Integrand size = 28, antiderivative size = 113 \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {(b e-a f) x \sqrt {c+d x^2}}{2 e (d e-c f) \left (e+f x^2\right )}-\frac {(b c e-2 a d e+a c f) \text {arctanh}\left (\frac {\sqrt {d e-c f} x}{\sqrt {e} \sqrt {c+d x^2}}\right )}{2 e^{3/2} (d e-c f)^{3/2}} \]
-1/2*(a*c*f-2*a*d*e+b*c*e)*arctanh(x*(-c*f+d*e)^(1/2)/e^(1/2)/(d*x^2+c)^(1 /2))/e^(3/2)/(-c*f+d*e)^(3/2)+1/2*(-a*f+b*e)*x*(d*x^2+c)^(1/2)/e/(-c*f+d*e )/(f*x^2+e)
Time = 0.48 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.16 \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {\frac {\sqrt {e} (b e-a f) x \sqrt {c+d x^2}}{(d e-c f) \left (e+f x^2\right )}-\frac {(b c e-2 a d e+a c f) \arctan \left (\frac {-f x \sqrt {c+d x^2}+\sqrt {d} \left (e+f x^2\right )}{\sqrt {e} \sqrt {-d e+c f}}\right )}{(-d e+c f)^{3/2}}}{2 e^{3/2}} \]
((Sqrt[e]*(b*e - a*f)*x*Sqrt[c + d*x^2])/((d*e - c*f)*(e + f*x^2)) - ((b*c *e - 2*a*d*e + a*c*f)*ArcTan[(-(f*x*Sqrt[c + d*x^2]) + Sqrt[d]*(e + f*x^2) )/(Sqrt[e]*Sqrt[-(d*e) + c*f])])/(-(d*e) + c*f)^(3/2))/(2*e^(3/2))
Time = 0.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {402, 25, 27, 291, 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.
| \(\displaystyle \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\int -\frac {b c e-2 a d e+a c f}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (d e-c f)}+\frac {x \sqrt {c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac {\int \frac {b c e-2 a d e+a c f}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac {(a c f-2 a d e+b c e) \int \frac {1}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{2 e (d e-c f)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {x \sqrt {c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac {(a c f-2 a d e+b c e) \int \frac {1}{e-\frac {(d e-c f) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{2 e (d e-c f)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x \sqrt {c+d x^2} (b e-a f)}{2 e \left (e+f x^2\right ) (d e-c f)}-\frac {(a c f-2 a d e+b c e) \text {arctanh}\left (\frac {x \sqrt {d e-c f}}{\sqrt {e} \sqrt {c+d x^2}}\right )}{2 e^{3/2} (d e-c f)^{3/2}}\) |
((b*e - a*f)*x*Sqrt[c + d*x^2])/(2*e*(d*e - c*f)*(e + f*x^2)) - ((b*c*e - 2*a*d*e + a*c*f)*ArcTanh[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])])/( 2*e^(3/2)*(d*e - c*f)^(3/2))
3.1.98.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Time = 3.48 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {\frac {\left (a f -b e \right ) \sqrt {d \,x^{2}+c}\, x}{f \,x^{2}+e}-\frac {\left (a c f -2 a d e +b c e \right ) \arctan \left (\frac {e \sqrt {d \,x^{2}+c}}{x \sqrt {\left (c f -d e \right ) e}}\right )}{\sqrt {\left (c f -d e \right ) e}}}{2 \left (c f -d e \right ) e}\) | \(100\) |
