Integrand size = 30, antiderivative size = 122 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx=\frac {b \arctan \left (\frac {\sqrt {b e-a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )}{\sqrt {a} (b c-a d) \sqrt {b e-a f}}-\frac {d \arctan \left (\frac {\sqrt {d e-c f} x}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d) \sqrt {d e-c f}} \]
b*arctan(x*(-a*f+b*e)^(1/2)/a^(1/2)/(f*x^2+e)^(1/2))/(-a*d+b*c)/a^(1/2)/(- a*f+b*e)^(1/2)-d*arctan(x*(-c*f+d*e)^(1/2)/c^(1/2)/(f*x^2+e)^(1/2))/(-a*d+ b*c)/c^(1/2)/(-c*f+d*e)^(1/2)
Time = 0.33 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx=\frac {-\frac {b \arctan \left (\frac {a \sqrt {f}+b x \left (\sqrt {f} x-\sqrt {e+f x^2}\right )}{\sqrt {a} \sqrt {b e-a f}}\right )}{\sqrt {a} \sqrt {b e-a f}}+\frac {d \arctan \left (\frac {c \sqrt {f}+d x \left (\sqrt {f} x-\sqrt {e+f x^2}\right )}{\sqrt {c} \sqrt {d e-c f}}\right )}{\sqrt {c} \sqrt {d e-c f}}}{b c-a d} \]
(-((b*ArcTan[(a*Sqrt[f] + b*x*(Sqrt[f]*x - Sqrt[e + f*x^2]))/(Sqrt[a]*Sqrt [b*e - a*f])])/(Sqrt[a]*Sqrt[b*e - a*f])) + (d*ArcTan[(c*Sqrt[f] + d*x*(Sq rt[f]*x - Sqrt[e + f*x^2]))/(Sqrt[c]*Sqrt[d*e - c*f])])/(Sqrt[c]*Sqrt[d*e - c*f]))/(b*c - a*d)
Time = 0.25 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {407, 291, 218}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx\) |
| \(\Big \downarrow \) 407 |
| \(\displaystyle \frac {b \int \frac {1}{\left (b x^2+a\right ) \sqrt {f x^2+e}}dx}{b c-a d}-\frac {d \int \frac {1}{\left (d x^2+c\right ) \sqrt {f x^2+e}}dx}{b c-a d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {b \int \frac {1}{a-\frac {(a f-b e) x^2}{f x^2+e}}d\frac {x}{\sqrt {f x^2+e}}}{b c-a d}-\frac {d \int \frac {1}{c-\frac {(c f-d e) x^2}{f x^2+e}}d\frac {x}{\sqrt {f x^2+e}}}{b c-a d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {b \arctan \left (\frac {x \sqrt {b e-a f}}{\sqrt {a} \sqrt {e+f x^2}}\right )}{\sqrt {a} (b c-a d) \sqrt {b e-a f}}-\frac {d \arctan \left (\frac {x \sqrt {d e-c f}}{\sqrt {c} \sqrt {e+f x^2}}\right )}{\sqrt {c} (b c-a d) \sqrt {d e-c f}}\) |
(b*ArcTan[(Sqrt[b*e - a*f]*x)/(Sqrt[a]*Sqrt[e + f*x^2])])/(Sqrt[a]*(b*c - a*d)*Sqrt[b*e - a*f]) - (d*ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f* x^2])])/(Sqrt[c]*(b*c - a*d)*Sqrt[d*e - c*f])
3.1.62.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2 ]), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/((a + b*x^2)*Sqrt[e + f*x^2]), x], x] - Simp[d/(b*c - a*d) Int[1/((c + d*x^2)*Sqrt[e + f*x^2]), x], x] / ; FreeQ[{a, b, c, d, e, f}, x]
Time = 3.43 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(\frac {d \,\operatorname {arctanh}\left (\frac {c \sqrt {f \,x^{2}+e}}{x \sqrt {\left (c f -d e \right ) c}}\right ) \sqrt {\left (a f -b e \right ) a}-b \,\operatorname {arctanh}\left (\frac {\sqrt {f \,x^{2}+e}\, a}{x \sqrt {\left (a f -b e \right ) a}}\right ) \sqrt {\left (c f -d e \right ) c}}{\left (a d -b c \right ) \sqrt {\left (c f -d e \right ) c}\, \sqrt {\left (a f -b e \right ) a}}\) | \(120\) |
| default | \(-\frac {b^{2} d \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}+\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} f +\frac {2 f \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-\frac {a f -b e}{b}}}+\frac {b^{2} d \ln \left (\frac {-\frac {2 \left (a f -b e \right )}{b}-\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a f -b e}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} f -\frac {2 f \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a f -b e}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-\frac {a f -b e}{b}}}-\frac {b \,d^{2} \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}-\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} f -\frac {2 f \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{2 \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}+\frac {b \,d^{2} \ln \left (\frac {-\frac {2 \left (c f -d e \right )}{d}+\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {-\frac {c f -d e}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} f +\frac {2 f \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}-\frac {c f -d e}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{2 \left (b \sqrt {-c d}+d \sqrt {-a b}\right ) \left (d \sqrt {-a b}-b \sqrt {-c d}\right ) \sqrt {-c d}\, \sqrt {-\frac {c f -d e}{d}}}\) | \(782\) |
(d*arctanh(c*(f*x^2+e)^(1/2)/x/((c*f-d*e)*c)^(1/2))*((a*f-b*e)*a)^(1/2)-b* arctanh((f*x^2+e)^(1/2)/x*a/((a*f-b*e)*a)^(1/2))*((c*f-d*e)*c)^(1/2))/(a*d -b*c)/((c*f-d*e)*c)^(1/2)/((a*f-b*e)*a)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (102) = 204\).
