Integrand size = 30, antiderivative size = 209 \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {(b e-a f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\sqrt {e} \sqrt {f} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {(b c-a d) \sqrt {e} \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
(-a*f+b*e)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticE(x*f^(1/2)/e^( 1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*(d*x^2+c)^(1/2)/(-c*f+d*e)/e^(1/ 2)/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-(-a*d+b*c)*(1/( 1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticF(x*f^(1/2)/e^(1/2)/(1+f*x^2/e )^(1/2),(1-d*e/c/f)^(1/2))*e^(1/2)*(d*x^2+c)^(1/2)/c/(-c*f+d*e)/f^(1/2)/(e *(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)
Result contains complex when optimal does not.
Time = 6.87 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.01 \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {d}{c}} f (-b e+a f) x \left (c+d x^2\right )-i d e (b e-a f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i b e (-d e+c f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{\sqrt {\frac {d}{c}} e f (-d e+c f) \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
(Sqrt[d/c]*f*(-(b*e) + a*f)*x*(c + d*x^2) - I*d*e*(b*e - a*f)*Sqrt[1 + (d* x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] - I*b*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[ I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(Sqrt[d/c]*e*f*(-(d*e) + c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {400, 313, 320}
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.
| \(\displaystyle \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {(b e-a f) \int \frac {\sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx}{d e-c f}-\frac {(b c-a d) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{d e-c f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\sqrt {e} \sqrt {f} \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {(b c-a d) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{d e-c f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {c+d x^2} (b e-a f) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\sqrt {e} \sqrt {f} \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {e} \sqrt {c+d x^2} (b c-a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\) |
((b*e - a*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d *e)/(c*f)])/(Sqrt[e]*Sqrt[f]*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^ 2))]*Sqrt[e + f*x^2]) - ((b*c - a*d)*Sqrt[e]*Sqrt[c + d*x^2]*EllipticF[Arc Tan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*Sqrt[f]*(d*e - c*f)*Sqrt[(e *(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2])
3.1.45.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Time = 3.91 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.67
| method | result | size |
| default | \(\frac {\left (\sqrt {-\frac {d}{c}}\, a d \,f^{2} x^{3}-\sqrt {-\frac {d}{c}}\, b d e f \,x^{3}+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c e f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b d \,e^{2}-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a d e f +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b d \,e^{2}+\sqrt {-\frac {d}{c}}\, a c \,f^{2} x -\sqrt {-\frac {d}{c}}\, b c e f x \right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{\sqrt {-\frac {d}{c}}\, e f \left (c f -d e \right ) \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\) | \(349\) |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {\left (d f \,x^{2}+c f \right ) x \left (a f -b e \right )}{f e \left (c f -d e \right ) \sqrt {\left (x^{2}+\frac {e}{f}\right ) \left (d f \,x^{2}+c f \right )}}+\frac {\left (\frac {b}{f}+\frac {a f -b e}{f e}-\frac {c \left (a f -b e \right )}{e \left (c f -d e \right )}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {d \left (a f -b e \right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\left (c f -d e \right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(367\) |
((-d/c)^(1/2)*a*d*f^2*x^3-(-d/c)^(1/2)*b*d*e*f*x^3+((d*x^2+c)/c)^(1/2)*((f *x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*c*e*f-((d*x^2 +c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2)) *b*d*e^2-((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(-d/c)^(1/2), (c*f/d/e)^(1/2))*a*d*e*f+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE (x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*b*d*e^2+(-d/c)^(1/2)*a*c*f^2*x-(-d/c)^(1/ 2)*b*c*e*f*x)*(f*x^2+e)^(1/2)*(d*x^2+c)^(1/2)/(-d/c)^(1/2)/e/f/(c*f-d*e)/( d*f*x^4+c*f*x^2+d*e*x^2+c*e)
Time = 0.10 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.23 \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {{\left (b c d e f - a c d f^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} x - {\left (b d^{2} e^{2} - a d^{2} e f + {\left (b d^{2} e f - a d^{2} f^{2}\right )} x^{2}\right )} \sqrt {c e} \sqrt {-\frac {d}{c}} E(\arcsin \left (x \sqrt {-\frac {d}{c}}\right )\,|\,\frac {c f}{d e}) + {\left (b d^{2} e^{2} + {\left (b c^{2} - a c d - a d^{2}\right )} e f + {\left (b d^{2} e f + {\left (b c^{2} - a c d - a d^{2}\right )} f^{2}\right )} x^{2}\right )} \sqrt {c e} \sqrt {-\frac {d}{c}} F(\arcsin \left (x \sqrt {-\frac {d}{c}}\right )\,|\,\frac {c f}{d e})}{c d^{2} e^{3} f - c^{2} d e^{2} f^{2} + {\left (c d^{2} e^{2} f^{2} - c^{2} d e f^{3}\right )} x^{2}} \]
((b*c*d*e*f - a*c*d*f^2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)*x - (b*d^2*e^2 - a*d^2*e*f + (b*d^2*e*f - a*d^2*f^2)*x^2)*sqrt(c*e)*sqrt(-d/c)*elliptic_e(a rcsin(x*sqrt(-d/c)), c*f/(d*e)) + (b*d^2*e^2 + (b*c^2 - a*c*d - a*d^2)*e*f + (b*d^2*e*f + (b*c^2 - a*c*d - a*d^2)*f^2)*x^2)*sqrt(c*e)*sqrt(-d/c)*ell iptic_f(arcsin(x*sqrt(-d/c)), c*f/(d*e)))/(c*d^2*e^3*f - c^2*d*e^2*f^2 + ( c*d^2*e^2*f^2 - c^2*d*e*f^3)*x^2)
\[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {a + b x^{2}}{\sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {b x^{2} + a}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {b x^{2} + a}{\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b x^2}{\sqrt {c+d x^2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {b\,x^2+a}{\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \]