Integrand size = 30, antiderivative size = 89 \[ \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d n}-\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{b^{3/2} d^{3/2} n} \] Output:
(a+b*x^n)^(1/2)*(c+d*x^n)^(1/2)/b/d/n-(a*d+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^(3/2)/n
Time = 0.23 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.38 \[ \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\frac {b \sqrt {d} \sqrt {a+b x^n} \left (c+d x^n\right )-\sqrt {b c-a d} (b c+a d) \sqrt {\frac {b \left (c+d x^n\right )}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b c-a d}}\right )}{b^2 d^{3/2} n \sqrt {c+d x^n}} \] 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) - Sqrt[b*c - a*d]*(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]])/(b^2*d^(3/2)*n*Sqrt[c + d*x^n])
Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {948, 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.
\(\displaystyle \int \frac {x^{2 n-1}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\int \frac {x^n}{\sqrt {b x^n+a} \sqrt {d x^n+c}}dx^n}{n}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d}-\frac {(a d+b c) \int \frac {1}{\sqrt {b x^n+a} \sqrt {d x^n+c}}dx^n}{2 b d}}{n}\) |
\(\Big \downarrow \) 66 |
\(\displaystyle \frac {\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d}-\frac {(a d+b c) \int \frac {1}{b-d x^{2 n}}d\frac {\sqrt {b x^n+a}}{\sqrt {d x^n+c}}}{b d}}{n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\sqrt {a+b x^n} \sqrt {c+d x^n}}{b d}-\frac {(a d+b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x^n}}{\sqrt {b} \sqrt {c+d x^n}}\right )}{b^{3/2} d^{3/2}}}{n}\) |
Input:
Int[x^(-1 + 2*n)/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),x]
Output:
((Sqrt[a + b*x^n]*Sqrt[c + d*x^n])/(b*d) - ((b*c + a*d)*ArcTanh[(Sqrt[d]*S qrt[a + b*x^n])/(Sqrt[b]*Sqrt[c + d*x^n])])/(b^(3/2)*d^(3/2)))/n
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]
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]
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]
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]]
\[\int \frac {x^{-1+2 n}}{\sqrt {a +b \,x^{n}}\, \sqrt {c +d \,x^{n}}}d x\]
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)
Time = 0.09 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.16 \[ \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\left [\frac {4 \, \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} b d + {\left (b c + a d\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2 \, n} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, \sqrt {b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}{4 \, b^{2} d^{2} n}, \frac {2 \, \sqrt {b x^{n} + a} \sqrt {d x^{n} + c} b d + {\left (b c + a d\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, \sqrt {-b d} b d x^{n} + {\left (b c + a d\right )} \sqrt {-b d}\right )} \sqrt {b x^{n} + a} \sqrt {d x^{n} + c}}{2 \, {\left (b^{2} d^{2} x^{2 \, n} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{n}\right )}}\right )}{2 \, b^{2} d^{2} n}\right ] \] Input:
integrate(x^(-1+2*n)/(a+b*x^n)^(1/2)/(c+d*x^n)^(1/2),x, algorithm="fricas" )
Output:
[1/4*(4*sqrt(b*x^n + a)*sqrt(d*x^n + c)*b*d + (b*c + a*d)*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)*sqrt(b*d))*sqrt(b*x^n + a)*sqrt(d*x^n + c) + 8*(b^2*c*d + a*b *d^2)*x^n))/(b^2*d^2*n), 1/2*(2*sqrt(b*x^n + a)*sqrt(d*x^n + c)*b*d + (b*c + a*d)*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)))/(b^2*d^2*n)]
\[ \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\int \frac {x^{2 n - 1}}{\sqrt {a + b x^{n}} \sqrt {c + d x^{n}}}\, dx \] 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)
\[ \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\int { \frac {x^{2 \, n - 1}}{\sqrt {b x^{n} + a} \sqrt {d x^{n} + c}} \,d x } \] Input:
integrate(x^(-1+2*n)/(a+b*x^n)^(1/2)/(c+d*x^n)^(1/2),x, algorithm="maxima" )
Output:
integrate(x^(2*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)), x)
\[ \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\int { \frac {x^{2 \, n - 1}}{\sqrt {b x^{n} + a} \sqrt {d x^{n} + c}} \,d x } \] Input:
integrate(x^(-1+2*n)/(a+b*x^n)^(1/2)/(c+d*x^n)^(1/2),x, algorithm="giac")
Output:
integrate(x^(2*n - 1)/(sqrt(b*x^n + a)*sqrt(d*x^n + c)), x)
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)
\[ \int \frac {x^{-1+2 n}}{\sqrt {a+b x^n} \sqrt {c+d x^n}} \, dx=\int \frac {x^{2 n} \sqrt {x^{n} d +c}\, \sqrt {x^{n} b +a}}{x^{2 n} b d x +x^{n} a d x +x^{n} b c x +a c x}d x \] Input:
int(x^(-1+2*n)/(a+b*x^n)^(1/2)/(c+d*x^n)^(1/2),x)
Output:
int((x**(2*n)*sqrt(x**n*d + c)*sqrt(x**n*b + a))/(x**(2*n)*b*d*x + x**n*a* d*x + x**n*b*c*x + a*c*x),x)