Integrand size = 26, antiderivative size = 391 \[ \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=-\frac {\left (16 a c-\frac {16 b c^2}{d}-\frac {a^2 d}{b}\right ) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 d^2}-\frac {e x^3 \left (a+b x^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{d}-\frac {(8 b c-7 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^3}+\frac {6 b e x^3 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 d^2}-\frac {\sqrt {c} \left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 b d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {c^{3/2} (8 b c-7 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{5 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
-1/5*(16*a*c-16*b*c^2/d-a^2*d/b)*e*x*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/d^2-e*x ^3*(b*x^2+a)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/d-1/5*(-7*a*d+8*b*c)*e*x*(d*x^2 +c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/d^3+6/5*b*e*x^3*(d*x^2+c)*(e*(b*x^2+a)/( d*x^2+c))^(1/2)/d^2+1/5*c^(3/2)*(-7*a*d+8*b*c)*e*(1/(1+d*x^2/c))^(1/2)*(1+ d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^( 1/2))*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/d^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2 )-1/5*(a^2*d^2-16*a*b*c*d+16*b^2*c^2)*e*(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)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/b/d^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/ 2)
Result contains complex when optimal does not.
Time = 5.61 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.68 \[ \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (a d \left (7 c+2 d x^2\right )+b \left (-8 c^2-2 c d x^2+d^2 x^4\right )\right )-i c \left (16 b^2 c^2-16 a b c d+a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+8 i c \left (2 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{5 \sqrt {\frac {b}{a}} d^4 \left (a+b x^2\right )} \]
(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*d*x*(a + b*x^2)*(a*d*(7*c + 2*d*x^2) + b*(-8*c^2 - 2*c*d*x^2 + d^2*x^4)) - I*c*(16*b^2*c^2 - 16*a*b* c*d + 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)] + (8*I)*c*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Sq rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a *d)/(b*c)]))/(5*Sqrt[b/a]*d^4*(a + b*x^2))
Time = 0.63 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2058, 369, 27, 443, 25, 444, 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.
| \(\displaystyle \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \int \frac {x^4 \left (b x^2+a\right )^{3/2}}{\left (d x^2+c\right )^{3/2}}dx}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 369 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\int \frac {3 x^2 \sqrt {b x^2+a} \left (2 b x^2+a\right )}{\sqrt {d x^2+c}}dx}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \int \frac {x^2 \sqrt {b x^2+a} \left (2 b x^2+a\right )}{\sqrt {d x^2+c}}dx}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 443 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \left (\frac {\int -\frac {x^2 \left (b (8 b c-7 a d) x^2+a (6 b c-5 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}+\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}\right )}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \left (\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\int \frac {x^2 \left (b (8 b c-7 a d) x^2+a (6 b c-5 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 d}\right )}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 444 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \left (\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (8 b c-7 a d)}{3 d}-\frac {\int \frac {b \left (\left (16 b^2 c^2-16 a b d c+a^2 d^2\right ) x^2+a c (8 b c-7 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 b d}}{5 d}\right )}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \left (\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (8 b c-7 a d)}{3 d}-\frac {\int \frac {\left (16 b^2 c^2-16 a b d c+a^2 d^2\right ) x^2+a c (8 b c-7 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}\right )}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \left (\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (8 b c-7 a d)}{3 d}-\frac {\left (a^2 d^2-16 a b c d+16 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (8 b c-7 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}}{5 d}\right )}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \left (\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (8 b c-7 a d)}{3 d}-\frac {\left (a^2 d^2-16 a b c d+16 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (8 b c-7 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{5 d}\right )}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \left (\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (8 b c-7 a d)}{3 d}-\frac {\left (a^2 d^2-16 a b c d+16 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (8 b c-7 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{5 d}\right )}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {3 \left (\frac {2 b x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{5 d}-\frac {\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (8 b c-7 a d)}{3 d}-\frac {\left (a^2 d^2-16 a b c d+16 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (8 b c-7 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 d}}{5 d}\right )}{d}-\frac {x^3 \left (a+b x^2\right )^{3/2}}{d \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-((x^3*(a + b*x^2)^( 3/2))/(d*Sqrt[c + d*x^2])) + (3*((2*b*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) /(5*d) - (((8*b*c - 7*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) - ((16 *b^2*c^2 - 16*a*b*c*d + 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*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2] )) + (c^(3/2)*(8*b*c - 7*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*d))/(5*d)))/d))/Sqrt[a + b*x^2]
