3.2650 \(\int x^{-1-4 n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=142 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{64 a^{7/2} n}-\frac{5 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^3 n}+\frac{5 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^2 n}-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 n}-\frac{b x^{-3 n} \sqrt{a+b x^n}}{24 a n} \]

[Out]

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

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Rubi [A]  time = 0.192269, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{5 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{64 a^{7/2} n}-\frac{5 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^3 n}+\frac{5 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^2 n}-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 n}-\frac{b x^{-3 n} \sqrt{a+b x^n}}{24 a n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.0638, size = 122, normalized size = 0.86 \[ - \frac{x^{- 4 n} \sqrt{a + b x^{n}}}{4 n} - \frac{b x^{- 3 n} \sqrt{a + b x^{n}}}{24 a n} + \frac{5 b^{2} x^{- 2 n} \sqrt{a + b x^{n}}}{96 a^{2} n} - \frac{5 b^{3} x^{- n} \sqrt{a + b x^{n}}}{64 a^{3} n} + \frac{5 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{64 a^{\frac{7}{2}} n} \]

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*n) - b*x**(-3*n)*sqrt(a + b*x**n)/(24*a*n) + 5*b*
*2*x**(-2*n)*sqrt(a + b*x**n)/(96*a**2*n) - 5*b**3*x**(-n)*sqrt(a + b*x**n)/(64*
a**3*n) + 5*b**4*atanh(sqrt(a + b*x**n)/sqrt(a))/(64*a**(7/2)*n)

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Mathematica [A]  time = 0.182649, size = 111, normalized size = 0.78 \[ \frac{15 b^4 \log \left (x^{-n} \left (2 \sqrt{a} \sqrt{a+b x^n}+2 a+b x^n\right )\right )-2 \sqrt{a} x^{-4 n} \sqrt{a+b x^n} \left (48 a^3+8 a^2 b x^n-10 a b^2 x^{2 n}+15 b^3 x^{3 n}\right )}{384 a^{7/2} n} \]

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 + 8*a^2*b*x^n - 10*a*b^2*x^(2*n) + 15*b^3*x
^(3*n)))/x^(4*n) + 15*b^4*Log[(2*a + b*x^n + 2*Sqrt[a]*Sqrt[a + b*x^n])/x^n])/(3
84*a^(7/2)*n)

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243192, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{4} x^{4 \, n} \log \left (\frac{\sqrt{a} b x^{n} + 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right ) - 2 \,{\left (15 \, \sqrt{a} b^{3} x^{3 \, n} - 10 \, a^{\frac{3}{2}} b^{2} x^{2 \, n} + 8 \, a^{\frac{5}{2}} b x^{n} + 48 \, a^{\frac{7}{2}}\right )} \sqrt{b x^{n} + a}}{384 \, a^{\frac{7}{2}} n x^{4 \, n}}, -\frac{15 \, b^{4} x^{4 \, n} \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right ) +{\left (15 \, \sqrt{-a} b^{3} x^{3 \, n} - 10 \, \sqrt{-a} a b^{2} x^{2 \, n} + 8 \, \sqrt{-a} a^{2} b x^{n} + 48 \, \sqrt{-a} a^{3}\right )} \sqrt{b x^{n} + a}}{192 \, \sqrt{-a} a^{3} n x^{4 \, n}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(15*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*(15*sqrt(a)*b^3*x^(3*n) - 10*a^(3/2)*b^2*x^(2*n) + 8*a^(5/2)*b*x^n + 48*a^
(7/2))*sqrt(b*x^n + a))/(a^(7/2)*n*x^(4*n)), -1/192*(15*b^4*x^(4*n)*arctan(a/(sq
rt(b*x^n + a)*sqrt(-a))) + (15*sqrt(-a)*b^3*x^(3*n) - 10*sqrt(-a)*a*b^2*x^(2*n)
+ 8*sqrt(-a)*a^2*b*x^n + 48*sqrt(-a)*a^3)*sqrt(b*x^n + a))/(sqrt(-a)*a^3*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. \[ \int \sqrt{b x^{n} + a} x^{-4 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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