Integrand size = 33, antiderivative size = 255 \[ \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {e \sqrt {c+d x^2}}{a c x \sqrt {a+b x^2}}-\frac {\sqrt {b} (2 b c e-a d e-a c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{a^{3/2} c (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {d (b e-a f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} c (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
-e*(d*x^2+c)^(1/2)/a/c/x/(b*x^2+a)^(1/2)-b^(1/2)*(-a*c*f-a*d*e+2*b*c*e)*(d *x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1 /2))/a^(3/2)/c/(-a*d+b*c)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+ d*(-a*f+b*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1- a*d/b/c)^(1/2))/a^(1/2)/b^(1/2)/c/(-a*d+b*c)/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/ c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 7.15 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.01 \[ \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (c+d x^2\right ) \left (a^2 d e-2 b^2 c e x^2+a b \left (-c e+d e x^2+c f x^2\right )\right )-i b c (-2 b c e+a d e+a c f) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) (-2 b e+a f) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{a^2 \sqrt {\frac {b}{a}} c (-b c+a d) x \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(e + f*x^2)/(x^2*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
(-(Sqrt[b/a]*(c + d*x^2)*(a^2*d*e - 2*b^2*c*e*x^2 + a*b*(-(c*e) + d*e*x^2 + c*f*x^2))) - I*b*c*(-2*b*c*e + a*d*e + a*c*f)*x*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)*(-2*b*e + a*f)*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipti cF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(a^2*Sqrt[b/a]*c*(-(b*c) + a*d)*x *Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.55 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.36, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {441, 25, 445, 25, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}-\frac {\int -\frac {d (b e-a f) x^2+2 b c e-a d e-a c f}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d (b e-a f) x^2+2 b c e-a (d e+c f)}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a (b c-a d)}+\frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {\int -\frac {d \left (b (2 b c e-a (d e+c f)) x^2+a c (b e-a f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-a c f-a d e+2 b c e)}{a c x}}{a (b c-a d)}+\frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {d \left (b (2 b c e-a (d e+c f)) x^2+a c (b e-a f)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-a c f-a d e+2 b c e)}{a c x}}{a (b c-a d)}+\frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \int \frac {b (2 b c e-a (d e+c f)) x^2+a c (b e-a f)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-a c f-a d e+2 b c e)}{a c x}}{a (b c-a d)}+\frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {d \left (a c (b e-a f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+b (2 b c e-a (c f+d e)) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-a c f-a d e+2 b c e)}{a c x}}{a (b c-a d)}+\frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {d \left (b (2 b c e-a (c f+d e)) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-a c f-a d e+2 b c e)}{a c x}}{a (b c-a d)}+\frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {d \left (b (2 b c e-a (c f+d e)) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-a c f-a d e+2 b c e)}{a c x}}{a (b c-a d)}+\frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {d \left (\frac {c^{3/2} \sqrt {a+b x^2} (b e-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b (2 b c e-a (c f+d e)) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (-a c f-a d e+2 b c e)}{a c x}}{a (b c-a d)}+\frac {\sqrt {c+d x^2} (b e-a f)}{a x \sqrt {a+b x^2} (b c-a d)}\) |
Input:
Int[(e + f*x^2)/(x^2*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
Output:
((b*e - a*f)*Sqrt[c + d*x^2])/(a*(b*c - a*d)*x*Sqrt[a + b*x^2]) + (-(((2*b *c*e - a*d*e - a*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x)) + (d*(b*(2 *b*c*e - a*(d*e + c*f))*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c ]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)]) /(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (c^( 3/2)*(b*e - a*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d* x^2])))/(a*c))/(a*(b*c - a*d))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Time = 10.29 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.61
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+b c \right ) x \left (a f -b e \right )}{a^{2} \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}-\frac {e \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{a^{2} c x}+\frac {\left (\frac {a f -b e}{a^{2}}+\frac {b c \left (a f -b e \right )}{a^{2} \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {b \left (a f -b e \right ) d}{a^{2} \left (a d -b c \right )}+\frac {b d e}{a^{2} c}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(411\) |
| risch | \(-\frac {e \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{a^{2} c x}+\frac {\left (a c \left (a f -b e \right ) \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )-\frac {b e c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{c \,a^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(497\) |