3.3.83.3.1 Defintions of rubi rules used
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[((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/(2* b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(m - 1) + d*(m + 2*q - 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && 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*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*g*(m + 2*(p + q + 1) + 1))), x] + Simp[1/(b*(m + 2* (p + q + 1) + 1)) Int[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(( b*e - a*f)*(m + 1) + b*e*2*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*2*q*(b *c - a*d) + b*e*d*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , g, m, p}, x] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^ 2])
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 = 9.88 (sec) , antiderivative size = 775, normalized size of antiderivative = 1.98
| method | result | size |
| risch | \(\frac {x \left (b d \,x^{2}+2 a d -3 b c \right ) \left (d \,x^{2}+c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{5 d^{3}}+\frac {\left (-\frac {2 \left (a^{2} d^{2}-11 a b c d +11 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 )}-\frac {c \left (7 a^{2} d^{2}-13 a b c d +5 b^{2} c^{2}\right ) \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 )}{d \sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {5 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (b d e \,x^{2}+e d a \right ) x}{c \left (a d -b c \right ) e \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d e \,x^{2}+e d a \right )}}+\frac {\left (\frac {1}{c}-\frac {d a}{c \left (a d -b c \right )}\right ) \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 b d a 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 )}{\left (a d -b c \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 )}{d}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{5 d^{3} \left (b \,x^{2}+a \right )}\) | \(775\) |
| default | \(\frac {{\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )}^{\frac {3}{2}} \left (d \,x^{2}+c \right ) \left (\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} d^{3} x^{7}+3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{3} x^{5}-2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{3} x^{3}-3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c \,d^{2} x^{3}-5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}-8 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \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}+24 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \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 -16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \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}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \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}-16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \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 +16 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \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}+2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -3 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x +5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} x -5 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} d x \right )}{5 d^{4} \left (b \,x^{2}+a \right )^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(933\) |
1/5*x*(b*d*x^2+2*a*d-3*b*c)*(d*x^2+c)/d^3*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)+ 1/5/d^3*(-2*(a^2*d^2-11*a*b*c*d+11*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)))-c*(7*a^2* d^2-13*a*b*c*d+5*b^2*c^2)/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))+5*c^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/d*((b*d* e*x^2+a*d*e)/c/(a*d-b*c)*x/e/((x^2+c/d)*(b*d*e*x^2+a*d*e))^(1/2)+(1/c-d*a/ c/(a*d-b*c))/(-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*b*d/(a*d-b*c)*a*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))-Ellipt icE(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2)))))*e/(b*x^2+a)*(e*(b*x^ 2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*e*(b*x^2+a))^(1/2)
Time = 0.12 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.70 \[ \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=-\frac {{\left (16 \, b^{2} c^{3} - 16 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {\frac {b e}{d}} e x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (16 \, b^{2} c^{3} - 16 \, a b c^{2} d - 7 \, a^{2} d^{3} + {\left (a^{2} + 8 \, a b\right )} c d^{2}\right )} \sqrt {\frac {b e}{d}} e x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} d^{3} e x^{6} - 2 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} e x^{4} + {\left (8 \, b^{2} c^{2} d - 9 \, a b c d^{2} + a^{2} d^{3}\right )} e x^{2} + {\left (16 \, b^{2} c^{3} - 16 \, a b c^{2} d + a^{2} c d^{2}\right )} e\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{5 \, b d^{4} x} \]
-1/5*((16*b^2*c^3 - 16*a*b*c^2*d + a^2*c*d^2)*sqrt(b*e/d)*e*x*sqrt(-c/d)*e lliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (16*b^2*c^3 - 16*a*b*c^2*d - 7*a^2*d^3 + (a^2 + 8*a*b)*c*d^2)*sqrt(b*e/d)*e*x*sqrt(-c/d)*elliptic_f(arc sin(sqrt(-c/d)/x), a*d/(b*c)) - (b^2*d^3*e*x^6 - 2*(b^2*c*d^2 - a*b*d^3)*e *x^4 + (8*b^2*c^2*d - 9*a*b*c*d^2 + a^2*d^3)*e*x^2 + (16*b^2*c^3 - 16*a*b* c^2*d + a^2*c*d^2)*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(b*d^4*x)
Timed out. \[ \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\int { \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} x^{4} \,d x } \]
\[ \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\int { \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} x^{4} \,d x } \]
Timed out. \[ \int x^4 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2} \, dx=\int x^4\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2} \,d x \]