Integrand size = 33, antiderivative size = 289 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {e \sqrt {a+b x^2}}{3 c x^3 \sqrt {c+d x^2}}-\frac {(b c e-4 a d e+3 a c f) \sqrt {a+b x^2}}{3 a c^2 x \sqrt {c+d x^2}}-\frac {\sqrt {d} (b c e-8 a d e+6 a c f) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b (4 d e-3 c f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
-1/3*e*(b*x^2+a)^(1/2)/c/x^3/(d*x^2+c)^(1/2)-1/3*(3*a*c*f-4*a*d*e+b*c*e)*( b*x^2+a)^(1/2)/a/c^2/x/(d*x^2+c)^(1/2)-1/3*d^(1/2)*(6*a*c*f-8*a*d*e+b*c*e) *(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d) ^(1/2))/a/c^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*b*(- 3*c*f+4*d*e)*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1- b*c/a/d)^(1/2))/a/c^(3/2)/d^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c )^(1/2)
Result contains complex when optimal does not.
Time = 3.36 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (-\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (b c e x^2 \left (c+d x^2\right )+a \left (-8 d^2 e x^4+c^2 \left (e+3 f x^2\right )+c d \left (-4 e x^2+6 f x^4\right )\right )\right )-i b c (b c e-8 a d e+6 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c (b c e-4 a d e+3 a c f) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b c^3 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[a + b*x^2]*(e + f*x^2))/(x^4*(c + d*x^2)^(3/2)),x]
Output:
(Sqrt[b/a]*(-(Sqrt[b/a]*(a + b*x^2)*(b*c*e*x^2*(c + d*x^2) + a*(-8*d^2*e*x ^4 + c^2*(e + 3*f*x^2) + c*d*(-4*e*x^2 + 6*f*x^4)))) - I*b*c*(b*c*e - 8*a* d*e + 6*a*c*f)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*Arc Sinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*b*c*(b*c*e - 4*a*d*e + 3*a*c*f)*x^3*Sq rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a *d)/(b*c)]))/(3*b*c^3*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.61 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.33, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {439, 25, 445, 25, 27, 445, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 439 |
| \(\displaystyle \frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}-\frac {\int -\frac {b (3 d e-2 c f) x^2+a (4 d e-3 c f)}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b (3 d e-2 c f) x^2+a (4 d e-3 c f)}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {\int -\frac {a d \left (-b (4 d e-3 c f) x^2+b c e-8 a d e+6 a c f\right )}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {a d \left (-b (4 d e-3 c f) x^2+b c e-8 a d e+6 a c f\right )}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \int \frac {-b (4 d e-3 c f) x^2+b c e-8 a d e+6 a c f}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\frac {d \left (-\frac {\int \frac {b \left (a c (4 d e-3 c f)-d (b c e-8 a d e+6 a c f) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 a c f-8 a d e+b c e)}{a c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d \left (-\frac {b \int \frac {a c (4 d e-3 c f)-d (b c e-8 a d e+6 a c f) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 a c f-8 a d e+b c e)}{a c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {d \left (-\frac {b \left (a c (4 d e-3 c f) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-d (6 a c f-8 a d e+b c e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 a c f-8 a d e+b c e)}{a c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {d \left (-\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} (4 d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-d (6 a c f-8 a d e+b c e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 a c f-8 a d e+b c e)}{a c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {d \left (-\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} (4 d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-d (6 a c f-8 a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 a c f-8 a d e+b c e)}{a c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {d \left (-\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} (4 d e-3 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-d (6 a c f-8 a d e+b c e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (6 a c f-8 a d e+b c e)}{a c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (4 d e-3 c f)}{3 c x^3}}{c d}+\frac {\sqrt {a+b x^2} (d e-c f)}{c d x^3 \sqrt {c+d x^2}}\) |
