3.205 \(\int \frac{\sqrt{a x^{23}}}{\sqrt{1+x^5}} \, dx\)

Optimal. Leaf size=75 \[ \frac{3 \sqrt{a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}}-\frac{3 \sqrt{x^5+1} \sqrt{a x^{23}}}{20 x^9}+\frac{\sqrt{x^5+1} \sqrt{a x^{23}}}{10 x^4} \]

[Out]

(-3*Sqrt[a*x^23]*Sqrt[1 + x^5])/(20*x^9) + (Sqrt[a*x^23]*Sqrt[1 + x^5])/(10*x^4)
+ (3*Sqrt[a*x^23]*ArcSinh[x^(5/2)])/(20*x^(23/2))

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Rubi [A]  time = 0.0557023, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{3 \sqrt{a x^{23}} \sinh ^{-1}\left (x^{5/2}\right )}{20 x^{23/2}}-\frac{3 \sqrt{x^5+1} \sqrt{a x^{23}}}{20 x^9}+\frac{\sqrt{x^5+1} \sqrt{a x^{23}}}{10 x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^23]/Sqrt[1 + x^5],x]

[Out]

(-3*Sqrt[a*x^23]*Sqrt[1 + x^5])/(20*x^9) + (Sqrt[a*x^23]*Sqrt[1 + x^5])/(10*x^4)
+ (3*Sqrt[a*x^23]*ArcSinh[x^(5/2)])/(20*x^(23/2))

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Rubi in Sympy [A]  time = 11.5955, size = 68, normalized size = 0.91 \[ \frac{\sqrt{a x^{23}} \sqrt{x^{5} + 1}}{10 x^{4}} - \frac{3 \sqrt{a x^{23}} \sqrt{x^{5} + 1}}{20 x^{9}} + \frac{3 \sqrt{a x^{23}} \operatorname{asinh}{\left (x^{\frac{5}{2}} \right )}}{20 x^{\frac{23}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**23)**(1/2)/(x**5+1)**(1/2),x)

[Out]

sqrt(a*x**23)*sqrt(x**5 + 1)/(10*x**4) - 3*sqrt(a*x**23)*sqrt(x**5 + 1)/(20*x**9
) + 3*sqrt(a*x**23)*asinh(x**(5/2))/(20*x**(23/2))

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Mathematica [A]  time = 0.109455, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{23}}}{\sqrt{1+x^5}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a*x^23]/Sqrt[1 + x^5],x]

[Out]

Integrate[Sqrt[a*x^23]/Sqrt[1 + x^5], x]

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Maple [A]  time = 0.193, size = 64, normalized size = 0.9 \[{\frac{2\,{x}^{5}-3}{20\,{x}^{9}}\sqrt{{x}^{5}+1}\sqrt{a{x}^{23}}}+{\frac{3}{20\,{x}^{12}}{\it Arcsinh} \left ({x}^{{\frac{5}{2}}} \right ) \sqrt{a{x}^{23}}\sqrt{ax \left ({x}^{5}+1 \right ) }{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{x}^{5}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^23)^(1/2)/(x^5+1)^(1/2),x)

[Out]

1/20/x^9*(2*x^5-3)*(x^5+1)^(1/2)*(a*x^23)^(1/2)+3/20/a^(1/2)*arcsinh(x^(5/2))*(a
*x^23)^(1/2)/x^12*(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. \[ \int \frac{\sqrt{a x^{23}}}{\sqrt{x^{5} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^23)/sqrt(x^5 + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^23)/sqrt(x^5 + 1), x)

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Fricas [A]  time = 0.395604, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} x^{9} \log \left (-\frac{8 \, a x^{19} + 8 \, a x^{14} + a x^{9} + 4 \, \sqrt{a x^{23}}{\left (2 \, x^{5} + 1\right )} \sqrt{x^{5} + 1} \sqrt{a}}{x^{9}}\right ) + 4 \, \sqrt{a x^{23}}{\left (2 \, x^{5} - 3\right )} \sqrt{x^{5} + 1}}{80 \, x^{9}}, -\frac{3 \, \sqrt{-a} x^{9} \arctan \left (\frac{{\left (2 \, x^{14} + x^{9}\right )} \sqrt{-a}}{2 \, \sqrt{a x^{23}} \sqrt{x^{5} + 1}}\right ) - 2 \, \sqrt{a x^{23}}{\left (2 \, x^{5} - 3\right )} \sqrt{x^{5} + 1}}{40 \, x^{9}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^23)/sqrt(x^5 + 1),x, algorithm="fricas")

[Out]

[1/80*(3*sqrt(a)*x^9*log(-(8*a*x^19 + 8*a*x^14 + a*x^9 + 4*sqrt(a*x^23)*(2*x^5 +
1)*sqrt(x^5 + 1)*sqrt(a))/x^9) + 4*sqrt(a*x^23)*(2*x^5 - 3)*sqrt(x^5 + 1))/x^9,
-1/40*(3*sqrt(-a)*x^9*arctan(1/2*(2*x^14 + x^9)*sqrt(-a)/(sqrt(a*x^23)*sqrt(x^5
+ 1))) - 2*sqrt(a*x^23)*(2*x^5 - 3)*sqrt(x^5 + 1))/x^9]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a x^{23}}}{\sqrt{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x**23)**(1/2)/(x**5+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**23)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x^23)/sqrt(x^5 + 1),x, algorithm="giac")

[Out]

Exception raised: RuntimeError