3.2647 \(\int x^{-1-n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=51 \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n}-\frac{x^{-n} \sqrt{a+b x^n}}{n} \]

[Out]

-(Sqrt[a + b*x^n]/(n*x^n)) - (b*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(Sqrt[a]*n)

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Rubi [A]  time = 0.0777012, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n}-\frac{x^{-n} \sqrt{a+b x^n}}{n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n)*Sqrt[a + b*x^n],x]

[Out]

-(Sqrt[a + b*x^n]/(n*x^n)) - (b*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(Sqrt[a]*n)

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Rubi in Sympy [A]  time = 8.45508, size = 41, normalized size = 0.8 \[ - \frac{x^{- n} \sqrt{a + b x^{n}}}{n} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{\sqrt{a} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-n)*(a+b*x**n)**(1/2),x)

[Out]

-x**(-n)*sqrt(a + b*x**n)/n - b*atanh(sqrt(a + b*x**n)/sqrt(a))/(sqrt(a)*n)

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Mathematica [A]  time = 0.120056, size = 67, normalized size = 1.31 \[ -\frac{2 x^{-n} \sqrt{a+b x^n}+\frac{b \log \left (x^{-n} \left (2 \sqrt{a} \sqrt{a+b x^n}+2 a+b x^n\right )\right )}{\sqrt{a}}}{2 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - n)*Sqrt[a + b*x^n],x]

[Out]

-((2*Sqrt[a + b*x^n])/x^n + (b*Log[(2*a + b*x^n + 2*Sqrt[a]*Sqrt[a + b*x^n])/x^n
])/Sqrt[a])/(2*n)

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \[ \int{x}^{-1-n}\sqrt{a+b{x}^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-n)*(a+b*x^n)^(1/2),x)

[Out]

int(x^(-1-n)*(a+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a)*x^(-n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231253, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{n} \log \left (\frac{\sqrt{a} b x^{n} - 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right ) - 2 \, \sqrt{b x^{n} + a} \sqrt{a}}{2 \, \sqrt{a} n x^{n}}, \frac{b x^{n} \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right ) - \sqrt{b x^{n} + a} \sqrt{-a}}{\sqrt{-a} n x^{n}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a)*x^(-n - 1),x, algorithm="fricas")

[Out]

[1/2*(b*x^n*log((sqrt(a)*b*x^n - 2*sqrt(b*x^n + a)*a + 2*a^(3/2))/x^n) - 2*sqrt(
b*x^n + a)*sqrt(a))/(sqrt(a)*n*x^n), (b*x^n*arctan(a/(sqrt(b*x^n + a)*sqrt(-a)))
- sqrt(b*x^n + a)*sqrt(-a))/(sqrt(-a)*n*x^n)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-n)*(a+b*x**n)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{n} + a} x^{-n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^n + a)*x^(-n - 1),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n + a)*x^(-n - 1), x)