Integrand size = 50, antiderivative size = 53 \[ \int \left (a+b x^2\right )^{-1-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac {a d}{2 b c-2 a d}} \, dx=\frac {x \left (a+b x^2\right )^{-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{\frac {a d}{2 b c-2 a d}}}{a c} \] Output:
x*(d*x^2+c)^(a*d/(-2*a*d+2*b*c))/a/c/((b*x^2+a)^(b*c/(-2*a*d+2*b*c)))
Time = 0.39 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^2\right )^{-1-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac {a d}{2 b c-2 a d}} \, dx=\frac {x \left (a+b x^2\right )^{\frac {b c}{-2 b c+2 a d}} \left (c+d x^2\right )^{\frac {a d}{2 b c-2 a d}}}{a c} \] Input:
Integrate[(a + b*x^2)^(-1 - (b*c)/(2*b*c - 2*a*d))*(c + d*x^2)^(-1 + (a*d) /(2*b*c - 2*a*d)),x]
Output:
(x*(a + b*x^2)^((b*c)/(-2*b*c + 2*a*d))*(c + d*x^2)^((a*d)/(2*b*c - 2*a*d) ))/(a*c)
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {295}
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.
\(\displaystyle \int \left (a+b x^2\right )^{-\frac {b c}{2 b c-2 a d}-1} \left (c+d x^2\right )^{\frac {a d}{2 b c-2 a d}-1} \, dx\) |
\(\Big \downarrow \) 295 |
\(\displaystyle \frac {x \left (a+b x^2\right )^{-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{\frac {a d}{2 b c-2 a d}}}{a c}\) |
Input:
Int[(a + b*x^2)^(-1 - (b*c)/(2*b*c - 2*a*d))*(c + d*x^2)^(-1 + (a*d)/(2*b* c - 2*a*d)),x]
Output:
(x*(c + d*x^2)^((a*d)/(2*b*c - 2*a*d)))/(a*c*(a + b*x^2)^((b*c)/(2*b*c - 2 *a*d)))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c)), x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && EqQ[a*d*( p + 1) + b*c*(q + 1), 0]
Time = 1.52 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.34
method | result | size |
gosper | \(\frac {x \left (b \,x^{2}+a \right )^{1-\frac {2 a d -3 b c}{2 \left (a d -b c \right )}} \left (x^{2} d +c \right )^{1-\frac {3 a d -2 b c}{2 \left (a d -b c \right )}}}{a c}\) | \(71\) |
orering | \(\frac {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right ) x \left (b \,x^{2}+a \right )^{-1-\frac {b c}{-2 a d +2 b c}} \left (x^{2} d +c \right )^{-1+\frac {a d}{-2 a d +2 b c}}}{a c}\) | \(72\) |
Input:
int((b*x^2+a)^(-1-b*c/(-2*a*d+2*b*c))*(d*x^2+c)^(-1+a*d/(-2*a*d+2*b*c)),x, method=_RETURNVERBOSE)
Output:
x/a/c*(b*x^2+a)^(1-1/2*(2*a*d-3*b*c)/(a*d-b*c))*(d*x^2+c)^(1-1/2*(3*a*d-2* b*c)/(a*d-b*c))
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.72 \[ \int \left (a+b x^2\right )^{-1-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac {a d}{2 b c-2 a d}} \, dx=\frac {b d x^{5} + {\left (b c + a d\right )} x^{3} + a c x}{{\left (b x^{2} + a\right )}^{\frac {3 \, b c - 2 \, a d}{2 \, {\left (b c - a d\right )}}} {\left (d x^{2} + c\right )}^{\frac {2 \, b c - 3 \, a d}{2 \, {\left (b c - a d\right )}}} a c} \] Input:
integrate((b*x^2+a)^(-1-b*c/(-2*a*d+2*b*c))*(d*x^2+c)^(-1+a*d/(-2*a*d+2*b* c)),x, algorithm="fricas")
Output:
(b*d*x^5 + (b*c + a*d)*x^3 + a*c*x)/((b*x^2 + a)^(1/2*(3*b*c - 2*a*d)/(b*c - a*d))*(d*x^2 + c)^(1/2*(2*b*c - 3*a*d)/(b*c - a*d))*a*c)
