Integrand size = 26, antiderivative size = 86 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=-\frac {(b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} (d e-c f)}+\frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} (d e-c f)} \] Output:
-(-a*d+b*c)*arctan(d^(1/2)*x/c^(1/2))/c^(1/2)/d^(1/2)/(-c*f+d*e)+(-a*f+b*e )*arctan(f^(1/2)*x/e^(1/2))/e^(1/2)/f^(1/2)/(-c*f+d*e)
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {(b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} (-d e+c f)}+\frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} (d e-c f)} \] Input:
Integrate[(a + b*x^2)/((c + d*x^2)*(e + f*x^2)),x]
Output:
((b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]*(-(d*e) + c*f)) + ((b*e - a*f)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]*(d*e - c*f))
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {397, 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.
\(\displaystyle \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx\) |
\(\Big \downarrow \) 397 |
\(\displaystyle \frac {(b e-a f) \int \frac {1}{f x^2+e}dx}{d e-c f}-\frac {(b c-a d) \int \frac {1}{d x^2+c}dx}{d e-c f}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(b e-a f) \arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {e} \sqrt {f} (d e-c f)}-\frac {(b c-a d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} \sqrt {d} (d e-c f)}\) |
Input:
Int[(a + b*x^2)/((c + d*x^2)*(e + f*x^2)),x]
Output:
-(((b*c - a*d)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*Sqrt[d]*(d*e - c*f))) + ((b*e - a*f)*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(Sqrt[e]*Sqrt[f]*(d*e - c*f))
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[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Time = 0.63 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {\left (a f -b e \right ) \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{\left (c f -d e \right ) \sqrt {e f}}+\frac {\left (-a d +b c \right ) \arctan \left (\frac {x d}{\sqrt {c d}}\right )}{\left (c f -d e \right ) \sqrt {c d}}\) | \(68\) |
risch | \(-\frac {\ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a f}{2 \left (c f -d e \right ) \sqrt {-e f}}+\frac {\ln \left (e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) b e}{2 \left (c f -d e \right ) \sqrt {-e f}}+\frac {\ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) a f}{2 \left (c f -d e \right ) \sqrt {-e f}}-\frac {\ln \left (-e \,f^{2} x -\left (-e f \right )^{\frac {3}{2}}\right ) b e}{2 \left (c f -d e \right ) \sqrt {-e f}}-\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a d}{2 \sqrt {-c d}\, \left (c f -d e \right )}+\frac {\ln \left (-c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b c}{2 \sqrt {-c d}\, \left (c f -d e \right )}+\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) a d}{2 \sqrt {-c d}\, \left (c f -d e \right )}-\frac {\ln \left (c \,d^{2} x -\left (-c d \right )^{\frac {3}{2}}\right ) b c}{2 \sqrt {-c d}\, \left (c f -d e \right )}\) | \(294\) |
Input:
int((b*x^2+a)/(d*x^2+c)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
(a*f-b*e)/(c*f-d*e)/(e*f)^(1/2)*arctan(f*x/(e*f)^(1/2))+(-a*d+b*c)/(c*f-d* e)/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))
Time = 0.14 (sec) , antiderivative size = 404, normalized size of antiderivative = 4.70 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\left [-\frac {{\left (b c - a d\right )} \sqrt {-c d} e f \log \left (\frac {d x^{2} + 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + {\left (b c d e - a c d f\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right )}{2 \, {\left (c d^{2} e^{2} f - c^{2} d e f^{2}\right )}}, -\frac {{\left (b c - a d\right )} \sqrt {-c d} e f \log \left (\frac {d x^{2} + 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - 2 \, {\left (b c d e - a c d f\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right )}{2 \, {\left (c d^{2} e^{2} f - c^{2} d e f^{2}\right )}}, -\frac {2 \, {\left (b c - a d\right )} \sqrt {c d} e f \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + {\left (b c d e - a c d f\right )} \sqrt {-e f} \log \left (\frac {f x^{2} - 2 \, \sqrt {-e f} x - e}{f x^{2} + e}\right )}{2 \, {\left (c d^{2} e^{2} f - c^{2} d e f^{2}\right )}}, -\frac {{\left (b c - a d\right )} \sqrt {c d} e f \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (b c d e - a c d f\right )} \sqrt {e f} \arctan \left (\frac {\sqrt {e f} x}{e}\right )}{c d^{2} e^{2} f - c^{2} d e f^{2}}\right ] \] Input:
