3.2658 \(\int \frac{x^{-1-3 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=116 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{8 a^{7/2} n}-\frac{5 b^2 x^{-n} \sqrt{a+b x^n}}{8 a^3 n}+\frac{5 b x^{-2 n} \sqrt{a+b x^n}}{12 a^2 n}-\frac{x^{-3 n} \sqrt{a+b x^n}}{3 a n} \]

[Out]

-Sqrt[a + b*x^n]/(3*a*n*x^(3*n)) + (5*b*Sqrt[a + b*x^n])/(12*a^2*n*x^(2*n)) - (5
*b^2*Sqrt[a + b*x^n])/(8*a^3*n*x^n) + (5*b^3*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(
8*a^(7/2)*n)

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Rubi [A]  time = 0.148356, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{8 a^{7/2} n}-\frac{5 b^2 x^{-n} \sqrt{a+b x^n}}{8 a^3 n}+\frac{5 b x^{-2 n} \sqrt{a+b x^n}}{12 a^2 n}-\frac{x^{-3 n} \sqrt{a+b x^n}}{3 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 3*n)/Sqrt[a + b*x^n],x]

[Out]

-Sqrt[a + b*x^n]/(3*a*n*x^(3*n)) + (5*b*Sqrt[a + b*x^n])/(12*a^2*n*x^(2*n)) - (5
*b^2*Sqrt[a + b*x^n])/(8*a^3*n*x^n) + (5*b^3*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(
8*a^(7/2)*n)

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Rubi in Sympy [A]  time = 16.3717, size = 100, normalized size = 0.86 \[ - \frac{x^{- 3 n} \sqrt{a + b x^{n}}}{3 a n} + \frac{5 b x^{- 2 n} \sqrt{a + b x^{n}}}{12 a^{2} n} - \frac{5 b^{2} x^{- n} \sqrt{a + b x^{n}}}{8 a^{3} n} + \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-3*n)/(a+b*x**n)**(1/2),x)

[Out]

-x**(-3*n)*sqrt(a + b*x**n)/(3*a*n) + 5*b*x**(-2*n)*sqrt(a + b*x**n)/(12*a**2*n)
- 5*b**2*x**(-n)*sqrt(a + b*x**n)/(8*a**3*n) + 5*b**3*atanh(sqrt(a + b*x**n)/sq
rt(a))/(8*a**(7/2)*n)

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Mathematica [A]  time = 0.144626, size = 98, normalized size = 0.84 \[ \frac{15 b^3 \log \left (x^{-n} \left (2 \sqrt{a} \sqrt{a+b x^n}+2 a+b x^n\right )\right )-2 \sqrt{a} x^{-3 n} \sqrt{a+b x^n} \left (8 a^2-10 a b x^n+15 b^2 x^{2 n}\right )}{48 a^{7/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 3*n)/Sqrt[a + b*x^n],x]

[Out]

((-2*Sqrt[a]*Sqrt[a + b*x^n]*(8*a^2 - 10*a*b*x^n + 15*b^2*x^(2*n)))/x^(3*n) + 15
*b^3*Log[(2*a + b*x^n + 2*Sqrt[a]*Sqrt[a + b*x^n])/x^n])/(48*a^(7/2)*n)

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{{x}^{-1-3\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3*n)/(a+b*x^n)^(1/2),x)

[Out]

int(x^(-1-3*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^(-3*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243697, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} x^{3 \, n} \log \left (\frac{\sqrt{a} b x^{n} + 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right ) - 2 \,{\left (15 \, \sqrt{a} b^{2} x^{2 \, n} - 10 \, a^{\frac{3}{2}} b x^{n} + 8 \, a^{\frac{5}{2}}\right )} \sqrt{b x^{n} + a}}{48 \, a^{\frac{7}{2}} n x^{3 \, n}}, -\frac{15 \, b^{3} x^{3 \, n} \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right ) +{\left (15 \, \sqrt{-a} b^{2} x^{2 \, n} - 10 \, \sqrt{-a} a b x^{n} + 8 \, \sqrt{-a} a^{2}\right )} \sqrt{b x^{n} + a}}{24 \, \sqrt{-a} a^{3} n x^{3 \, n}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

[1/48*(15*b^3*x^(3*n)*log((sqrt(a)*b*x^n + 2*sqrt(b*x^n + a)*a + 2*a^(3/2))/x^n)
- 2*(15*sqrt(a)*b^2*x^(2*n) - 10*a^(3/2)*b*x^n + 8*a^(5/2))*sqrt(b*x^n + a))/(a
^(7/2)*n*x^(3*n)), -1/24*(15*b^3*x^(3*n)*arctan(a/(sqrt(b*x^n + a)*sqrt(-a))) +
(15*sqrt(-a)*b^2*x^(2*n) - 10*sqrt(-a)*a*b*x^n + 8*sqrt(-a)*a^2)*sqrt(b*x^n + a)
)/(sqrt(-a)*a^3*n*x^(3*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**(-1-3*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. \[ \int \frac{x^{-3 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-3*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(-3*n - 1)/sqrt(b*x^n + a), x)