Integrand size = 31, antiderivative size = 237 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}-\frac {(d e+c f) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {e \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
(c*f+d*e)*x*(b*x^2+a)^(1/2)/c/(a*d+b*c)/(-d*x^2+c)^(1/2)-(c*f+d*e)*Ellipti cE(x*d^(1/2)/c^(1/2),(-b*c/a/d)^(1/2))*(b*x^2+a)^(1/2)*(1-d*x^2/c)^(1/2)/( a*d+b*c)/c^(1/2)/d^(1/2)/(1+b*x^2/a)^(1/2)/(-d*x^2+c)^(1/2)+e*EllipticF(x* d^(1/2)/c^(1/2),(-b*c/a/d)^(1/2))*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)/c^(1 /2)/d^(1/2)/(b*x^2+a)^(1/2)/(-d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 6.75 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.90 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d (d e+c f) x \left (a+b x^2\right )-i b c (d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c (b c+a d) f \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c d (b c+a d) \sqrt {a+b x^2} \sqrt {c-d x^2}} \]
(Sqrt[b/a]*d*(d*e + c*f)*x*(a + b*x^2) - I*b*c*(d*e + c*f)*Sqrt[1 + (b*x^2 )/a]*Sqrt[1 - (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))] + I*c*(b*c + a*d)*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*EllipticF[I*A rcSinh[Sqrt[b/a]*x], -((a*d)/(b*c))])/(Sqrt[b/a]*c*d*(b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c - d*x^2])
Time = 0.41 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {402, 399, 323, 323, 321, 331, 330, 327}
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.
\(\displaystyle \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\int \frac {c (b e-a f)-b (d e+c f) x^2}{\sqrt {b x^2+a} \sqrt {c-d x^2}}dx}{c (a d+b c)}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)}\) |
\(\Big \downarrow \) 399 |
\(\displaystyle \frac {e (a d+b c) \int \frac {1}{\sqrt {b x^2+a} \sqrt {c-d x^2}}dx-(c f+d e) \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2}}dx}{c (a d+b c)}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)}\) |
\(\Big \downarrow \) 323 |
\(\displaystyle \frac {\frac {e \sqrt {1-\frac {d x^2}{c}} (a d+b c) \int \frac {1}{\sqrt {b x^2+a} \sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {c-d x^2}}-(c f+d e) \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2}}dx}{c (a d+b c)}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)}\) |
\(\Big \downarrow \) 323 |
\(\displaystyle \frac {\frac {e \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {a+b x^2} \sqrt {c-d x^2}}-(c f+d e) \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2}}dx}{c (a d+b c)}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {\frac {\sqrt {c} e \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}-(c f+d e) \int \frac {\sqrt {b x^2+a}}{\sqrt {c-d x^2}}dx}{c (a d+b c)}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)}\) |
\(\Big \downarrow \) 331 |
\(\displaystyle \frac {\frac {\sqrt {c} e \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}-\frac {\sqrt {1-\frac {d x^2}{c}} (c f+d e) \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {c-d x^2}}}{c (a d+b c)}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)}\) |
\(\Big \downarrow \) 330 |
\(\displaystyle \frac {\frac {\sqrt {c} e \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} (c f+d e) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{\sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}}{c (a d+b c)}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\frac {\sqrt {c} e \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} (c f+d e) E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {\frac {b x^2}{a}+1} \sqrt {c-d x^2}}}{c (a d+b c)}+\frac {x \sqrt {a+b x^2} (c f+d e)}{c \sqrt {c-d x^2} (a d+b c)}\) |
