\(\int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx\) [684]

Optimal result

Integrand size = 23, antiderivative size = 153 \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{7 a c n}+\frac {12 b x^n (c x)^{-7 n/2} \sqrt {a+b x^n}}{35 a^2 c n}-\frac {16 b^2 x^{2 n} (c x)^{-7 n/2} \sqrt {a+b x^n}}{35 a^3 c n}+\frac {32 b^3 x^{3 n} (c x)^{-7 n/2} \sqrt {a+b x^n}}{35 a^4 c n} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45 \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n} \left (5 a^3-6 a^2 b x^n+8 a b^2 x^{2 n}-16 b^3 x^{3 n}\right )}{35 a^4 c n} \] Input:

Integrate[(c*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*c*n*(c*x)^((7*n)/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {805, 805, 805, 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.

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

\(\Big \downarrow \) 805

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

\(\Big \downarrow \) 805

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

\(\Big \downarrow \) 805

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

\(\Big \downarrow \) 796

\(\displaystyle -\frac {6 \left (-\frac {4 \left (\frac {4 (c x)^{-7 n/2} \left (a+b x^n\right )^{7/2}}{35 a^2 c n}-\frac {2 (c x)^{-7 n/2} \left (a+b x^n\right )^{5/2}}{5 a c n}\right )}{3 a}-\frac {2 (c x)^{-7 n/2} \left (a+b x^n\right )^{3/2}}{3 a c n}\right )}{a}-\frac {2 (c x)^{-7 n/2} \sqrt {a+b x^n}}{a c n}\)

Input:

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

Output:

(-2*Sqrt[a + b*x^n])/(a*c*n*(c*x)^((7*n)/2)) - (6*((-2*(a + b*x^n)^(3/2))/ 
(3*a*c*n*(c*x)^((7*n)/2)) - (4*((-2*(a + b*x^n)^(5/2))/(5*a*c*n*(c*x)^((7* 
n)/2)) + (4*(a + b*x^n)^(7/2))/(35*a^2*c*n*(c*x)^((7*n)/2))))/(3*a)))/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 805

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

integrate((c*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. 677 vs. \(2 (134) = 268\).

Time = 1.18 (sec) , antiderivative size = 677, normalized size of antiderivative = 4.42 \[ \int \frac {(c x)^{-1-\frac {7 n}{2}}}{\sqrt {a+b x^n}} \, dx=- \frac {10 a^{6} b^{\frac {19}{2}} c^{- \frac {7 n}{2} - 1} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac {18 a^{5} b^{\frac {21}{2}} c^{- \frac {7 n}{2} - 1} x^{n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac {10 a^{4} b^{\frac {23}{2}} c^{- \frac {7 n}{2} - 1} x^{2 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {10 a^{3} b^{\frac {25}{2}} c^{- \frac {7 n}{2} - 1} x^{3 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {60 a^{2} b^{\frac {27}{2}} c^{- \frac {7 n}{2} - 1} x^{4 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {80 a b^{\frac {29}{2}} c^{- \frac {7 n}{2} - 1} x^{5 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac {32 b^{\frac {31}{2}} c^{- \frac {7 n}{2} - 1} x^{6 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} \] Input:

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

Output:

-10*a**6*b**(19/2)*c**(-7*n/2 - 1)*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)*c**(-7*n/2 - 1)*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)*c**(-7*n/2 - 1 
)*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)*c**(-7*n/2 - 1)*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)*c**(-7*n/2 - 1)*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**11*n*x**(5*n) + 35*a**4*b**12*n*x**(6*n)) + 80*a*b**(29/2)*c**(-7*n/2 
 - 1)*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)) + 3 
2*b**(31/2)*c**(-7*n/2 - 1)*x**(6*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))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((c*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)/(c**((7*n)/2 
)*c)