Input:
Int[(Sqrt[a + b*x^2]*(e + f*x^2))/(x^4*(c + d*x^2)^(3/2)),x]
Output:
((d*e - c*f)*Sqrt[a + b*x^2])/(c*d*x^3*Sqrt[c + d*x^2]) + (-1/3*((4*d*e - 3*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x^3) + (d*(-(((b*c*e - 8*a*d*e + 6*a*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(a*c*x)) - (b*(-(d*(b*c*e - 8* a*d*e + 6*a*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*Sqr t[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))) + (c^(3/2)*( 4*d*e - 3*c*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)))/(3*c))/(c*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/(2*a*b*g*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*b*e*( p + 1) + (b*e - a*f)*(m + 1)) + d*(2*b*e*(p + 1) + (b*e - a*f)*(m + 2*q + 1 ))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && G tQ[q, 0] && !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
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 = 9.86 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.59
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 c^{2} x^{3}}-\frac {\left (3 a c f -5 a d e +b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 a \,c^{3} x}-\frac {\left (b d \,x^{2}+a d \right ) \left (c f -d e \right ) x}{c^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (-\frac {d b e}{3 c^{2}}-\frac {\left (c f -d e \right ) \left (a d -b c \right )}{c^{3}}+\frac {a d \left (c f -d e \right )}{c^{3}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {b d \left (3 a c f -5 a d e +b c e \right )}{3 a \,c^{3}}+\frac {b d \left (c f -d e \right )}{c^{3}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(460\) |
| default | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (6 \sqrt {-\frac {b}{a}}\, a b c d f \,x^{6}-8 \sqrt {-\frac {b}{a}}\, a b \,d^{2} e \,x^{6}+\sqrt {-\frac {b}{a}}\, b^{2} c d e \,x^{6}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f \,x^{3}-4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e \,x^{3}-6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f \,x^{3}+8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e \,x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} e \,x^{3}+6 \sqrt {-\frac {b}{a}}\, a^{2} c d f \,x^{4}-8 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} e \,x^{4}+3 \sqrt {-\frac {b}{a}}\, a b \,c^{2} f \,x^{4}-3 \sqrt {-\frac {b}{a}}\, a b c d e \,x^{4}+\sqrt {-\frac {b}{a}}\, b^{2} c^{2} e \,x^{4}+3 \sqrt {-\frac {b}{a}}\, a^{2} c^{2} f \,x^{2}-4 \sqrt {-\frac {b}{a}}\, a^{2} c d e \,x^{2}+2 \sqrt {-\frac {b}{a}}\, a b \,c^{2} e \,x^{2}+\sqrt {-\frac {b}{a}}\, a^{2} c^{2} e \right )}{3 \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) c^{3} x^{3} a \sqrt {-\frac {b}{a}}}\) | \(640\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (3 a c f \,x^{2}-5 a d e \,x^{2}+b c e \,x^{2}+a c e \right )}{3 c^{3} x^{3} a}+\frac {\left (-\frac {b \left (3 a c f -5 a d e +b c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-3 \left (a c d f -a \,d^{2} e -b \,c^{2} f +b c d e \right ) c a \left (\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{\left (a d -b c \right ) c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )-\frac {a c d e b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 a \,c^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(644\) |
Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)/x^4/(d*x^2+c)^(3/2),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)*(-1/3/c^2*e*(b *d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3-1/3/a/c^3*(3*a*c*f-5*a*d*e+b*c*e)*(b *d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x-(b*d*x^2+a*d)*(c*f-d*e)/c^3*x/((x^2+c/ d)*(b*d*x^2+a*d))^(1/2)+(-1/3*d*b/c^2*e-(c*f-d*e)*(a*d-b*c)/c^3+a*d*(c*f-d *e)/c^3)/(-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))-(1/ 3*b*d*(3*a*c*f-5*a*d*e+b*c*e)/a/c^3+b*d*(c*f-d*e)/c^3)*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*(Ell ipticF(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 = 0.09 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (6 \, a b c d f + {\left (b^{2} c d - 8 \, a b d^{2}\right )} e\right )} x^{5} + {\left (6 \, a b c^{2} f + {\left (b^{2} c^{2} - 8 \, a b c d\right )} e\right )} x^{3}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (3 \, {\left (a^{2} + 2 \, a b\right )} c d f + {\left (b^{2} c d - 4 \, {\left (a^{2} + 2 \, a b\right )} d^{2}\right )} e\right )} x^{5} + {\left (3 \, {\left (a^{2} + 2 \, a b\right )} c^{2} f + {\left (b^{2} c^{2} - 4 \, {\left (a^{2} + 2 \, a b\right )} c d\right )} e\right )} x^{3}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a^{2} c^{2} e + {\left (6 \, a^{2} c d f + {\left (a b c d - 8 \, a^{2} d^{2}\right )} e\right )} x^{4} + {\left (3 \, a^{2} c^{2} f + {\left (a b c^{2} - 4 \, a^{2} c d\right )} e\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{2} c^{3} d x^{5} + a^{2} c^{4} x^{3}\right )}} \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)/x^4/(d*x^2+c)^(3/2),x, algorithm="fric as")
Output:
1/3*(((6*a*b*c*d*f + (b^2*c*d - 8*a*b*d^2)*e)*x^5 + (6*a*b*c^2*f + (b^2*c^ 2 - 8*a*b*c*d)*e)*x^3)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a) ), a*d/(b*c)) - ((3*(a^2 + 2*a*b)*c*d*f + (b^2*c*d - 4*(a^2 + 2*a*b)*d^2)* e)*x^5 + (3*(a^2 + 2*a*b)*c^2*f + (b^2*c^2 - 4*(a^2 + 2*a*b)*c*d)*e)*x^3)* sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (a^2*c^ 2*e + (6*a^2*c*d*f + (a*b*c*d - 8*a^2*d^2)*e)*x^4 + (3*a^2*c^2*f + (a*b*c^ 2 - 4*a^2*c*d)*e)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^2*c^3*d*x^5 + a ^2*c^4*x^3)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (e + f x^{2}\right )}{x^{4} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(f*x**2+e)/x**4/(d*x**2+c)**(3/2),x)
Output:
Integral(sqrt(a + b*x**2)*(e + f*x**2)/(x**4*(c + d*x**2)**(3/2)), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)/x^4/(d*x^2+c)^(3/2),x, algorithm="maxi ma")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)/((d*x^2 + c)^(3/2)*x^4), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(f*x^2+e)/x^4/(d*x^2+c)^(3/2),x, algorithm="giac ")
Output:
integrate(sqrt(b*x^2 + a)*(f*x^2 + e)/((d*x^2 + c)^(3/2)*x^4), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (f\,x^2+e\right )}{x^4\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int(((a + b*x^2)^(1/2)*(e + f*x^2))/(x^4*(c + d*x^2)^(3/2)),x)
Output:
int(((a + b*x^2)^(1/2)*(e + f*x^2))/(x^4*(c + d*x^2)^(3/2)), x)
\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^4 \left (c+d x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(1/2)*(f*x^2+e)/x^4/(d*x^2+c)^(3/2),x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*e + 6*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(2*a**2*c**2*d*x**2 + 4*a**2*c*d**2*x**4 + 2*a**2*d**3*x**6 + a* b*c**3*x**2 + 4*a*b*c**2*d*x**4 + 5*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b* *2*c**3*x**4 + 2*b**2*c**2*d*x**6 + b**2*c*d**2*x**8),x)*a**2*c**2*d*f*x** 3 - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(2*a**2*c**2*d*x**2 + 4*a**2 *c*d**2*x**4 + 2*a**2*d**3*x**6 + a*b*c**3*x**2 + 4*a*b*c**2*d*x**4 + 5*a* b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 + 2*b**2*c**2*d*x**6 + b* *2*c*d**2*x**8),x)*a**2*c*d**2*e*x**3 + 6*int((sqrt(c + d*x**2)*sqrt(a + b *x**2))/(2*a**2*c**2*d*x**2 + 4*a**2*c*d**2*x**4 + 2*a**2*d**3*x**6 + a*b* c**3*x**2 + 4*a*b*c**2*d*x**4 + 5*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2 *c**3*x**4 + 2*b**2*c**2*d*x**6 + b**2*c*d**2*x**8),x)*a**2*c*d**2*f*x**5 - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(2*a**2*c**2*d*x**2 + 4*a**2*c *d**2*x**4 + 2*a**2*d**3*x**6 + a*b*c**3*x**2 + 4*a*b*c**2*d*x**4 + 5*a*b* c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 + 2*b**2*c**2*d*x**6 + b**2 *c*d**2*x**8),x)*a**2*d**3*e*x**5 + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x** 2))/(2*a**2*c**2*d*x**2 + 4*a**2*c*d**2*x**4 + 2*a**2*d**3*x**6 + a*b*c**3 *x**2 + 4*a*b*c**2*d*x**4 + 5*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c** 3*x**4 + 2*b**2*c**2*d*x**6 + b**2*c*d**2*x**8),x)*a*b*c**3*f*x**3 - 2*int ((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(2*a**2*c**2*d*x**2 + 4*a**2*c*d**2*x **4 + 2*a**2*d**3*x**6 + a*b*c**3*x**2 + 4*a*b*c**2*d*x**4 + 5*a*b*c*d*...