integrate((b*x^2+a)/(d*x^2+c)/(f*x^2+e),x, algorithm="fricas")
Output:
[-1/2*((b*c - a*d)*sqrt(-c*d)*e*f*log((d*x^2 + 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) + (b*c*d*e - a*c*d*f)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x - e)/(f *x^2 + e)))/(c*d^2*e^2*f - c^2*d*e*f^2), -1/2*((b*c - a*d)*sqrt(-c*d)*e*f* log((d*x^2 + 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) - 2*(b*c*d*e - a*c*d*f)*sqrt (e*f)*arctan(sqrt(e*f)*x/e))/(c*d^2*e^2*f - c^2*d*e*f^2), -1/2*(2*(b*c - a *d)*sqrt(c*d)*e*f*arctan(sqrt(c*d)*x/c) + (b*c*d*e - a*c*d*f)*sqrt(-e*f)*l og((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)))/(c*d^2*e^2*f - c^2*d*e*f^2), -((b*c - a*d)*sqrt(c*d)*e*f*arctan(sqrt(c*d)*x/c) - (b*c*d*e - a*c*d*f)*s qrt(e*f)*arctan(sqrt(e*f)*x/e))/(c*d^2*e^2*f - c^2*d*e*f^2)]
Timed out. \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x**2+a)/(d*x**2+c)/(f*x**2+e),x)
Output:
Timed out
Exception generated. \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x^2+a)/(d*x^2+c)/(f*x^2+e),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=-\frac {{\left (b c - a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} {\left (d e - c f\right )}} + \frac {{\left (b e - a f\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{{\left (d e - c f\right )} \sqrt {e f}} \] Input:
integrate((b*x^2+a)/(d*x^2+c)/(f*x^2+e),x, algorithm="giac")
Output:
-(b*c - a*d)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*(d*e - c*f)) + (b*e - a*f)*a rctan(f*x/sqrt(e*f))/((d*e - c*f)*sqrt(e*f))
Time = 2.35 (sec) , antiderivative size = 1245, normalized size of antiderivative = 14.48 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\text {Too large to display} \] Input:
int((a + b*x^2)/((c + d*x^2)*(e + f*x^2)),x)
Output:
(a*d*e*f*atan((b^2*c^3*f^2*x*(-c*d)^(3/2)*1i + a^2*c*d^2*f^2*x*(-c*d)^(3/2 )*1i + b^2*c^4*d*f^2*x*(-c*d)^(1/2)*1i + a^2*c^2*d^3*f^2*x*(-c*d)^(1/2)*2i + b^2*c^2*d^3*e^2*x*(-c*d)^(1/2)*1i + a^2*d^3*e*f*x*(-c*d)^(3/2)*1i - a*b *c^2*d*f^2*x*(-c*d)^(3/2)*2i + b^2*c^2*d*e*f*x*(-c*d)^(3/2)*1i - a*b*c^3*d ^2*f^2*x*(-c*d)^(1/2)*2i - a*b*c*d^2*e*f*x*(-c*d)^(3/2)*2i - a*b*c^2*d^3*e *f*x*(-c*d)^(1/2)*2i)/(a^2*c^3*d^3*f^2 + b^2*c^3*d^3*e^2 - a^2*c^2*d^4*e*f - b^2*c^4*d^2*e*f))*(-c*d)^(1/2)*1i)/(c*d^2*e^2*f - c^2*d*e*f^2) - (b*c*e *f*atan((b^2*c^3*f^2*x*(-c*d)^(3/2)*1i + a^2*c*d^2*f^2*x*(-c*d)^(3/2)*1i + b^2*c^4*d*f^2*x*(-c*d)^(1/2)*1i + a^2*c^2*d^3*f^2*x*(-c*d)^(1/2)*2i + b^2 *c^2*d^3*e^2*x*(-c*d)^(1/2)*1i + a^2*d^3*e*f*x*(-c*d)^(3/2)*1i - a*b*c^2*d *f^2*x*(-c*d)^(3/2)*2i + b^2*c^2*d*e*f*x*(-c*d)^(3/2)*1i - a*b*c^3*d^2*f^2 *x*(-c*d)^(1/2)*2i - a*b*c*d^2*e*f*x*(-c*d)^(3/2)*2i - a*b*c^2*d^3*e*f*x*( -c*d)^(1/2)*2i)/(a^2*c^3*d^3*f^2 + b^2*c^3*d^3*e^2 - a^2*c^2*d^4*e*f - b^2 *c^4*d^2*e*f))*(-c*d)^(1/2)*1i)/(c*d^2*e^2*f - c^2*d*e*f^2) - (a*c*d*f*ata n((b^2*d^2*e^3*x*(-e*f)^(3/2)*1i + a^2*d^2*e*f^2*x*(-e*f)^(3/2)*1i + b^2*d ^2*e^4*f*x*(-e*f)^(1/2)*1i + a^2*d^2*e^2*f^3*x*(-e*f)^(1/2)*2i + b^2*c^2*e ^2*f^3*x*(-e*f)^(1/2)*1i + a^2*c*d*f^3*x*(-e*f)^(3/2)*1i - a*b*d^2*e^2*f*x *(-e*f)^(3/2)*2i + b^2*c*d*e^2*f*x*(-e*f)^(3/2)*1i - a*b*d^2*e^3*f^2*x*(-e *f)^(1/2)*2i - a*b*c*d*e*f^2*x*(-e*f)^(3/2)*2i - a*b*c*d*e^2*f^3*x*(-e*f)^ (1/2)*2i)/(a^2*d^2*e^3*f^3 + b^2*c^2*e^3*f^3 - a^2*c*d*e^2*f^4 - b^2*c*...
Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28 \[ \int \frac {a+b x^2}{\left (c+d x^2\right ) \left (e+f x^2\right )} \, dx=\frac {-\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) a d e f +\sqrt {d}\, \sqrt {c}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {c}}\right ) b c e f +\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) a c d f -\sqrt {f}\, \sqrt {e}\, \mathit {atan} \left (\frac {f x}{\sqrt {f}\, \sqrt {e}}\right ) b c d e}{c d e f \left (c f -d e \right )} \] Input:
int((b*x^2+a)/(d*x^2+c)/(f*x^2+e),x)
Output:
( - sqrt(d)*sqrt(c)*atan((d*x)/(sqrt(d)*sqrt(c)))*a*d*e*f + sqrt(d)*sqrt(c )*atan((d*x)/(sqrt(d)*sqrt(c)))*b*c*e*f + sqrt(f)*sqrt(e)*atan((f*x)/(sqrt (f)*sqrt(e)))*a*c*d*f - sqrt(f)*sqrt(e)*atan((f*x)/(sqrt(f)*sqrt(e)))*b*c* d*e)/(c*d*e*f*(c*f - d*e))