\(\int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx\) [551]

Optimal result

Integrand size = 19, antiderivative size = 116 \[ \int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 a n}+\frac {5 b x^{-2 n} \sqrt {a+b x^n}}{12 a^2 n}-\frac {5 b^2 x^{-n} \sqrt {a+b x^n}}{8 a^3 n}+\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{7/2} n} \] Output:

-1/3*(a+b*x^n)^(1/2)/a/n/(x^(3*n))+5/12*b*(a+b*x^n)^(1/2)/a^2/n/(x^(2*n))- 
5/8*b^2*(a+b*x^n)^(1/2)/a^3/n/(x^n)+5/8*b^3*arctanh((a+b*x^n)^(1/2)/a^(1/2 
))/a^(7/2)/n
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.70 \[ \int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx=\frac {\sqrt {a} x^{-3 n} \sqrt {a+b x^n} \left (-8 a^2+10 a b x^n-15 b^2 x^{2 n}\right )+15 b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{24 a^{7/2} n} \] Input:

Integrate[x^(-1 - 3*n)/Sqrt[a + b*x^n],x]
 

Output:

((Sqrt[a]*Sqrt[a + b*x^n]*(-8*a^2 + 10*a*b*x^n - 15*b^2*x^(2*n)))/x^(3*n) 
+ 15*b^3*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(24*a^(7/2)*n)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {798, 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.

\(\displaystyle \int \frac {x^{-3 n-1}}{\sqrt {a+b x^n}} \, dx\)

\(\Big \downarrow \) 798

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[x^(-1 - 3*n)/Sqrt[a + b*x^n],x]
 

Output:

(-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))/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]

\[\int \frac {x^{-1-3 n}}{\sqrt {a +b \,x^{n}}}d x\]

Input:

int(x^(-1-3*n)/(a+b*x^n)^(1/2),x)
 

Output:

int(x^(-1-3*n)/(a+b*x^n)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.55 \[ \int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx=\left [\frac {15 \, \sqrt {a} b^{3} x^{3 \, n} \log \left (\frac {b x^{n} + 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) - 2 \, {\left (15 \, a b^{2} x^{2 \, n} - 10 \, a^{2} b x^{n} + 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{48 \, a^{4} n x^{3 \, n}}, -\frac {15 \, \sqrt {-a} b^{3} x^{3 \, n} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (15 \, a b^{2} x^{2 \, n} - 10 \, a^{2} b x^{n} + 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{24 \, a^{4} n x^{3 \, n}}\right ] \] Input:

integrate(x^(-1-3*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")
 

Output:

[1/48*(15*sqrt(a)*b^3*x^(3*n)*log((b*x^n + 2*sqrt(b*x^n + a)*sqrt(a) + 2*a 
)/x^n) - 2*(15*a*b^2*x^(2*n) - 10*a^2*b*x^n + 8*a^3)*sqrt(b*x^n + a))/(a^4 
*n*x^(3*n)), -1/24*(15*sqrt(-a)*b^3*x^(3*n)*arctan(sqrt(-a)/sqrt(b*x^n + a 
)) + (15*a*b^2*x^(2*n) - 10*a^2*b*x^n + 8*a^3)*sqrt(b*x^n + a))/(a^4*n*x^( 
3*n))]
 

Sympy [A] (verification not implemented)

Time = 29.55 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.29 \[ \int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx=- \frac {x^{- \frac {7 n}{2}}}{3 \sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {\sqrt {b} x^{- \frac {5 n}{2}}}{12 a n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {5 b^{\frac {3}{2}} x^{- \frac {3 n}{2}}}{24 a^{2} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {5 b^{\frac {5}{2}} x^{- \frac {n}{2}}}{8 a^{3} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {5 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{8 a^{\frac {7}{2}} n} \] Input:

integrate(x**(-1-3*n)/(a+b*x**n)**(1/2),x)
 

Output:

-1/(3*sqrt(b)*n*x**(7*n/2)*sqrt(a/(b*x**n) + 1)) + sqrt(b)/(12*a*n*x**(5*n 
/2)*sqrt(a/(b*x**n) + 1)) - 5*b**(3/2)/(24*a**2*n*x**(3*n/2)*sqrt(a/(b*x** 
n) + 1)) - 5*b**(5/2)/(8*a**3*n*x**(n/2)*sqrt(a/(b*x**n) + 1)) + 5*b**3*as 
inh(sqrt(a)/(sqrt(b)*x**(n/2)))/(8*a**(7/2)*n)
 

Maxima [F]

\[ \int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-3 \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:

integrate(x^(-1-3*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^(-3*n - 1)/sqrt(b*x^n + a), x)
 

Giac [F]

\[ \int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-3 \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:

integrate(x^(-1-3*n)/(a+b*x^n)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^(-3*n - 1)/sqrt(b*x^n + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx=\int \frac {1}{x^{3\,n+1}\,\sqrt {a+b\,x^n}} \,d x \] Input:

int(1/(x^(3*n + 1)*(a + b*x^n)^(1/2)),x)
 

Output:

int(1/(x^(3*n + 1)*(a + b*x^n)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^{-1-3 n}}{\sqrt {a+b x^n}} \, dx=\frac {-30 x^{2 n} \sqrt {x^{n} b +a}\, b^{2}+20 x^{n} \sqrt {x^{n} b +a}\, a b -16 \sqrt {x^{n} b +a}\, a^{2}-15 x^{3 n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right ) b^{3} n}{48 x^{3 n} a^{3} n} \] Input:

int(x^(-1-3*n)/(a+b*x^n)^(1/2),x)
 

Output:

( - 30*x**(2*n)*sqrt(x**n*b + a)*b**2 + 20*x**n*sqrt(x**n*b + a)*a*b - 16* 
sqrt(x**n*b + a)*a**2 - 15*x**(3*n)*int(sqrt(x**n*b + a)/(x**n*b*x + a*x), 
x)*b**3*n)/(48*x**(3*n)*a**3*n)