Integrand size = 26, antiderivative size = 403 \[ \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {(b c-4 a d) x \left (a+b x^2\right )}{15 b^2 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {x^3 \left (a+b x^2\right )}{5 b \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) x \left (a+b x^2\right )}{15 b^3 d \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}+\frac {\sqrt {c} \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \left (a+b x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^3 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}-\frac {c^{3/2} (b c-4 a d) \left (a+b x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b^2 d^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \]
1/15*(-4*a*d+b*c)*x*(b*x^2+a)/b^2/d/(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/5*x^3* (b*x^2+a)/b/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/15*(-8*a^2*d^2+3*a*b*c*d+2*b^2 *c^2)*x*(b*x^2+a)/b^3/d/(d*x^2+c)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/15*c^(3/ 2)*(-4*a*d+b*c)*(b*x^2+a)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*Elliptic F(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/b^2/d^(3/2)/(d*x^ 2+c)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/15*(- 8*a^2*d^2+3*a*b*c*d+2*b^2*c^2)*(b*x^2+a)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c) ^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^ (1/2)/b^3/d^(3/2)/(d*x^2+c)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(e*(b*x^2+a)/( d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 2.52 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.64 \[ \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a d-b \left (c+3 d x^2\right )\right )-i c \left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i c \left (-b^2 c^2-a b c d+2 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \left (\frac {b}{a}\right )^{5/2} d^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )} \]
(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a*d - b*(c + 3*d*x^2))) - I*c* (-2*b^2*c^2 - 3*a*b*c*d + 8*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/ c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*c*(-(b^2*c^2) - a*b*c*d + 2*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*A rcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*a^2*(b/a)^(5/2)*d^2*Sqrt[(e*(a + b* x^2))/(c + d*x^2)]*(c + d*x^2))
Time = 0.57 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2058, 380, 444, 25, 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.
| \(\displaystyle \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \int \frac {x^4 \sqrt {d x^2+c}}{\sqrt {b x^2+a}}dx}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 380 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}-\frac {\int \frac {x^2 \left (3 a c-(b c-4 a d) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 b}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}-\frac {-\frac {\int -\frac {\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{5 b}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}-\frac {\frac {\int \frac {\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{5 b}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}-\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (b c-4 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{5 b}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}-\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{5 b}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}-\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{5 b}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {a+b x^2} \left (\frac {x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 b}-\frac {\frac {\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 b d}-\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 b d}}{5 b}\right )}{\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}\) |
(Sqrt[a + b*x^2]*((x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b) - (-1/3*((b* c - 4*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(b*d) + ((2*b^2*c^2 + 3*a*b* c*d - 8*a^2*d^2)*((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)*(b *c - 4*a*d)*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]) )/(3*b*d))/(5*b)))/(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])
3.4.2.3.1 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_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b* (m + 2*(p + q) + 1))), x] - Simp[e^2/(b*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[a*c*(m - 1) + (a*d*(m - 1) - 2 *q*(b*c - a*d))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Time = 5.14 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(-\frac {x \left (-3 b d \,x^{2}+4 a d -b c \right ) \left (b \,x^{2}+a \right )}{15 d \,b^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (\frac {4 a^{2} c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {a b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 \left (8 a^{2} d^{2}-3 a b c d -2 b^{2} c^{2}\right ) a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{15 d \,b^{2} \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(492\) |
| default | \(-\frac {\left (b \,x^{2}+a \right ) \left (-3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+\sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}-4 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-\sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3}+4 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -\sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{15 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b^{2} d^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(554\) |
-1/15*x*(-3*b*d*x^2+4*a*d-b*c)*(b*x^2+a)/d/b^2/(e*(b*x^2+a)/(d*x^2+c))^(1/ 2)+1/15/d/b^2*(4*a^2*c*d/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2 )/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1 +(a*d*e+b*c*e)/c/b/e)^(1/2))-a*b*c^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c *d*x^2)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/ a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))-2*(8*a^2*d^2-3*a*b*c*d-2*b^2*c^2) *a*c*e/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2)/(b*d*e*x^4+a*d*e *x^2+b*c*e*x^2+a*c*e)^(1/2)/(e*d*a+e*b*c+e*(a*d-b*c))*(EllipticF(x*(-b/a)^ (1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d*e+ b*c*e)/c/b/e)^(1/2))))/(e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*e*(b*x^2+a ))^(1/2)/(d*x^2+c)
Time = 0.12 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.68 \[ \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\frac {{\left (2 \, b^{2} c^{3} + 3 \, a b c^{2} d - 8 \, a^{2} c d^{2}\right )} \sqrt {\frac {b e}{d}} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{2} c^{3} + 3 \, a b c^{2} d - 4 \, a^{2} d^{3} - {\left (8 \, a^{2} - a b\right )} c d^{2}\right )} \sqrt {\frac {b e}{d}} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (3 \, b^{2} d^{3} x^{6} - 2 \, b^{2} c^{3} - 3 \, a b c^{2} d + 8 \, a^{2} c d^{2} + 4 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{4} - {\left (b^{2} c^{2} d + 7 \, a b c d^{2} - 8 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{15 \, b^{3} d^{2} e x} \]
1/15*((2*b^2*c^3 + 3*a*b*c^2*d - 8*a^2*c*d^2)*sqrt(b*e/d)*x*sqrt(-c/d)*ell iptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (2*b^2*c^3 + 3*a*b*c^2*d - 4*a^ 2*d^3 - (8*a^2 - a*b)*c*d^2)*sqrt(b*e/d)*x*sqrt(-c/d)*elliptic_f(arcsin(sq rt(-c/d)/x), a*d/(b*c)) + (3*b^2*d^3*x^6 - 2*b^2*c^3 - 3*a*b*c^2*d + 8*a^2 *c*d^2 + 4*(b^2*c*d^2 - a*b*d^3)*x^4 - (b^2*c^2*d + 7*a*b*c*d^2 - 8*a^2*d^ 3)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b^3*d^2*e*x)
Timed out. \[ \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\text {Timed out} \]
\[ \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int { \frac {x^{4}}{\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}} \,d x } \]
\[ \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int { \frac {x^{4}}{\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}} \,d x } \]
Timed out. \[ \int \frac {x^4}{\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \, dx=\int \frac {x^4}{\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \]