3.2694 \(\int \frac{x^{-1-\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=120 \[ -\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n} \]

[Out]

(-2*Sqrt[a + b*x^n])/(7*a*n*x^((7*n)/2)) + (12*b*Sqrt[a + b*x^n])/(35*a^2*n*x^((5*n)/2)) - (16*b^2*Sqrt[a + b*
x^n])/(35*a^3*n*x^((3*n)/2)) + (32*b^3*Sqrt[a + b*x^n])/(35*a^4*n*x^(n/2))

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Rubi [A]  time = 0.0391954, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {271, 264} \[ -\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - (7*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n])/(7*a*n*x^((7*n)/2)) + (12*b*Sqrt[a + b*x^n])/(35*a^2*n*x^((5*n)/2)) - (16*b^2*Sqrt[a + b*
x^n])/(35*a^3*n*x^((3*n)/2)) + (32*b^3*Sqrt[a + b*x^n])/(35*a^4*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{7 n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n}-\frac{(6 b) \int \frac{x^{-1-\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx}{7 a}\\ &=-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}+\frac{\left (24 b^2\right ) \int \frac{x^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx}{35 a^2}\\ &=-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}-\frac{\left (16 b^3\right ) \int \frac{x^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx}{35 a^3}\\ &=-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}\\ \end{align*}

Mathematica [A]  time = 0.0217774, size = 64, normalized size = 0.53 \[ -\frac{2 x^{-7 n/2} \sqrt{a+b x^n} \left (-6 a^2 b x^n+5 a^3+8 a b^2 x^{2 n}-16 b^3 x^{3 n}\right )}{35 a^4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - (7*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-2*Sqrt[a + b*x^n]*(5*a^3 - 6*a^2*b*x^n + 8*a*b^2*x^(2*n) - 16*b^3*x^(3*n)))/(35*a^4*n*x^((7*n)/2))

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-{\frac{7\,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^(-1-7/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int(x^(-1-7/2*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{7}{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^(-1-7/2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^(-7/2*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^(-1-7/2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [B]  time = 17.4394, size = 605, normalized size = 5.04 \begin{align*} - \frac{10 a^{6} b^{\frac{19}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac{18 a^{5} b^{\frac{21}{2}} x^{n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac{10 a^{4} b^{\frac{23}{2}} x^{2 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac{10 a^{3} b^{\frac{25}{2}} x^{3 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac{60 a^{2} b^{\frac{27}{2}} x^{4 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac{80 a b^{\frac{29}{2}} x^{5 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac{32 b^{\frac{31}{2}} x^{6 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-7/2*n)/(a+b*x**n)**(1/2),x)

[Out]

-10*a**6*b**(19/2)*sqrt(a*x**(-n)/b + 1)/(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11
*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n)) - 18*a**5*b**(21/2)*x**n*sqrt(a*x**(-n)/b + 1)/(35*a**7*b**9*n*x**(3*n
) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n)) - 10*a**4*b**(23/2)*x**(
2*n)*sqrt(a*x**(-n)/b + 1)/(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) +
35*a**4*b**12*n*x**(6*n)) + 10*a**3*b**(25/2)*x**(3*n)*sqrt(a*x**(-n)/b + 1)/(35*a**7*b**9*n*x**(3*n) + 105*a*
*6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n)) + 60*a**2*b**(27/2)*x**(4*n)*sqrt(
a*x**(-n)/b + 1)/(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b*
*12*n*x**(6*n)) + 80*a*b**(29/2)*x**(5*n)*sqrt(a*x**(-n)/b + 1)/(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x*
*(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n)) + 32*b**(31/2)*x**(6*n)*sqrt(a*x**(-n)/b + 1)/(
35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{7}{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^(-1-7/2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(x^(-7/2*n - 1)/sqrt(b*x^n + a), x)