Timed out. \[ \int \left (a+b x^2\right )^{-1-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac {a d}{2 b c-2 a d}} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)**(-1-b*c/(-2*a*d+2*b*c))*(d*x**2+c)**(-1+a*d/(-2*a*d+ 2*b*c)),x)
Output:
Timed out
\[ \int \left (a+b x^2\right )^{-1-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac {a d}{2 b c-2 a d}} \, dx=\int { {\left (b x^{2} + a\right )}^{-\frac {b c}{2 \, {\left (b c - a d\right )}} - 1} {\left (d x^{2} + c\right )}^{\frac {a d}{2 \, {\left (b c - a d\right )}} - 1} \,d x } \] Input:
integrate((b*x^2+a)^(-1-b*c/(-2*a*d+2*b*c))*(d*x^2+c)^(-1+a*d/(-2*a*d+2*b* c)),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)^(-1/2*b*c/(b*c - a*d) - 1)*(d*x^2 + c)^(1/2*a*d/(b*c - a*d) - 1), x)
\[ \int \left (a+b x^2\right )^{-1-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac {a d}{2 b c-2 a d}} \, dx=\int { {\left (b x^{2} + a\right )}^{-\frac {b c}{2 \, {\left (b c - a d\right )}} - 1} {\left (d x^{2} + c\right )}^{\frac {a d}{2 \, {\left (b c - a d\right )}} - 1} \,d x } \] Input:
integrate((b*x^2+a)^(-1-b*c/(-2*a*d+2*b*c))*(d*x^2+c)^(-1+a*d/(-2*a*d+2*b* c)),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(-1/2*b*c/(b*c - a*d) - 1)*(d*x^2 + c)^(1/2*a*d/(b*c - a*d) - 1), x)
Time = 1.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.47 \[ \int \left (a+b x^2\right )^{-1-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac {a d}{2 b c-2 a d}} \, dx=\frac {x\,{\left (b\,x^2+a\right )}^{\frac {b\,c}{2\,a\,d-2\,b\,c}-1}+\frac {x^3\,{\left (b\,x^2+a\right )}^{\frac {b\,c}{2\,a\,d-2\,b\,c}-1}\,\left (a\,d+b\,c\right )}{a\,c}+\frac {b\,d\,x^5\,{\left (b\,x^2+a\right )}^{\frac {b\,c}{2\,a\,d-2\,b\,c}-1}}{a\,c}}{{\left (d\,x^2+c\right )}^{\frac {a\,d}{2\,a\,d-2\,b\,c}+1}} \] Input:
int((a + b*x^2)^((b*c)/(2*a*d - 2*b*c) - 1)/(c + d*x^2)^((a*d)/(2*a*d - 2* b*c) + 1),x)
Output:
(x*(a + b*x^2)^((b*c)/(2*a*d - 2*b*c) - 1) + (x^3*(a + b*x^2)^((b*c)/(2*a* d - 2*b*c) - 1)*(a*d + b*c))/(a*c) + (b*d*x^5*(a + b*x^2)^((b*c)/(2*a*d - 2*b*c) - 1))/(a*c))/(c + d*x^2)^((a*d)/(2*a*d - 2*b*c) + 1)
\[ \int \left (a+b x^2\right )^{-1-\frac {b c}{2 b c-2 a d}} \left (c+d x^2\right )^{-1+\frac {a d}{2 b c-2 a d}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {b c}{2 a d -2 b c}}}{\left (d \,x^{2}+c \right )^{\frac {a d}{2 a d -2 b c}} a c +\left (d \,x^{2}+c \right )^{\frac {a d}{2 a d -2 b c}} a d \,x^{2}+\left (d \,x^{2}+c \right )^{\frac {a d}{2 a d -2 b c}} b c \,x^{2}+\left (d \,x^{2}+c \right )^{\frac {a d}{2 a d -2 b c}} b d \,x^{4}}d x \] Input:
int((b*x^2+a)^(-1-b*c/(-2*a*d+2*b*c))*(d*x^2+c)^(-1+a*d/(-2*a*d+2*b*c)),x)
Output:
int((a + b*x**2)**((b*c)/(2*a*d - 2*b*c))/((c + d*x**2)**((a*d)/(2*a*d - 2 *b*c))*a*c + (c + d*x**2)**((a*d)/(2*a*d - 2*b*c))*a*d*x**2 + (c + d*x**2) **((a*d)/(2*a*d - 2*b*c))*b*c*x**2 + (c + d*x**2)**((a*d)/(2*a*d - 2*b*c)) *b*d*x**4),x)