\(\int \frac {x^{-1+3 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx\) [488]

Optimal result

Integrand size = 30, antiderivative size = 154 \[ \int \frac {x^{-1+3 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=-\frac {(3 b c+5 a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b^2 d^2 n}+\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b^2 d n}+\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{4 b^{5/2} d^{5/2} n} \] Output:

-1/4*(5*a*d+3*b*c)*(a+b*x^n)^(1/2)*(c+d*x^n)^(1/2)/b^2/d^2/n+1/2*(a+b*x^n) 
^(3/2)*(c+d*x^n)^(1/2)/b^2/d/n+1/4*(3*a^2*d^2+2*a*b*c*d+3*b^2*c^2)*arctanh 
(d^(1/2)*(a+b*x^n)^(1/2)/b^(1/2)/(c+d*x^n)^(1/2))/b^(5/2)/d^(5/2)/n
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.02 \[ \int \frac {x^{-1+3 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\frac {b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right ) \left (-3 b c-3 a d+2 b d x^n\right )+\sqrt {b c-a d} \left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )}{4 b^3 d^{5/2} n \sqrt {c+d x^n}} \] Input:

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

Output:

(b*Sqrt[d]*Sqrt[a + b*x^n]*(c + d*x^n)*(-3*b*c - 3*a*d + 2*b*d*x^n) + Sqrt 
[b*c - a*d]*(3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*Sqrt[(b*(c + d*x^n))/(b*c 
- a*d)]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^n])/Sqrt[b*c - a*d]])/(4*b^3*d^(5/2) 
*n*Sqrt[c + d*x^n])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {948, 101, 27, 90, 66, 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.

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

\(\Big \downarrow \) 948

\(\displaystyle \frac {\int \frac {x^{2 n}}{\sqrt {b x^n+a} \sqrt {d x^n+c}}dx^n}{n}\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {\frac {\int -\frac {3 (b c+a d) x^n+2 a c}{2 \sqrt {b x^n+a} \sqrt {d x^n+c}}dx^n}{2 b d}+\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d}}{n}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {x^n \sqrt {a+b x^n} \sqrt {c+d x^n}}{2 b d}-\frac {\int \frac {3 (b c+a d) x^n+2 a c}{\sqrt {b x^n+a} \sqrt {d x^n+c}}dx^n}{4 b d}}{n}\)

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((x^n*Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(2*b*d) - ((3*(b*c + a*d)*Sqrt[a + 
b*x^n]*Sqrt[c + d*x^n])/(b*d) + ((4*a*b*c*d - 3*(b*c + a*d)^2)*ArcTanh[(Sq 
rt[d]*Sqrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(3/2)*d^(3/2)))/(4*b 
*d))/n
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

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 948

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.34 \[ \int \frac {x^{-1+3 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\left [\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, \sqrt {b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right ) + 4 \, {\left (2 \, b^{2} d^{2} x^{n} - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{16 \, b^{3} d^{3} n}, -\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, \sqrt {-b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {-b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{2 \, {\left (b^{2} d^{2} x^{2 \, n} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{n} - 3 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{8 \, b^{3} d^{3} n}\right ] \] Input:

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

Output:

[1/16*((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*sqrt(b*d)*log(8*b^2*d^2*x^(2*n) 
 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*sqrt(b*d)*b*d*x^n + (b*c + a*d)*sq 
rt(b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c) + 8*(b^2*c*d + a*b*d^2)*x^n) + 4* 
(2*b^2*d^2*x^n - 3*b^2*c*d - 3*a*b*d^2)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/( 
b^3*d^3*n), -1/8*((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*sqrt(-b*d)*arctan(1/ 
2*(2*sqrt(-b*d)*b*d*x^n + (b*c + a*d)*sqrt(-b*d))*sqrt(b*x^n + a)*sqrt(d*x 
^n + c)/(b^2*d^2*x^(2*n) + a*b*c*d + (b^2*c*d + a*b*d^2)*x^n)) - 2*(2*b^2* 
d^2*x^n - 3*b^2*c*d - 3*a*b*d^2)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^3*d^3 
*n)]
 