| default | \(\frac {\left (-a f +b e \right ) \left (-\frac {f \sqrt {d \left (x +\frac {\sqrt {-e f}}{f}\right )^{2}-\frac {2 d \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {c f -d e}{f}}}{\left (c f -d e \right ) \left (x +\frac {\sqrt {-e f}}{f}\right )}-\frac {d \sqrt {-e f}\, \ln \left (\frac {\frac {2 c f -2 d e}{f}-\frac {2 d \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {c f -d e}{f}}\, \sqrt {d \left (x +\frac {\sqrt {-e f}}{f}\right )^{2}-\frac {2 d \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {c f -d e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{\left (c f -d e \right ) \sqrt {\frac {c f -d e}{f}}}\right )}{4 e \,f^{2}}+\frac {\left (-a f +b e \right ) \left (-\frac {f \sqrt {d \left (x -\frac {\sqrt {-e f}}{f}\right )^{2}+\frac {2 d \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {c f -d e}{f}}}{\left (c f -d e \right ) \left (x -\frac {\sqrt {-e f}}{f}\right )}+\frac {d \sqrt {-e f}\, \ln \left (\frac {\frac {2 c f -2 d e}{f}+\frac {2 d \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {c f -d e}{f}}\, \sqrt {d \left (x -\frac {\sqrt {-e f}}{f}\right )^{2}+\frac {2 d \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {c f -d e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{\left (c f -d e \right ) \sqrt {\frac {c f -d e}{f}}}\right )}{4 e \,f^{2}}-\frac {\left (a f +b e \right ) \ln \left (\frac {\frac {2 c f -2 d e}{f}+\frac {2 d \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {c f -d e}{f}}\, \sqrt {d \left (x -\frac {\sqrt {-e f}}{f}\right )^{2}+\frac {2 d \sqrt {-e f}\, \left (x -\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {c f -d e}{f}}}{x -\frac {\sqrt {-e f}}{f}}\right )}{4 e \sqrt {-e f}\, f \sqrt {\frac {c f -d e}{f}}}-\frac {\left (-a f -b e \right ) \ln \left (\frac {\frac {2 c f -2 d e}{f}-\frac {2 d \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+2 \sqrt {\frac {c f -d e}{f}}\, \sqrt {d \left (x +\frac {\sqrt {-e f}}{f}\right )^{2}-\frac {2 d \sqrt {-e f}\, \left (x +\frac {\sqrt {-e f}}{f}\right )}{f}+\frac {c f -d e}{f}}}{x +\frac {\sqrt {-e f}}{f}}\right )}{4 e \sqrt {-e f}\, f \sqrt {\frac {c f -d e}{f}}}\) | \(848\) |
1/2/(c*f-d*e)/e*((a*f-b*e)*(d*x^2+c)^(1/2)*x/(f*x^2+e)-(a*c*f-2*a*d*e+b*c* e)/((c*f-d*e)*e)^(1/2)*arctan(e*(d*x^2+c)^(1/2)/x/((c*f-d*e)*e)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (97) = 194\).
Time = 1.11 (sec) , antiderivative size = 513, normalized size of antiderivative = 4.54 \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\left [\frac {4 \, {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} \sqrt {d x^{2} + c} x - {\left (a c e f + {\left (b c - 2 \, a d\right )} e^{2} + {\left (a c f^{2} + {\left (b c - 2 \, a d\right )} e f\right )} x^{2}\right )} \sqrt {d e^{2} - c e f} \log \left (\frac {{\left (8 \, d^{2} e^{2} - 8 \, c d e f + c^{2} f^{2}\right )} x^{4} + c^{2} e^{2} + 2 \, {\left (4 \, c d e^{2} - 3 \, c^{2} e f\right )} x^{2} + 4 \, {\left ({\left (2 \, d e - c f\right )} x^{3} + c e x\right )} \sqrt {d e^{2} - c e f} \sqrt {d x^{2} + c}}{f^{2} x^{4} + 2 \, e f x^{2} + e^{2}}\right )}{8 \, {\left (d^{2} e^{5} - 2 \, c d e^{4} f + c^{2} e^{3} f^{2} + {\left (d^{2} e^{4} f - 2 \, c d e^{3} f^{2} + c^{2} e^{2} f^{3}\right )} x^{2}\right )}}, \frac {2 \, {\left (b d e^{3} + a c e f^{2} - {\left (b c + a d\right )} e^{2} f\right )} \sqrt {d x^{2} + c} x + {\left (a c e f + {\left (b c - 2 \, a d\right )} e^{2} + {\left (a c f^{2} + {\left (b c - 2 \, a d\right )} e f\right )} x^{2}\right )} \sqrt {-d e^{2} + c e f} \arctan \left (\frac {\sqrt {-d e^{2} + c e f} {\left ({\left (2 \, d e - c f\right )} x^{2} + c e\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (d^{2} e^{2} - c d e f\right )} x^{3} + {\left (c d e^{2} - c^{2} e f\right )} x\right )}}\right )}{4 \, {\left (d^{2} e^{5} - 2 \, c d e^{4} f + c^{2} e^{3} f^{2} + {\left (d^{2} e^{4} f - 2 \, c d e^{3} f^{2} + c^{2} e^{2} f^{3}\right )} x^{2}\right )}}\right ] \]