((d*e + c*f)*x*Sqrt[a + b*x^2])/(c*(b*c + a*d)*Sqrt[c - d*x^2]) + (-((Sqrt [c]*(d*e + c*f)*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt [d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d* x^2])) + (Sqrt[c]*(b*c + a*d)*e*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*El lipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[a + b* x^2]*Sqrt[c - d*x^2]))/(c*(b*c + a*d))
3.1.50.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
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 = 4.51 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.41
method | result | size |
default | \(\frac {\left (\sqrt {\frac {d}{c}}\, b c f \,x^{3}+\sqrt {\frac {d}{c}}\, b d e \,x^{3}+\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b c e -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a c f -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {d}{c}}\, a c f x +\sqrt {\frac {d}{c}}\, a d e x \right ) \sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}{\sqrt {\frac {d}{c}}\, c \left (a d +b c \right ) \left (-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c \right )}\) | \(333\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (-\frac {\left (-b d \,x^{2}-a d \right ) x \left (c f +d e \right )}{d c \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {c}{d}\right ) \left (-b d \,x^{2}-a d \right )}}+\frac {\left (-\frac {f}{d}+\frac {c f +d e}{d c}-\frac {a \left (c f +d e \right )}{c \left (a d +b c \right )}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c}}+\frac {\left (c f +d e \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-E\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{c \left (a d +b c \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}\) | \(376\) |
((d/c)^(1/2)*b*c*f*x^3+(d/c)^(1/2)*b*d*e*x^3+((-d*x^2+c)/c)^(1/2)*((b*x^2+ a)/a)^(1/2)*EllipticF(x*(d/c)^(1/2),(-b*c/a/d)^(1/2))*a*d*e+((-d*x^2+c)/c) ^(1/2)*((b*x^2+a)/a)^(1/2)*EllipticF(x*(d/c)^(1/2),(-b*c/a/d)^(1/2))*b*c*e -((-d*x^2+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)*EllipticE(x*(d/c)^(1/2),(-b*c/a/ d)^(1/2))*a*c*f-((-d*x^2+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)*EllipticE(x*(d/c) ^(1/2),(-b*c/a/d)^(1/2))*a*d*e+(d/c)^(1/2)*a*c*f*x+(d/c)^(1/2)*a*d*e*x)*(b *x^2+a)^(1/2)*(-d*x^2+c)^(1/2)/(d/c)^(1/2)/c/(a*d+b*c)/(-b*d*x^4-a*d*x^2+b *c*x^2+a*c)
Time = 0.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.07 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {{\left (a c d^{2} e + a c^{2} d f\right )} \sqrt {b x^{2} + a} \sqrt {-d x^{2} + c} x - {\left (a c d^{2} e + a c^{2} d f - {\left (a d^{3} e + a c d^{2} f\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {d}{c}} E(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d}) - {\left ({\left ({\left (b c^{2} d + a d^{3}\right )} e - {\left (a c^{2} d - a c d^{2}\right )} f\right )} x^{2} - {\left (b c^{3} + a c d^{2}\right )} e + {\left (a c^{3} - a c^{2} d\right )} f\right )} \sqrt {a c} \sqrt {\frac {d}{c}} F(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d})}{a b c^{4} d + a^{2} c^{3} d^{2} - {\left (a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}} \]
((a*c*d^2*e + a*c^2*d*f)*sqrt(b*x^2 + a)*sqrt(-d*x^2 + c)*x - (a*c*d^2*e + a*c^2*d*f - (a*d^3*e + a*c*d^2*f)*x^2)*sqrt(a*c)*sqrt(d/c)*elliptic_e(arc sin(x*sqrt(d/c)), -b*c/(a*d)) - (((b*c^2*d + a*d^3)*e - (a*c^2*d - a*c*d^2 )*f)*x^2 - (b*c^3 + a*c*d^2)*e + (a*c^3 - a*c^2*d)*f)*sqrt(a*c)*sqrt(d/c)* elliptic_f(arcsin(x*sqrt(d/c)), -b*c/(a*d)))/(a*b*c^4*d + a^2*c^3*d^2 - (a *b*c^3*d^2 + a^2*c^2*d^3)*x^2)
\[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {e + f x^{2}}{\sqrt {a + b x^{2}} \left (c - d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {f x^{2} + e}{\sqrt {b x^{2} + a} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {f x^{2} + e}{\sqrt {b x^{2} + a} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {f\,x^2+e}{\sqrt {b\,x^2+a}\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]