Integrand size = 26, antiderivative size = 383 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^3}+\frac {d (7 b c-8 a d) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{3 c^3}+\frac {(3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^2 d x^3}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{3 c^3 x}-\frac {\sqrt {d} (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b (3 b c-4 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
-(-a*d+b*c)*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c/d/x^3+1/3*d*(-8*a*d+7*b*c)*e *x*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^3+1/3*(-4*a*d+3*b*c)*e*(d*x^2+c)*(e*(b* x^2+a)/(d*x^2+c))^(1/2)/c^2/d/x^3-1/3*(-8*a*d+7*b*c)*e*(d*x^2+c)*(e*(b*x^2 +a)/(d*x^2+c))^(1/2)/c^3/x+1/3*b*(-4*a*d+3*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)/a/c^(3/2)/d^(1/2)/(c*(b*x^2+a)/a/(d*x ^2+c))^(1/2)-1/3*(-8*a*d+7*b*c)*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))*d^(1/2)*( e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 5.63 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.66 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (-b c x^2 \left (4 c+7 d x^2\right )+a \left (-c^2+4 c d x^2+8 d^2 x^4\right )\right )+i b c (-7 b c+8 a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-4 i b c (-b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 \sqrt {\frac {b}{a}} c^3 x^3 \left (a+b x^2\right )} \]
(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(a + b*x^2)*(-(b*c*x^2*(4* c + 7*d*x^2)) + a*(-c^2 + 4*c*d*x^2 + 8*d^2*x^4)) + I*b*c*(-7*b*c + 8*a*d) *x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a] *x], (a*d)/(b*c)] - (4*I)*b*c*(-(b*c) + a*d)*x^3*Sqrt[1 + (b*x^2)/a]*Sqrt[ 1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*Sqrt[b/ a]*c^3*x^3*(a + b*x^2))
Time = 0.63 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {2058, 370, 445, 27, 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.
| \(\displaystyle \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \int \frac {\left (b x^2+a\right )^{3/2}}{x^4 \left (d x^2+c\right )^{3/2}}dx}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 370 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {\int \frac {b (2 b c-3 a d) x^2+a (3 b c-4 a d)}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {\int \frac {a d \left (b (3 b c-4 a d) x^2+a (7 b c-8 a d)\right )}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {d \int \frac {b (3 b c-4 a d) x^2+a (7 b c-8 a d)}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {d \left (-\frac {\int -\frac {a b \left (d (7 b c-8 a d) x^2+c (3 b c-4 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {d \left (\frac {\int \frac {a b \left (d (7 b c-8 a d) x^2+c (3 b c-4 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {d \left (\frac {b \int \frac {d (7 b c-8 a d) x^2+c (3 b c-4 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {d \left (\frac {b \left (c (3 b c-4 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+d (7 b c-8 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {d \left (\frac {b \left (d (7 b c-8 a d) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (3 b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {d \left (\frac {b \left (d (7 b c-8 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (3 b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {d \left (\frac {b \left (\frac {c^{3/2} \sqrt {a+b x^2} (3 b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+d (7 b c-8 a d) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{c x}\right )}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (3 b c-4 a d)}{3 c x^3}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^3 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-(((b*c - a*d)*Sqrt[ a + b*x^2])/(c*d*x^3*Sqrt[c + d*x^2])) - (-1/3*((3*b*c - 4*a*d)*Sqrt[a + b *x^2]*Sqrt[c + d*x^2])/(c*x^3) - (d*(-(((7*b*c - 8*a*d)*Sqrt[a + b*x^2]*Sq rt[c + d*x^2])/(c*x)) + (b*(d*(7*b*c - 8*a*d)*((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))]*Sq rt[c + d*x^2])) + (c^(3/2)*(3*b*c - 4*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTa n[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/ (a*(c + d*x^2))]*Sqrt[c + d*x^2])))/c))/(3*c))/(c*d)))/Sqrt[a + b*x^2]
3.3.87.3.1 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[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(e*x) ^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a *d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 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[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]
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 = 8.28 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.06
| method | result | size |
| 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 (-5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}+4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}-3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}+3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}-4 \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 d \,x^{3}+4 \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^{2} 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}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{3}-7 \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^{2} x^{3}-5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}+4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}-3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}+3 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c d \,x^{4}-4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}+5 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \left (b \,x^{2}+a \right )^{2} c^{3} x^{3} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(790\) |
| risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (-5 a d \,x^{2}+4 b c \,x^{2}+a c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{3 c^{3} x^{3}}-\frac {\left (\frac {a b 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 {10 a^{2} b \,d^{2} 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 {8 b^{2} c^{2} 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 )}{\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 )}-3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) c \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 )\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 )}}{3 c^{3} \left (b \,x^{2}+a \right )}\) | \(898\) |
-1/3*(e*(b*x^2+a)/(d*x^2+c))^(3/2)*(d*x^2+c)*(-5*((d*x^2+c)*(b*x^2+a))^(1/ 2)*(-b/a)^(1/2)*a*b*d^2*x^6+4*((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a)^(1/2)*b^2 *c*d*x^6-3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a*b*d^2*x^6+3* (b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*b^2*c*d*x^6-4*((d*x^2+c)* (b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a )^(1/2),(a*d/b/c)^(1/2))*a*b*c*d*x^3+4*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2 +a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2)) *b^2*c^2*x^3+8*((d*x^2+c)*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/ c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c*d*x^3-7*((d*x^2+c )*(b*x^2+a))^(1/2)*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b /a)^(1/2),(a*d/b/c)^(1/2))*b^2*c^2*x^3-5*((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a )^(1/2)*a^2*d^2*x^4+4*((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a)^(1/2)*b^2*c^2*x^4 -3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a^2*d^2*x^4+3*(b*d*x^4 +a*d*x^2+b*c*x^2+a*c)^(1/2)*(-b/a)^(1/2)*a*b*c*d*x^4-4*((d*x^2+c)*(b*x^2+a ))^(1/2)*(-b/a)^(1/2)*a^2*c*d*x^2+5*((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a)^(1/ 2)*a*b*c^2*x^2+((d*x^2+c)*(b*x^2+a))^(1/2)*(-b/a)^(1/2)*a^2*c^2)/(b*x^2+a) ^2/c^3/x^3/(-b/a)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)
Time = 0.12 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.56 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\frac {{\left (7 \, b^{2} c d - 8 \, a b d^{2}\right )} \sqrt {\frac {a c e}{d^{2}}} e x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (3 \, a b + 7 \, b^{2}\right )} c d - 4 \, {\left (a^{2} + 2 \, a b\right )} d^{2}\right )} \sqrt {\frac {a c e}{d^{2}}} e x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (7 \, a b c d - 8 \, a^{2} d^{2}\right )} e x^{4} + a^{2} c^{2} e + 4 \, {\left (a b c^{2} - a^{2} c d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{3 \, a c^{3} x^{3}} \]
1/3*((7*b^2*c*d - 8*a*b*d^2)*sqrt(a*c*e/d^2)*e*x^3*sqrt(-b/a)*elliptic_e(a rcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((3*a*b + 7*b^2)*c*d - 4*(a^2 + 2*a*b)*d ^2)*sqrt(a*c*e/d^2)*e*x^3*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/ (b*c)) - ((7*a*b*c*d - 8*a^2*d^2)*e*x^4 + a^2*c^2*e + 4*(a*b*c^2 - a^2*c*d )*e*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a*c^3*x^3)
Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\text {Timed out} \]
\[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\int { \frac {\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
\[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\int { \frac {\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^4} \,d x \]