[1/8*(4*(b*d*e^3 + a*c*e*f^2 - (b*c + a*d)*e^2*f)*sqrt(d*x^2 + c)*x - (a*c *e*f + (b*c - 2*a*d)*e^2 + (a*c*f^2 + (b*c - 2*a*d)*e*f)*x^2)*sqrt(d*e^2 - c*e*f)*log(((8*d^2*e^2 - 8*c*d*e*f + c^2*f^2)*x^4 + c^2*e^2 + 2*(4*c*d*e^ 2 - 3*c^2*e*f)*x^2 + 4*((2*d*e - c*f)*x^3 + c*e*x)*sqrt(d*e^2 - c*e*f)*sqr t(d*x^2 + c))/(f^2*x^4 + 2*e*f*x^2 + e^2)))/(d^2*e^5 - 2*c*d*e^4*f + c^2*e ^3*f^2 + (d^2*e^4*f - 2*c*d*e^3*f^2 + c^2*e^2*f^3)*x^2), 1/4*(2*(b*d*e^3 + a*c*e*f^2 - (b*c + a*d)*e^2*f)*sqrt(d*x^2 + c)*x + (a*c*e*f + (b*c - 2*a* d)*e^2 + (a*c*f^2 + (b*c - 2*a*d)*e*f)*x^2)*sqrt(-d*e^2 + c*e*f)*arctan(1/ 2*sqrt(-d*e^2 + c*e*f)*((2*d*e - c*f)*x^2 + c*e)*sqrt(d*x^2 + c)/((d^2*e^2 - c*d*e*f)*x^3 + (c*d*e^2 - c^2*e*f)*x)))/(d^2*e^5 - 2*c*d*e^4*f + c^2*e^ 3*f^2 + (d^2*e^4*f - 2*c*d*e^3*f^2 + c^2*e^2*f^3)*x^2)]
\[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {a + b x^{2}}{\sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}\, dx \]
\[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int { \frac {b x^{2} + a}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (97) = 194\).
Time = 0.94 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.91 \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\frac {{\left (b c \sqrt {d} e - 2 \, a d^{\frac {3}{2}} e + a c \sqrt {d} f\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} f + 2 \, d e - c f}{2 \, \sqrt {-d^{2} e^{2} + c d e f}}\right )}{2 \, \sqrt {-d^{2} e^{2} + c d e f} {\left (d e^{2} - c e f\right )}} + \frac {2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b d^{\frac {3}{2}} e^{2} - {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c \sqrt {d} e f - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d^{\frac {3}{2}} e f + {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a c \sqrt {d} f^{2} + b c^{2} \sqrt {d} e f - a c^{2} \sqrt {d} f^{2}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} f + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} d e - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} c f + c^{2} f\right )} {\left (d e^{2} f - c e f^{2}\right )}} \]
1/2*(b*c*sqrt(d)*e - 2*a*d^(3/2)*e + a*c*sqrt(d)*f)*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*f + 2*d*e - c*f)/sqrt(-d^2*e^2 + c*d*e*f))/(sqrt(-d^ 2*e^2 + c*d*e*f)*(d*e^2 - c*e*f)) + (2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*d ^(3/2)*e^2 - (sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c*sqrt(d)*e*f - 2*(sqrt(d)* x - sqrt(d*x^2 + c))^2*a*d^(3/2)*e*f + (sqrt(d)*x - sqrt(d*x^2 + c))^2*a*c *sqrt(d)*f^2 + b*c^2*sqrt(d)*e*f - a*c^2*sqrt(d)*f^2)/(((sqrt(d)*x - sqrt( d*x^2 + c))^4*f + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*d*e - 2*(sqrt(d)*x - s qrt(d*x^2 + c))^2*c*f + c^2*f)*(d*e^2*f - c*e*f^2))
Timed out. \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^2} \, dx=\int \frac {b\,x^2+a}{\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2} \,d x \]