \sqrt{1+x^5}\\ \end{align*}

Mathematica [A]  time = 0.0062115, size = 30, normalized size = 0.61 \[ -\frac{2}{15} x \left (1-2 x^5\right ) \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x^17]/Sqrt[1 + x^5],x]

[Out]

(-2*Sqrt[a/x^17]*x*(1 - 2*x^5)*Sqrt[1 + x^5])/15

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Maple [A]  time = 0.004, size = 44, normalized size = 0.9 \begin{align*}{\frac{2\,x \left ( 1+x \right ) \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) \left ( 2\,{x}^{5}-1 \right ) }{15}\sqrt{{\frac{a}{{x}^{17}}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/x^17)^(1/2)/(x^5+1)^(1/2),x)

[Out]

2/15*x*(1+x)*(x^4-x^3+x^2-x+1)*(2*x^5-1)*(a/x^17)^(1/2)/(x^5+1)^(1/2)

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Maxima [A]  time = 1.61558, size = 68, normalized size = 1.39 \begin{align*} \frac{2 \,{\left (2 \, \sqrt{a} x^{11} + \sqrt{a} x^{6} - \sqrt{a} x\right )}}{15 \, \sqrt{x^{4} - x^{3} + x^{2} - x + 1} \sqrt{x + 1} x^{\frac{17}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(2*sqrt(a)*x^11 + sqrt(a)*x^6 - sqrt(a)*x)/(sqrt(x^4 - x^3 + x^2 - x + 1)*sqrt(x + 1)*x^(17/2))

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Fricas [A]  time = 1.30168, size = 61, normalized size = 1.24 \begin{align*} \frac{2}{15} \,{\left (2 \, x^{6} - x\right )} \sqrt{x^{5} + 1} \sqrt{\frac{a}{x^{17}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(2*x^6 - x)*sqrt(x^5 + 1)*sqrt(a/x^17)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{17}}}}{\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**17)**(1/2)/(x**5+1)**(1/2),x)

[Out]

Integral(sqrt(a/x**17)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

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Giac [A]  time = 1.16706, size = 63, normalized size = 1.29 \begin{align*} -\frac{2 \, a^{3}{\left (\frac{2}{a^{\frac{3}{2}}} + \frac{{\left (a + \frac{a}{x^{5}}\right )}^{\frac{3}{2}} a - 3 \, \sqrt{a + \frac{a}{x^{5}}} a^{2}}{a^{4}}\right )} \mathrm{sgn}\left (x\right )}{15 \,{\left | a \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*a^3*(2/a^(3/2) + ((a + a/x^5)^(3/2)*a - 3*sqrt(a + a/x^5)*a^2)/a^4)*sgn(x)/abs(a)