\(\int \frac {x^{-1+2 n} \sqrt {a+b x^n}}{\sqrt {c+d x^n}} \, dx\) [481]

Optimal result

Integrand size = 30, antiderivative size = 146 \[ \int \frac {x^{-1+2 n} \sqrt {a+b x^n}}{\sqrt {c+d x^n}} \, dx=-\frac {(3 b c+a d) \sqrt {a+b x^n} \sqrt {c+d x^n}}{4 b d^2 n}+\frac {\left (a+b x^n\right )^{3/2} \sqrt {c+d x^n}}{2 b d n}+\frac {(b c-a d) (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{4 b^{3/2} d^{5/2} n} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int \frac {x^{-1+2 n} \sqrt {a+b x^n}}{\sqrt {c+d x^n}} \, dx=\frac {b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right ) \left (-3 b c+a d+2 b d x^n\right )+(b c-a d)^{3/2} (3 b c+a d) \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )}{4 b^2 d^{5/2} n \sqrt {c+d x^n}} \] Input:

Integrate[(x^(-1 + 2*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 + a*d + 2*b*d*x^n) + (b*c - 
 a*d)^(3/2)*(3*b*c + a*d)*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^2*d^(5/2)*n*Sqrt[c + d*x^n])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {948, 90, 60, 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.

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 60

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

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

Input:

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

Output:

int(x^(-1+2*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 = 359, normalized size of antiderivative = 2.46 \[ \int \frac {x^{-1+2 n} \sqrt {a+b x^n}}{\sqrt {c+d x^n}} \, dx=\left [-\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, \sqrt {b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right ) - 4 \, {\left (2 \, b^{2} d^{2} x^{n} - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{16 \, b^{2} d^{3} n}, -\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, \sqrt {-b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {-b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{2 \, {\left (b^{2} d^{2} x^{2 \, n} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x^{n} - 3 \, b^{2} c d + a b d^{2}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{8 \, b^{2} d^{3} n}\right ] \] Input:

integrate(x^(-1+2*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 - 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)*sqr 
t(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 + a*b*d^2)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^2 
*d^3*n), -1/8*((3*b^2*c^2 - 2*a*b*c*d - 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 + a*b*d^2)*sqrt(b*x^n + a)*sqrt(d*x^n + c))/(b^2*d^3*n)]
 

Sympy [F]

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

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

Output:

Integral(x**(2*n - 1)*sqrt(a + b*x**n)/sqrt(c + d*x**n), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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