3.206 \(\int \frac{\sqrt{a x^{13}}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=50 \[ \frac{\sqrt{x^5+1} \sqrt{a x^{13}}}{5 x^4}-\frac{\sqrt{a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}} \]

[Out]

(Sqrt[a*x^13]*Sqrt[1 + x^5])/(5*x^4) - (Sqrt[a*x^13]*ArcSinh[x^(5/2)])/(5*x^(13/
2))

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Rubi [A]  time = 0.0400788, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{\sqrt{x^5+1} \sqrt{a x^{13}}}{5 x^4}-\frac{\sqrt{a x^{13}} \sinh ^{-1}\left (x^{5/2}\right )}{5 x^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^13]/Sqrt[1 + x^5],x]

[Out]

(Sqrt[a*x^13]*Sqrt[1 + x^5])/(5*x^4) - (Sqrt[a*x^13]*ArcSinh[x^(5/2)])/(5*x^(13/
2))

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Rubi in Sympy [A]  time = 10.2477, size = 42, normalized size = 0.84 \[ \frac{\sqrt{a x^{13}} \sqrt{x^{5} + 1}}{5 x^{4}} - \frac{\sqrt{a x^{13}} \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{5 x^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**13)**(1/2)/(x**5+1)**(1/2),x)

[Out]

sqrt(a*x**13)*sqrt(x**5 + 1)/(5*x**4) - sqrt(a*x**13)*asinh(x**(5/2))/(5*x**(13/
2))

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Mathematica [A]  time = 0.0880008, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{13}}}{\sqrt{1+x^5}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a*x^13]/Sqrt[1 + x^5],x]

[Out]

Integrate[Sqrt[a*x^13]/Sqrt[1 + x^5], x]

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Maple [A]  time = 0.066, size = 57, normalized size = 1.1 \[{\frac{1}{5\,{x}^{4}}\sqrt{a{x}^{13}}\sqrt{{x}^{5}+1}}-{\frac{1}{5\,{x}^{7}}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{a{x}^{13}}\sqrt{ax \left ({x}^{5}+1 \right ) }{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^13)^(1/2)/(x^5+1)^(1/2),x)

[Out]

1/5*(a*x^13)^(1/2)*(x^5+1)^(1/2)/x^4-1/5/a^(1/2)*arcsinh(x^(5/2))*(a*x^13)^(1/2)
/x^7*(a*x*(x^5+1))^(1/2)/(x^5+1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{13}}}{\sqrt{x^{5} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^13)/sqrt(x^5 + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^13)/sqrt(x^5 + 1), x)

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Fricas [A]  time = 0.392661, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} x^{4} \log \left (-\frac{8 \, a x^{14} + 8 \, a x^{9} + a x^{4} - 4 \, \sqrt{a x^{13}}{\left (2 \, x^{5} + 1\right )} \sqrt{x^{5} + 1} \sqrt{a}}{x^{4}}\right ) + 4 \, \sqrt{a x^{13}} \sqrt{x^{5} + 1}}{20 \, x^{4}}, \frac{\sqrt{-a} x^{4} \arctan \left (\frac{{\left (2 \, x^{9} + x^{4}\right )} \sqrt{-a}}{2 \, \sqrt{a x^{13}} \sqrt{x^{5} + 1}}\right ) + 2 \, \sqrt{a x^{13}} \sqrt{x^{5} + 1}}{10 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^13)/sqrt(x^5 + 1),x, algorithm="fricas")

[Out]

[1/20*(sqrt(a)*x^4*log(-(8*a*x^14 + 8*a*x^9 + a*x^4 - 4*sqrt(a*x^13)*(2*x^5 + 1)
*sqrt(x^5 + 1)*sqrt(a))/x^4) + 4*sqrt(a*x^13)*sqrt(x^5 + 1))/x^4, 1/10*(sqrt(-a)
*x^4*arctan(1/2*(2*x^9 + x^4)*sqrt(-a)/(sqrt(a*x^13)*sqrt(x^5 + 1))) + 2*sqrt(a*
x^13)*sqrt(x^5 + 1))/x^4]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{13}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**13)**(1/2)/(x**5+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**13)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

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GIAC/XCAS [A]  time = 0.282395, size = 92, normalized size = 1.84 \[ \frac{a^{\frac{11}{2}}{\rm ln}\left (-\sqrt{a x} a^{\frac{5}{2}} x^{2} + \sqrt{a^{6} x^{5} + a^{6}}\right )}{5 \,{\left | a \right |}^{5}} + \frac{\sqrt{a^{6} x^{5} + a^{6}} \sqrt{a x} x^{2}}{5 \, a^{2}{\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^13)/sqrt(x^5 + 1),x, algorithm="giac")

[Out]

1/5*a^(11/2)*ln(-sqrt(a*x)*a^(5/2)*x^2 + sqrt(a^6*x^5 + a^6))/abs(a)^5 + 1/5*sqr
t(a^6*x^5 + a^6)*sqrt(a*x)*x^2/(a^2*abs(a))