Sympy [F]

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

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

Output:

Integral(x**(3*n - 1)/(sqrt(a + b*x**n)*sqrt(c + d*x**n)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^{-1+3 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\frac {2 x^{n} \sqrt {x^{n} d +c}\, \sqrt {x^{n} b +a}\, a d +2 x^{n} \sqrt {x^{n} d +c}\, \sqrt {x^{n} b +a}\, b c -4 \sqrt {x^{n} d +c}\, \sqrt {x^{n} b +a}\, a c -3 \left (\int \frac {x^{2 n} \sqrt {x^{n} d +c}\, \sqrt {x^{n} b +a}}{x^{2 n} a b \,d^{2} x +x^{2 n} b^{2} c d x +x^{n} a^{2} d^{2} x +2 x^{n} a b c d x +x^{n} b^{2} c^{2} x +a^{2} c d x +a b \,c^{2} x}d x \right ) a^{3} d^{3} n -5 \left (\int \frac {x^{2 n} \sqrt {x^{n} d +c}\, \sqrt {x^{n} b +a}}{x^{2 n} a b \,d^{2} x +x^{2 n} b^{2} c d x +x^{n} a^{2} d^{2} x +2 x^{n} a b c d x +x^{n} b^{2} c^{2} x +a^{2} c d x +a b \,c^{2} x}d x \right ) a^{2} b c \,d^{2} n -5 \left (\int \frac {x^{2 n} \sqrt {x^{n} d +c}\, \sqrt {x^{n} b +a}}{x^{2 n} a b \,d^{2} x +x^{2 n} b^{2} c d x +x^{n} a^{2} d^{2} x +2 x^{n} a b c d x +x^{n} b^{2} c^{2} x +a^{2} c d x +a b \,c^{2} x}d x \right ) a \,b^{2} c^{2} d n -3 \left (\int \frac {x^{2 n} \sqrt {x^{n} d +c}\, \sqrt {x^{n} b +a}}{x^{2 n} a b \,d^{2} x +x^{2 n} b^{2} c d x +x^{n} a^{2} d^{2} x +2 x^{n} a b c d x +x^{n} b^{2} c^{2} x +a^{2} c d x +a b \,c^{2} x}d x \right ) b^{3} c^{3} n}{4 b d n \left (a d +b c \right )} \] Input:

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

Output:

(2*x**n*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a*d + 2*x**n*sqrt(x**n*d + c)*sq 
rt(x**n*b + a)*b*c - 4*sqrt(x**n*d + c)*sqrt(x**n*b + a)*a*c - 3*int((x**( 
2*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a))/(x**(2*n)*a*b*d**2*x + x**(2*n)*b* 
*2*c*d*x + x**n*a**2*d**2*x + 2*x**n*a*b*c*d*x + x**n*b**2*c**2*x + a**2*c 
*d*x + a*b*c**2*x),x)*a**3*d**3*n - 5*int((x**(2*n)*sqrt(x**n*d + c)*sqrt( 
x**n*b + a))/(x**(2*n)*a*b*d**2*x + x**(2*n)*b**2*c*d*x + x**n*a**2*d**2*x 
 + 2*x**n*a*b*c*d*x + x**n*b**2*c**2*x + a**2*c*d*x + a*b*c**2*x),x)*a**2* 
b*c*d**2*n - 5*int((x**(2*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a))/(x**(2*n)* 
a*b*d**2*x + x**(2*n)*b**2*c*d*x + x**n*a**2*d**2*x + 2*x**n*a*b*c*d*x + x 
**n*b**2*c**2*x + a**2*c*d*x + a*b*c**2*x),x)*a*b**2*c**2*d*n - 3*int((x** 
(2*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a))/(x**(2*n)*a*b*d**2*x + x**(2*n)*b 
**2*c*d*x + x**n*a**2*d**2*x + 2*x**n*a*b*c*d*x + x**n*b**2*c**2*x + a**2* 
c*d*x + a*b*c**2*x),x)*b**3*c**3*n)/(4*b*d*n*(a*d + b*c))