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

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 = 0.0302922, 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 \[ \frac{4}{15} x^6 \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}}-\frac{2}{15} x \sqrt{x^5+1} \sqrt{\frac{a}{x^{17}}} \]

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

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Rubi in Sympy [A]  time = 8.22471, size = 44, normalized size = 0.9 \[ \frac{4 x^{6} \sqrt{\frac{a}{x^{17}}} \sqrt{x^{5} + 1}}{15} - \frac{2 x \sqrt{\frac{a}{x^{17}}} \sqrt{x^{5} + 1}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.014932, size = 30, normalized size = 0.61 \[ \frac{2}{15} x \sqrt{x^5+1} \left (2 x^5-1\right ) \sqrt{\frac{a}{x^{17}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 44, normalized size = 0.9 \[{\frac{2\,x \left ( 1+x \right ) \left ({x}^{4}-{x}^{3}+{x}^{2}-x+1 \right ) \left ( 2\,{x}^{5}-1 \right ) }{15}\sqrt{{\frac{a}{{x}^{17}}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \]

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 = 0.782597, size = 68, normalized size = 1.39 \[ \frac{2 \,{\left (2 \, \sqrt{a} x^{11} + \sqrt{a} x^{6} - \sqrt{a} x\right )}}{15 \, \sqrt{x^{4} - x^{3} + x^{2} - x + 1} \sqrt{x + 1} x^{\frac{17}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^17)/sqrt(x^5 + 1),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)*s
qrt(x + 1)*x^(17/2))

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Fricas [A]  time = 0.276903, size = 34, normalized size = 0.69 \[ \frac{2}{15} \,{\left (2 \, x^{6} - x\right )} \sqrt{x^{5} + 1} \sqrt{\frac{a}{x^{17}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a/x^17)/sqrt(x^5 + 1),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. \[ \int \frac{\sqrt{\frac{a}{x^{17}}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]

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/XCAS [A]  time = 0.279049, size = 58, normalized size = 1.18 \[ -\frac{2 \, a^{3}{\left (\frac{2}{a^{\frac{3}{2}}} + \frac{{\left (a + \frac{a}{x^{5}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{a + \frac{a}{x^{5}}} a}{a^{3}}\right )}{\rm sign}\left (x\right )}{15 \,{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*a^3*(2/a^(3/2) + ((a + a/x^5)^(3/2) - 3*sqrt(a + a/x^5)*a)/a^3)*sign(x)/ab
s(a)