Integrand size = 23, antiderivative size = 87 \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\frac {x \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{3/4} \sqrt [4]{c+d x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (c+d x^2\right )}\right )}{c \left (a+b x^2\right )^{3/4}} \] Output:
x*(c*(b*x^2+a)/a/(d*x^2+c))^(3/4)*(d*x^2+c)^(1/4)*hypergeom([1/2, 3/4],[3/ 2],-(-a*d+b*c)*x^2/a/(d*x^2+c))/c/(b*x^2+a)^(3/4)
Time = 4.46 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\frac {x \left (1+\frac {b x^2}{a}\right )^{3/4} \sqrt [4]{1+\frac {d x^2}{c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {(-b c+a d) x^2}{a \left (c+d x^2\right )}\right )}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \] Input:
Integrate[1/((a + b*x^2)^(3/4)*(c + d*x^2)^(3/4)),x]
Output:
(x*(1 + (b*x^2)/a)^(3/4)*(1 + (d*x^2)/c)^(1/4)*Hypergeometric2F1[1/2, 3/4, 3/2, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/((a + b*x^2)^(3/4)*(c + d*x^2 )^(3/4))
Time = 0.18 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {294}
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.
\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 294 |
\(\displaystyle \frac {x \sqrt [4]{c+d x^2} \left (\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {(b c-a d) x^2}{a \left (d x^2+c\right )}\right )}{c \left (a+b x^2\right )^{3/4}}\) |
Input:
Int[1/((a + b*x^2)^(3/4)*(c + d*x^2)^(3/4)),x]
Output:
(x*((c*(a + b*x^2))/(a*(c + d*x^2)))^(3/4)*(c + d*x^2)^(1/4)*Hypergeometri c2F1[1/2, 3/4, 3/2, -(((b*c - a*d)*x^2)/(a*(c + d*x^2)))])/(c*(a + b*x^2)^ (3/4))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[x*((a + b*x^2)^p/(c*(c*((a + b*x^2)/(a*(c + d*x^2))))^p*(c + d*x^2)^(1/2 + p)))*Hypergeometric2F1[1/2, -p, 3/2, (-(b*c - a*d))*(x^2/(a*(c + d*x^2))) ], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0]
\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (x^{2} d +c \right )^{\frac {3}{4}}}d x\]
Input:
int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x)
Output:
int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x, algorithm="fricas")
Output:
integral((b*x^2 + a)^(1/4)*(d*x^2 + c)^(1/4)/(b*d*x^4 + (b*c + a*d)*x^2 + a*c), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {3}{4}} \left (c + d x^{2}\right )^{\frac {3}{4}}}\, dx \] Input:
integrate(1/(b*x**2+a)**(3/4)/(d*x**2+c)**(3/4),x)
Output:
Integral(1/((a + b*x**2)**(3/4)*(c + d*x**2)**(3/4)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^(3/4)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} {\left (d x^{2} + c\right )}^{\frac {3}{4}}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(3/4)*(d*x^2 + c)^(3/4)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{3/4}\,{\left (d\,x^2+c\right )}^{3/4}} \,d x \] Input:
int(1/((a + b*x^2)^(3/4)*(c + d*x^2)^(3/4)),x)
Output:
int(1/((a + b*x^2)^(3/4)*(c + d*x^2)^(3/4)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{3/4} \left (c+d x^2\right )^{3/4}} \, dx=\int \frac {1}{\left (d \,x^{2}+c \right )^{\frac {3}{4}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}d x \] Input:
int(1/(b*x^2+a)^(3/4)/(d*x^2+c)^(3/4),x)
Output:
int(1/((c + d*x**2)**(3/4)*(a + b*x**2)**(3/4)),x)