\(\int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx\) [612]

Optimal result

Integrand size = 21, antiderivative size = 120 \[ \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}+\frac {12 b x^{-5 n/2} \sqrt {a+b x^n}}{35 a^2 n}-\frac {16 b^2 x^{-3 n/2} \sqrt {a+b x^n}}{35 a^3 n}+\frac {32 b^3 x^{-n/2} \sqrt {a+b x^n}}{35 a^4 n} \] Output:

-2/7*(a+b*x^n)^(1/2)/a/n/(x^(7/2*n))+12/35*b*(a+b*x^n)^(1/2)/a^2/n/(x^(5/2 
*n))-16/35*b^2*(a+b*x^n)^(1/2)/a^3/n/(x^(3/2*n))+32/35*b^3*(a+b*x^n)^(1/2) 
/a^4/n/(x^(1/2*n))
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.53 \[ \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 x^{-7 n/2} \sqrt {a+b x^n} \left (5 a^3-6 a^2 b x^n+8 a b^2 x^{2 n}-16 b^3 x^{3 n}\right )}{35 a^4 n} \] Input:

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

Output:

(-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))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {803, 803, 803, 796}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 803

\(\displaystyle -\frac {6 b \int \frac {x^{-\frac {5 n}{2}-1}}{\sqrt {b x^n+a}}dx}{7 a}-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}\)

\(\Big \downarrow \) 803

\(\displaystyle -\frac {6 b \left (-\frac {4 b \int \frac {x^{-\frac {3 n}{2}-1}}{\sqrt {b x^n+a}}dx}{5 a}-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n}\right )}{7 a}-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}\)

\(\Big \downarrow \) 803

\(\displaystyle -\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {x^{-\frac {n}{2}-1}}{\sqrt {b x^n+a}}dx}{3 a}-\frac {2 x^{-3 n/2} \sqrt {a+b x^n}}{3 a n}\right )}{5 a}-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n}\right )}{7 a}-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}\)

\(\Big \downarrow \) 796

\(\displaystyle -\frac {6 b \left (-\frac {4 b \left (\frac {4 b x^{-n/2} \sqrt {a+b x^n}}{3 a^2 n}-\frac {2 x^{-3 n/2} \sqrt {a+b x^n}}{3 a n}\right )}{5 a}-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n}\right )}{7 a}-\frac {2 x^{-7 n/2} \sqrt {a+b x^n}}{7 a n}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 796

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 803

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

Maple [F]

\[\int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a +b \,x^{n}}}d x\]

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (105) = 210\).

Time = 1.09 (sec) , antiderivative size = 605, normalized size of antiderivative = 5.04 \[ \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=- \frac {10 a^{6} b^{\frac {19}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac {18 a^{5} b^{\frac {21}{2}} x^{n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac {10 a^{4} b^{\frac {23}{2}} x^{2 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {10 a^{3} b^{\frac {25}{2}} x^{3 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {60 a^{2} b^{\frac {27}{2}} x^{4 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {80 a b^{\frac {29}{2}} x^{5 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {32 b^{\frac {31}{2}} x^{6 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} \] Input:

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

Output:

-10*a**6*b**(19/2)*sqrt(a/(b*x**n) + 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/(b*x**n) + 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/(b*x**n) + 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/(b*x**n) + 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/(b*x**n) 
 + 1)/(35*a**7*b**9*n*x**(3*n) + 105*a**6*b**10*n*x**(4*n) + 105*a**5*b**1 
1*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n)) + 80*a*b**(29/2)*x**(5*n)*sqrt(a/ 
(b*x**n) + 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)*s 
qrt(a/(b*x**n) + 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))
 

Maxima [F]

\[ \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-\frac {7}{2} \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {1}{x^{\frac {7\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \] Input:

int(1/(x^((7*n)/2 + 1)*(a + b*x^n)^(1/2)),x)
 

Output:

int(1/(x^((7*n)/2 + 1)*(a + b*x^n)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=\int \frac {\sqrt {x^{n} b +a}}{x^{\frac {9 n}{2}} b x +x^{\frac {7 n}{2}} a x}d x \] Input:

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

Output:

int(sqrt(x**n*b + a)/(x**((9*n)/2)*b*x + x**((7*n)/2)*a*x),x)