\(\int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx\) [552]

Optimal result

Integrand size = 19, antiderivative size = 145 \[ \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a n}+\frac {7 b x^{-3 n} \sqrt {a+b x^n}}{24 a^2 n}-\frac {35 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^3 n}+\frac {35 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^4 n}-\frac {35 b^4 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{64 a^{9/2} n} \] Output:

-1/4*(a+b*x^n)^(1/2)/a/n/(x^(4*n))+7/24*b*(a+b*x^n)^(1/2)/a^2/n/(x^(3*n))- 
35/96*b^2*(a+b*x^n)^(1/2)/a^3/n/(x^(2*n))+35/64*b^3*(a+b*x^n)^(1/2)/a^4/n/ 
(x^n)-35/64*b^4*arctanh((a+b*x^n)^(1/2)/a^(1/2))/a^(9/2)/n
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.65 \[ \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx=\frac {\sqrt {a} x^{-4 n} \sqrt {a+b x^n} \left (-48 a^3+56 a^2 b x^n-70 a b^2 x^{2 n}+105 b^3 x^{3 n}\right )-105 b^4 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{192 a^{9/2} n} \] Input:

Integrate[x^(-1 - 4*n)/Sqrt[a + b*x^n],x]
 

Output:

((Sqrt[a]*Sqrt[a + b*x^n]*(-48*a^3 + 56*a^2*b*x^n - 70*a*b^2*x^(2*n) + 105 
*b^3*x^(3*n)))/x^(4*n) - 105*b^4*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(192*a^ 
(9/2)*n)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {798, 52, 52, 52, 52, 73, 221}

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.

\(\displaystyle \int \frac {x^{-4 n-1}}{\sqrt {a+b x^n}} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\int \frac {x^{-5 n}}{\sqrt {b x^n+a}}dx^n}{n}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {-\frac {7 b \int \frac {x^{-4 n}}{\sqrt {b x^n+a}}dx^n}{8 a}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a}}{n}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {-\frac {7 b \left (-\frac {5 b \int \frac {x^{-3 n}}{\sqrt {b x^n+a}}dx^n}{6 a}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 a}\right )}{8 a}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a}}{n}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \int \frac {x^{-2 n}}{\sqrt {b x^n+a}}dx^n}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}\right )}{6 a}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 a}\right )}{8 a}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a}}{n}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {b \int \frac {x^{-n}}{\sqrt {b x^n+a}}dx^n}{2 a}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}\right )}{6 a}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 a}\right )}{8 a}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a}}{n}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {x^{2 n}}{b}-\frac {a}{b}}d\sqrt {b x^n+a}}{a}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}\right )}{6 a}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 a}\right )}{8 a}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a}}{n}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}\right )}{6 a}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 a}\right )}{8 a}-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 a}}{n}\)

Input:

Int[x^(-1 - 4*n)/Sqrt[a + b*x^n],x]
 

Output:

(-1/4*Sqrt[a + b*x^n]/(a*x^(4*n)) - (7*b*(-1/3*Sqrt[a + b*x^n]/(a*x^(3*n)) 
 - (5*b*(-1/2*Sqrt[a + b*x^n]/(a*x^(2*n)) - (3*b*(-(Sqrt[a + b*x^n]/(a*x^n 
)) + (b*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/a^(3/2)))/(4*a)))/(6*a)))/(8*a)) 
/n
 

Defintions of rubi rules used

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 798

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

Maple [F]

\[\int \frac {x^{-4 n -1}}{\sqrt {a +b \,x^{n}}}d x\]

Input:

int(x^(-4*n-1)/(a+b*x^n)^(1/2),x)
 

Output:

int(x^(-4*n-1)/(a+b*x^n)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.42 \[ \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx=\left [\frac {105 \, \sqrt {a} b^{4} x^{4 \, n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (105 \, a b^{3} x^{3 \, n} - 70 \, a^{2} b^{2} x^{2 \, n} + 56 \, a^{3} b x^{n} - 48 \, a^{4}\right )} \sqrt {b x^{n} + a}}{384 \, a^{5} n x^{4 \, n}}, \frac {105 \, \sqrt {-a} b^{4} x^{4 \, n} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (105 \, a b^{3} x^{3 \, n} - 70 \, a^{2} b^{2} x^{2 \, n} + 56 \, a^{3} b x^{n} - 48 \, a^{4}\right )} \sqrt {b x^{n} + a}}{192 \, a^{5} n x^{4 \, n}}\right ] \] Input:

integrate(x^(-1-4*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")
 

