3.2659 \(\int \frac{x^{-1-4 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{64 a^{9/2} n}+\frac{35 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^4 n}-\frac{35 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^3 n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n} \]

[Out]

-Sqrt[a + b*x^n]/(4*a*n*x^(4*n)) + (7*b*Sqrt[a + b*x^n])/(24*a^2*n*x^(3*n)) - (3
5*b^2*Sqrt[a + b*x^n])/(96*a^3*n*x^(2*n)) + (35*b^3*Sqrt[a + b*x^n])/(64*a^4*n*x
^n) - (35*b^4*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(64*a^(9/2)*n)

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Rubi [A]  time = 0.193634, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{64 a^{9/2} n}+\frac{35 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^4 n}-\frac{35 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^3 n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 4*n)/Sqrt[a + b*x^n],x]

[Out]

-Sqrt[a + b*x^n]/(4*a*n*x^(4*n)) + (7*b*Sqrt[a + b*x^n])/(24*a^2*n*x^(3*n)) - (3
5*b^2*Sqrt[a + b*x^n])/(96*a^3*n*x^(2*n)) + (35*b^3*Sqrt[a + b*x^n])/(64*a^4*n*x
^n) - (35*b^4*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(64*a^(9/2)*n)

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Rubi in Sympy [A]  time = 21.1643, size = 128, normalized size = 0.88 \[ - \frac{x^{- 4 n} \sqrt{a + b x^{n}}}{4 a n} + \frac{7 b x^{- 3 n} \sqrt{a + b x^{n}}}{24 a^{2} n} - \frac{35 b^{2} x^{- 2 n} \sqrt{a + b x^{n}}}{96 a^{3} n} + \frac{35 b^{3} x^{- n} \sqrt{a + b x^{n}}}{64 a^{4} n} - \frac{35 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{64 a^{\frac{9}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-4*n)/(a+b*x**n)**(1/2),x)

[Out]

-x**(-4*n)*sqrt(a + b*x**n)/(4*a*n) + 7*b*x**(-3*n)*sqrt(a + b*x**n)/(24*a**2*n)
- 35*b**2*x**(-2*n)*sqrt(a + b*x**n)/(96*a**3*n) + 35*b**3*x**(-n)*sqrt(a + b*x
**n)/(64*a**4*n) - 35*b**4*atanh(sqrt(a + b*x**n)/sqrt(a))/(64*a**(9/2)*n)

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Mathematica [A]  time = 0.181982, size = 111, normalized size = 0.77 \[ \frac{-2 \sqrt{a} x^{-4 n} \sqrt{a+b x^n} \left (48 a^3-56 a^2 b x^n+70 a b^2 x^{2 n}-105 b^3 x^{3 n}\right )-105 b^4 \log \left (x^{-n} \left (2 \sqrt{a} \sqrt{a+b x^n}+2 a+b x^n\right )\right )}{384 a^{9/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 4*n)/Sqrt[a + b*x^n],x]

[Out]

((-2*Sqrt[a]*Sqrt[a + b*x^n]*(48*a^3 - 56*a^2*b*x^n + 70*a*b^2*x^(2*n) - 105*b^3
*x^(3*n)))/x^(4*n) - 105*b^4*Log[(2*a + b*x^n + 2*Sqrt[a]*Sqrt[a + b*x^n])/x^n])
/(384*a^(9/2)*n)

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{{x}^{-1-4\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-4*n)/(a+b*x^n)^(1/2),x)

[Out]

int(x^(-1-4*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(x^(-4*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244343, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, b^{4} x^{4 \, n} \log \left (\frac{\sqrt{a} b x^{n} - 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right ) + 2 \,{\left (105 \, \sqrt{a} b^{3} x^{3 \, n} - 70 \, a^{\frac{3}{2}} b^{2} x^{2 \, n} + 56 \, a^{\frac{5}{2}} b x^{n} - 48 \, a^{\frac{7}{2}}\right )} \sqrt{b x^{n} + a}}{384 \, a^{\frac{9}{2}} n x^{4 \, n}}, \frac{105 \, b^{4} x^{4 \, n} \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right ) +{\left (105 \, \sqrt{-a} b^{3} x^{3 \, n} - 70 \, \sqrt{-a} a b^{2} x^{2 \, n} + 56 \, \sqrt{-a} a^{2} b x^{n} - 48 \, \sqrt{-a} a^{3}\right )} \sqrt{b x^{n} + a}}{192 \, \sqrt{-a} a^{4} n x^{4 \, n}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-4*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

[1/384*(105*b^4*x^(4*n)*log((sqrt(a)*b*x^n - 2*sqrt(b*x^n + a)*a + 2*a^(3/2))/x^
n) + 2*(105*sqrt(a)*b^3*x^(3*n) - 70*a^(3/2)*b^2*x^(2*n) + 56*a^(5/2)*b*x^n - 48
*a^(7/2))*sqrt(b*x^n + a))/(a^(9/2)*n*x^(4*n)), 1/192*(105*b^4*x^(4*n)*arctan(a/
(sqrt(b*x^n + a)*sqrt(-a))) + (105*sqrt(-a)*b^3*x^(3*n) - 70*sqrt(-a)*a*b^2*x^(2
*n) + 56*sqrt(-a)*a^2*b*x^n - 48*sqrt(-a)*a^3)*sqrt(b*x^n + a))/(sqrt(-a)*a^4*n*
x^(4*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**(-1-4*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. \[ \int \frac{x^{-4 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-4*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(-4*n - 1)/sqrt(b*x^n + a), x)