Time = 61.43 (sec) , antiderivative size = 1305, normalized size of antiderivative = 10.70 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx=\text {Too large to display} \]
[1/4*((b*c*d*e - b*c^2*f)*sqrt(-a*b*e + a^2*f)*log(((b^2*e^2 - 8*a*b*e*f + 8*a^2*f^2)*x^4 + a^2*e^2 - 2*(3*a*b*e^2 - 4*a^2*e*f)*x^2 + 4*((b*e - 2*a* f)*x^3 - a*e*x)*sqrt(-a*b*e + a^2*f)*sqrt(f*x^2 + e))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + (a*b*d*e - a^2*d*f)*sqrt(-c*d*e + c^2*f)*log(((d^2*e^2 - 8*c*d* e*f + 8*c^2*f^2)*x^4 + c^2*e^2 - 2*(3*c*d*e^2 - 4*c^2*e*f)*x^2 - 4*((d*e - 2*c*f)*x^3 - c*e*x)*sqrt(-c*d*e + c^2*f)*sqrt(f*x^2 + e))/(d^2*x^4 + 2*c* d*x^2 + c^2)))/((a*b^2*c^2*d - a^2*b*c*d^2)*e^2 - (a*b^2*c^3 - a^3*c*d^2)* e*f + (a^2*b*c^3 - a^3*c^2*d)*f^2), 1/4*(2*(b*c*d*e - b*c^2*f)*sqrt(a*b*e - a^2*f)*arctan(1/2*sqrt(a*b*e - a^2*f)*((b*e - 2*a*f)*x^2 - a*e)*sqrt(f*x ^2 + e)/((a*b*e*f - a^2*f^2)*x^3 + (a*b*e^2 - a^2*e*f)*x)) + (a*b*d*e - a^ 2*d*f)*sqrt(-c*d*e + c^2*f)*log(((d^2*e^2 - 8*c*d*e*f + 8*c^2*f^2)*x^4 + c ^2*e^2 - 2*(3*c*d*e^2 - 4*c^2*e*f)*x^2 - 4*((d*e - 2*c*f)*x^3 - c*e*x)*sqr t(-c*d*e + c^2*f)*sqrt(f*x^2 + e))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/((a*b^2*c ^2*d - a^2*b*c*d^2)*e^2 - (a*b^2*c^3 - a^3*c*d^2)*e*f + (a^2*b*c^3 - a^3*c ^2*d)*f^2), -1/4*(2*(a*b*d*e - a^2*d*f)*sqrt(c*d*e - c^2*f)*arctan(1/2*sqr t(c*d*e - c^2*f)*((d*e - 2*c*f)*x^2 - c*e)*sqrt(f*x^2 + e)/((c*d*e*f - c^2 *f^2)*x^3 + (c*d*e^2 - c^2*e*f)*x)) - (b*c*d*e - b*c^2*f)*sqrt(-a*b*e + a^ 2*f)*log(((b^2*e^2 - 8*a*b*e*f + 8*a^2*f^2)*x^4 + a^2*e^2 - 2*(3*a*b*e^2 - 4*a^2*e*f)*x^2 + 4*((b*e - 2*a*f)*x^3 - a*e*x)*sqrt(-a*b*e + a^2*f)*sqrt( f*x^2 + e))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/((a*b^2*c^2*d - a^2*b*c*d^2)*...
\[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right ) \sqrt {e + f x^{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )} \sqrt {f x^{2} + e}} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx=-f^{\frac {3}{2}} {\left (\frac {b \arctan \left (\frac {{\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} b - b e + 2 \, a f}{2 \, \sqrt {a b e f - a^{2} f^{2}}}\right )}{\sqrt {a b e f - a^{2} f^{2}} {\left (b c f - a d f\right )}} - \frac {d \arctan \left (\frac {{\left (\sqrt {f} x - \sqrt {f x^{2} + e}\right )}^{2} d - d e + 2 \, c f}{2 \, \sqrt {c d e f - c^{2} f^{2}}}\right )}{\sqrt {c d e f - c^{2} f^{2}} {\left (b c f - a d f\right )}}\right )} \]
-f^(3/2)*(b*arctan(1/2*((sqrt(f)*x - sqrt(f*x^2 + e))^2*b - b*e + 2*a*f)/s qrt(a*b*e*f - a^2*f^2))/(sqrt(a*b*e*f - a^2*f^2)*(b*c*f - a*d*f)) - d*arct an(1/2*((sqrt(f)*x - sqrt(f*x^2 + e))^2*d - d*e + 2*c*f)/sqrt(c*d*e*f - c^ 2*f^2))/(sqrt(c*d*e*f - c^2*f^2)*(b*c*f - a*d*f)))
Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right ) \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (b\,x^2+a\right )\,\left (d\,x^2+c\right )\,\sqrt {f\,x^2+e}} \,d x \]