Output:

[1/384*(105*sqrt(a)*b^4*x^(4*n)*log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2 
*a)/x^n) + 2*(105*a*b^3*x^(3*n) - 70*a^2*b^2*x^(2*n) + 56*a^3*b*x^n - 48*a 
^4)*sqrt(b*x^n + a))/(a^5*n*x^(4*n)), 1/192*(105*sqrt(-a)*b^4*x^(4*n)*arct 
an(sqrt(-a)/sqrt(b*x^n + a)) + (105*a*b^3*x^(3*n) - 70*a^2*b^2*x^(2*n) + 5 
6*a^3*b*x^n - 48*a^4)*sqrt(b*x^n + a))/(a^5*n*x^(4*n))]
 

Sympy [A] (verification not implemented)

Time = 95.59 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.26 \[ \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx=- \frac {x^{- \frac {9 n}{2}}}{4 \sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {\sqrt {b} x^{- \frac {7 n}{2}}}{24 a n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {7 b^{\frac {3}{2}} x^{- \frac {5 n}{2}}}{96 a^{2} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {35 b^{\frac {5}{2}} x^{- \frac {3 n}{2}}}{192 a^{3} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {35 b^{\frac {7}{2}} x^{- \frac {n}{2}}}{64 a^{4} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {35 b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{64 a^{\frac {9}{2}} n} \] Input:

integrate(x**(-1-4*n)/(a+b*x**n)**(1/2),x)
 

Output:

-1/(4*sqrt(b)*n*x**(9*n/2)*sqrt(a/(b*x**n) + 1)) + sqrt(b)/(24*a*n*x**(7*n 
/2)*sqrt(a/(b*x**n) + 1)) - 7*b**(3/2)/(96*a**2*n*x**(5*n/2)*sqrt(a/(b*x** 
n) + 1)) + 35*b**(5/2)/(192*a**3*n*x**(3*n/2)*sqrt(a/(b*x**n) + 1)) + 35*b 
**(7/2)/(64*a**4*n*x**(n/2)*sqrt(a/(b*x**n) + 1)) - 35*b**4*asinh(sqrt(a)/ 
(sqrt(b)*x**(n/2)))/(64*a**(9/2)*n)
 

Maxima [F]

\[ \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-4 \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:

integrate(x^(-1-4*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^(-4*n - 1)/sqrt(b*x^n + a), x)
 

Giac [F]

\[ \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-4 \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:

integrate(x^(-1-4*n)/(a+b*x^n)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^(-4*n - 1)/sqrt(b*x^n + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx=\int \frac {1}{x^{4\,n+1}\,\sqrt {a+b\,x^n}} \,d x \] Input:

int(1/(x^(4*n + 1)*(a + b*x^n)^(1/2)),x)
 

Output:

int(1/(x^(4*n + 1)*(a + b*x^n)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^{-1-4 n}}{\sqrt {a+b x^n}} \, dx=\frac {210 x^{3 n} \sqrt {x^{n} b +a}\, b^{3}-140 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2}+112 x^{n} \sqrt {x^{n} b +a}\, a^{2} b -96 \sqrt {x^{n} b +a}\, a^{3}+105 x^{4 n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right ) b^{4} n}{384 x^{4 n} a^{4} n} \] Input:

int(x^(-1-4*n)/(a+b*x^n)^(1/2),x)
 

Output:

(210*x**(3*n)*sqrt(x**n*b + a)*b**3 - 140*x**(2*n)*sqrt(x**n*b + a)*a*b**2 
 + 112*x**n*sqrt(x**n*b + a)*a**2*b - 96*sqrt(x**n*b + a)*a**3 + 105*x**(4 
*n)*int(sqrt(x**n*b + a)/(x**n*b*x + a*x),x)*b**4*n)/(384*x**(4*n)*a**4*n)