| default | \(\frac {\left (-\sqrt {-\frac {b}{a}}\, a b c d f \,x^{4}-\sqrt {-\frac {b}{a}}\, a b \,d^{2} e \,x^{4}+2 \sqrt {-\frac {b}{a}}\, b^{2} c d e \,x^{4}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d f x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f x -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e x +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e x -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e x -\sqrt {-\frac {b}{a}}\, a^{2} d^{2} e \,x^{2}-\sqrt {-\frac {b}{a}}\, a b \,c^{2} f \,x^{2}+2 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} e \,x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c d e +\sqrt {-\frac {b}{a}}\, a b \,c^{2} e \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, x c \,a^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(614\) |
Input:
int((f*x^2+e)/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-(b*d*x^2+b*c )/a^2/(a*d-b*c)*x*(a*f-b*e)/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)-1/a^2*e/c*(b*d *x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x+((a*f-b*e)/a^2+b*c/a^2/(a*d-b*c)*(a*f-b* e))/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c* x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(b*(a*f- b*e)*d/a^2/(a*d-b*c)+b*d*e/a^2/c)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^ 2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2) ,(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/ 2))))
Time = 0.10 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.38 \[ \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {{\left ({\left (a b^{3} c f - {\left (2 \, b^{4} c - a b^{3} d\right )} e\right )} x^{3} + {\left (a^{2} b^{2} c f - {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} e\right )} x\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left ({\left ({\left (2 \, b^{4} c + {\left (a^{2} b^{2} - a b^{3}\right )} d\right )} e - {\left (a b^{3} c + a^{3} b d\right )} f\right )} x^{3} + {\left ({\left (2 \, a b^{3} c + {\left (a^{3} b - a^{2} b^{2}\right )} d\right )} e - {\left (a^{2} b^{2} c + a^{4} d\right )} f\right )} x\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (a^{2} b^{2} c f - {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} e\right )} x^{2} - {\left (a^{2} b^{2} c - a^{3} b d\right )} e\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (a^{3} b^{3} c^{2} - a^{4} b^{2} c d\right )} x^{3} + {\left (a^{4} b^{2} c^{2} - a^{5} b c d\right )} x} \] Input:
integrate((f*x^2+e)/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fric as")
Output:
-(((a*b^3*c*f - (2*b^4*c - a*b^3*d)*e)*x^3 + (a^2*b^2*c*f - (2*a*b^3*c - a ^2*b^2*d)*e)*x)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/ (b*c)) + (((2*b^4*c + (a^2*b^2 - a*b^3)*d)*e - (a*b^3*c + a^3*b*d)*f)*x^3 + ((2*a*b^3*c + (a^3*b - a^2*b^2)*d)*e - (a^2*b^2*c + a^4*d)*f)*x)*sqrt(a* c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((a^2*b^2*c*f - (2*a*b^3*c - a^2*b^2*d)*e)*x^2 - (a^2*b^2*c - a^3*b*d)*e)*sqrt(b*x^2 + a )*sqrt(d*x^2 + c))/((a^3*b^3*c^2 - a^4*b^2*c*d)*x^3 + (a^4*b^2*c^2 - a^5*b *c*d)*x)
\[ \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {e + f x^{2}}{x^{2} \left (a + b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate((f*x**2+e)/x**2/(b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
Output:
Integral((e + f*x**2)/(x**2*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
\[ \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} x^{2}} \,d x } \] Input:
integrate((f*x^2+e)/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="maxi ma")
Output:
integrate((f*x^2 + e)/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*x^2), x)
\[ \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c} x^{2}} \,d x } \] Input:
integrate((f*x^2+e)/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="giac ")
Output:
integrate((f*x^2 + e)/((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*x^2), x)
Timed out. \[ \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {f\,x^2+e}{x^2\,{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((e + f*x^2)/(x^2*(a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
Output:
int((e + f*x^2)/(x^2*(a + b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {e+f x^2}{x^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, e -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b d e x -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} d e \,x^{3}+\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} c f x -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b c e x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) a b c f \,x^{3}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}+a^{2} d \,x^{2}+2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} c e \,x^{3}}{a c x \left (b \,x^{2}+a \right )} \] Input:
int((f*x^2+e)/x^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*e - int((sqrt(c + d*x**2)*sqrt(a + b *x**2)*x**2)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c* x**4 + b**2*d*x**6),x)*a*b*d*e*x - int((sqrt(c + d*x**2)*sqrt(a + b*x**2)* x**2)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*b**2*d*e*x**3 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a **2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x **6),x)*a**2*c*f*x - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a **2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a *b*c*e*x + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b*c*f*x**3 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c + a**2*d*x**2 + 2*a*b* c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*b**2*c*e*x**3)/(a*c* x*(a + b*x**2))