\(\int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx\) [2693]

Optimal result

Integrand size = 21, antiderivative size = 89 \[ \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n}+\frac {8 b x^{-3 n/2} \sqrt {a+b x^n}}{15 a^2 n}-\frac {16 b^2 x^{-n/2} \sqrt {a+b x^n}}{15 a^3 n} \]

[Out]

-2/5*(a+b*x^n)^(1/2)/a/n/(x^(5/2*n))+8/15*b*(a+b*x^n)^(1/2)/a^2/n/(x^(3/2*n))-16/15*b^2*(a+b*x^n)^(1/2)/a^3/n/
(x^(1/2*n))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {277, 270} \[ \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {16 b^2 x^{-n/2} \sqrt {a+b x^n}}{15 a^3 n}+\frac {8 b x^{-3 n/2} \sqrt {a+b x^n}}{15 a^2 n}-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n} \]

[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 270

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]

Rule 277

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]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.57 \[ \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 x^{-5 n/2} \sqrt {a+b x^n} \left (3 a^2-4 a b x^n+8 b^2 x^{2 n}\right )}{15 a^3 n} \]

[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))

Maple [F]

\[\int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a +b \,x^{n}}}d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (76) = 152\).

Time = 0.84 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.98 \[ \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=- \frac {6 a^{4} b^{\frac {9}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac {4 a^{3} b^{\frac {11}{2}} x^{n} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac {6 a^{2} b^{\frac {13}{2}} x^{2 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac {24 a b^{\frac {15}{2}} x^{3 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac {16 b^{\frac {17}{2}} x^{4 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} \]

[In]

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

[Out]

-6*a**4*b**(9/2)*sqrt(a/(b*x**n) + 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/(b*x**n) + 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/(b*x**n) + 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/(b*x**n) + 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/(b*x**n) + 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))

Maxima [F]

\[ \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-\frac {5}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-\frac {5}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {1}{x^{\frac {5\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \]

[In]

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

[Out]

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