3.2686 \(\int \frac{x^{-1-\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{16 b^2 x^{-n/2} \sqrt{a+b x^n}}{15 a^3 n}+\frac{8 b x^{-3 n/2} \sqrt{a+b x^n}}{15 a^2 n}-\frac{2 x^{-5 n/2} \sqrt{a+b x^n}}{5 a n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(5*a*n*x^((5*n)/2)) + (8*b*Sqrt[a + b*x^n])/(15*a^2*n*x^((3
*n)/2)) - (16*b^2*Sqrt[a + b*x^n])/(15*a^3*n*x^(n/2))

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Rubi [A]  time = 0.0889409, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{16 b^2 x^{-n/2} \sqrt{a+b x^n}}{15 a^3 n}+\frac{8 b x^{-3 n/2} \sqrt{a+b x^n}}{15 a^2 n}-\frac{2 x^{-5 n/2} \sqrt{a+b x^n}}{5 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - (5*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n])/(5*a*n*x^((5*n)/2)) + (8*b*Sqrt[a + b*x^n])/(15*a^2*n*x^((3
*n)/2)) - (16*b^2*Sqrt[a + b*x^n])/(15*a^3*n*x^(n/2))

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Rubi in Sympy [A]  time = 8.75745, size = 76, normalized size = 0.85 \[ - \frac{2 x^{- \frac{5 n}{2}} \sqrt{a + b x^{n}}}{5 a n} + \frac{8 b x^{- \frac{3 n}{2}} \sqrt{a + b x^{n}}}{15 a^{2} n} - \frac{16 b^{2} x^{- \frac{n}{2}} \sqrt{a + b x^{n}}}{15 a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-5/2*n)/(a+b*x**n)**(1/2),x)

[Out]

-2*x**(-5*n/2)*sqrt(a + b*x**n)/(5*a*n) + 8*b*x**(-3*n/2)*sqrt(a + b*x**n)/(15*a
**2*n) - 16*b**2*x**(-n/2)*sqrt(a + b*x**n)/(15*a**3*n)

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Mathematica [A]  time = 0.0540237, size = 51, normalized size = 0.57 \[ -\frac{2 x^{-5 n/2} \sqrt{a+b x^n} \left (3 a^2-4 a b x^n+8 b^2 x^{2 n}\right )}{15 a^3 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - (5*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n]*(3*a^2 - 4*a*b*x^n + 8*b^2*x^(2*n)))/(15*a^3*n*x^((5*n)/2))

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{1{x}^{-1-{\frac{5\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^(-1-5/2*n)/(a+b*x^n)^(1/2),x)

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Maxima [A]  time = 1.5411, size = 96, normalized size = 1.08 \[ -\frac{2 \, \sqrt{b x^{n} + a} b^{2} x^{-\frac{1}{2} \, n}}{a^{3} n} + \frac{4 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}} b x^{-\frac{3}{2} \, n}}{3 \, a^{3} n} - \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{5}{2}} x^{-\frac{5}{2} \, n}}{5 \, a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

-2*sqrt(b*x^n + a)*b^2*x^(-1/2*n)/(a^3*n) + 4/3*(b*x^n + a)^(3/2)*b*x^(-3/2*n)/(
a^3*n) - 2/5*(b*x^n + a)^(5/2)*x^(-5/2*n)/(a^3*n)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-5/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-\frac{5}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(-5/2*n - 1)/sqrt(b*x^n + a), x)