3.2693 \(\int \frac{x^{-1-\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

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 = 0.0265203, 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, Rules used = {271, 264} \[ -\frac{16 b^2 x^{-n/2} \sqrt{a+b x^n}}{15 a^3 n}+\frac{8 b x^{-3 n/2} \sqrt{a+b x^n}}{15 a^2 n}-\frac{2 x^{-5 n/2} \sqrt{a+b x^n}}{5 a n} \]

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))

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^{-1-\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 x^{-5 n/2} \sqrt{a+b x^n}}{5 a n}-\frac{(4 b) \int \frac{x^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx}{5 a}\\ &=-\frac{2 x^{-5 n/2} \sqrt{a+b x^n}}{5 a n}+\frac{8 b x^{-3 n/2} \sqrt{a+b x^n}}{15 a^2 n}+\frac{\left (8 b^2\right ) \int \frac{x^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx}{15 a^2}\\ &=-\frac{2 x^{-5 n/2} \sqrt{a+b x^n}}{5 a n}+\frac{8 b x^{-3 n/2} \sqrt{a+b x^n}}{15 a^2 n}-\frac{16 b^2 x^{-n/2} \sqrt{a+b x^n}}{15 a^3 n}\\ \end{align*}

Mathematica [A]  time = 0.0174706, 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 = 0.079, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-{\frac{5\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{5}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 21.5919, size = 354, normalized size = 3.98 \begin{align*} - \frac{6 a^{4} b^{\frac{9}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac{4 a^{3} b^{\frac{11}{2}} x^{n} \sqrt{\frac{a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac{6 a^{2} b^{\frac{13}{2}} x^{2 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac{24 a b^{\frac{15}{2}} x^{3 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac{16 b^{\frac{17}{2}} x^{4 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} \end{align*}

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]

-6*a**4*b**(9/2)*sqrt(a*x**(-n)/b + 1)/(15*a**5*b**4*n*x**(2*n) + 30*a**4*b**5*n*x**(3*n) + 15*a**3*b**6*n*x**
(4*n)) - 4*a**3*b**(11/2)*x**n*sqrt(a*x**(-n)/b + 1)/(15*a**5*b**4*n*x**(2*n) + 30*a**4*b**5*n*x**(3*n) + 15*a
**3*b**6*n*x**(4*n)) - 6*a**2*b**(13/2)*x**(2*n)*sqrt(a*x**(-n)/b + 1)/(15*a**5*b**4*n*x**(2*n) + 30*a**4*b**5
*n*x**(3*n) + 15*a**3*b**6*n*x**(4*n)) - 24*a*b**(15/2)*x**(3*n)*sqrt(a*x**(-n)/b + 1)/(15*a**5*b**4*n*x**(2*n
) + 30*a**4*b**5*n*x**(3*n) + 15*a**3*b**6*n*x**(4*n)) - 16*b**(17/2)*x**(4*n)*sqrt(a*x**(-n)/b + 1)/(15*a**5*
b**4*n*x**(2*n) + 30*a**4*b**5*n*x**(3*n) + 15*a**3*b**6*n*x**(4*n))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{5}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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