\(\int x^{-1-4 n} \sqrt {a+b x^n} \, dx\) [543]

Optimal result

Integrand size = 19, antiderivative size = 142 \[ \int x^{-1-4 n} \sqrt {a+b x^n} \, dx=-\frac {x^{-4 n} \sqrt {a+b x^n}}{4 n}-\frac {b x^{-3 n} \sqrt {a+b x^n}}{24 a n}+\frac {5 b^2 x^{-2 n} \sqrt {a+b x^n}}{96 a^2 n}-\frac {5 b^3 x^{-n} \sqrt {a+b x^n}}{64 a^3 n}+\frac {5 b^4 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{64 a^{7/2} n} \] Output:

-1/4*(a+b*x^n)^(1/2)/n/(x^(4*n))-1/24*b*(a+b*x^n)^(1/2)/a/n/(x^(3*n))+5/96 
*b^2*(a+b*x^n)^(1/2)/a^2/n/(x^(2*n))-5/64*b^3*(a+b*x^n)^(1/2)/a^3/n/(x^n)+ 
5/64*b^4*arctanh((a+b*x^n)^(1/2)/a^(1/2))/a^(7/2)/n
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.67 \[ \int x^{-1-4 n} \sqrt {a+b x^n} \, dx=\frac {-\sqrt {a} x^{-4 n} \sqrt {a+b x^n} \left (48 a^3+8 a^2 b x^n-10 a b^2 x^{2 n}+15 b^3 x^{3 n}\right )+15 b^4 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{192 a^{7/2} n} \] Input:

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

Output:

(-((Sqrt[a]*Sqrt[a + b*x^n]*(48*a^3 + 8*a^2*b*x^n - 10*a*b^2*x^(2*n) + 15* 
b^3*x^(3*n)))/x^(4*n)) + 15*b^4*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(192*a^( 
7/2)*n)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {798, 51, 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.

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

\(\Big \downarrow \) 798

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

\(\Big \downarrow \) 51

\(\displaystyle \frac {\frac {1}{8} b \int \frac {x^{-4 n}}{\sqrt {b x^n+a}}dx^n-\frac {1}{4} x^{-4 n} \sqrt {a+b x^n}}{n}\)

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 51

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

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]

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

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 = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.45 \[ \int x^{-1-4 n} \sqrt {a+b x^n} \, dx=\left [\frac {15 \, \sqrt {a} b^{4} x^{4 \, n} \log \left (\frac {b x^{n} + 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) - 2 \, {\left (15 \, a b^{3} x^{3 \, n} - 10 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} + 48 \, a^{4}\right )} \sqrt {b x^{n} + a}}{384 \, a^{4} n x^{4 \, n}}, -\frac {15 \, \sqrt {-a} b^{4} x^{4 \, n} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (15 \, a b^{3} x^{3 \, n} - 10 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} + 48 \, a^{4}\right )} \sqrt {b x^{n} + a}}{192 \, a^{4} n x^{4 \, n}}\right ] \] Input:

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

Output:

[1/384*(15*sqrt(a)*b^4*x^(4*n)*log((b*x^n + 2*sqrt(b*x^n + a)*sqrt(a) + 2* 
a)/x^n) - 2*(15*a*b^3*x^(3*n) - 10*a^2*b^2*x^(2*n) + 8*a^3*b*x^n + 48*a^4) 
*sqrt(b*x^n + a))/(a^4*n*x^(4*n)), -1/192*(15*sqrt(-a)*b^4*x^(4*n)*arctan( 
sqrt(-a)/sqrt(b*x^n + a)) + (15*a*b^3*x^(3*n) - 10*a^2*b^2*x^(2*n) + 8*a^3 
*b*x^n + 48*a^4)*sqrt(b*x^n + a))/(a^4*n*x^(4*n))]
 

Sympy [A] (verification not implemented)

Time = 54.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.25 \[ \int x^{-1-4 n} \sqrt {a+b x^n} \, dx=- \frac {a x^{- \frac {9 n}{2}}}{4 \sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {7 \sqrt {b} x^{- \frac {7 n}{2}}}{24 n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {b^{\frac {3}{2}} x^{- \frac {5 n}{2}}}{96 a n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {5 b^{\frac {5}{2}} x^{- \frac {3 n}{2}}}{192 a^{2} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {5 b^{\frac {7}{2}} x^{- \frac {n}{2}}}{64 a^{3} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {5 b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{64 a^{\frac {7}{2}} n} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^{-1-4 n} \sqrt {a+b x^n} \, dx=\frac {-30 x^{3 n} \sqrt {x^{n} b +a}\, b^{3}+20 x^{2 n} \sqrt {x^{n} b +a}\, a \,b^{2}-16 x^{n} \sqrt {x^{n} b +a}\, a^{2} b -96 \sqrt {x^{n} b +a}\, a^{3}-15 x^{4 n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right ) b^{4} n}{384 x^{4 n} a^{3} n} \] Input:

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

Output:

( - 30*x**(3*n)*sqrt(x**n*b + a)*b**3 + 20*x**(2*n)*sqrt(x**n*b + a)*a*b** 
2 - 16*x**n*sqrt(x**n*b + a)*a**2*b - 96*sqrt(x**n*b + a)*a**3 - 15*x**(4* 
n)*int(sqrt(x**n*b + a)/(x**n*b*x + a*x),x)*b**4*n)/(384*x**(4*n)*a**3*n)