Integrand size = 30, antiderivative size = 628 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}-\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}} \]
1/2*d*(2*a*d+b*c)/a/c/(-a*d+b*c)^2/e/(e*x)^(1/2)/(-d*x^2+c)^(1/2)+1/2*b/a/ (-a*d+b*c)/e/(-b*x^2+a)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)-1/2*(6*a^2*d^2-8*a*b* c*d+5*b^2*c^2)*(-d*x^2+c)^(1/2)/a^2/c^2/(-a*d+b*c)^2/e/(e*x)^(1/2)-1/2*d^( 1/4)*(6*a^2*d^2-8*a*b*c*d+5*b^2*c^2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4) /e^(1/2),I)*(1-d*x^2/c)^(1/2)/a^2/c^(5/4)/(-a*d+b*c)^2/e^(3/2)/(-d*x^2+c)^ (1/2)+1/2*d^(1/4)*(6*a^2*d^2-8*a*b*c*d+5*b^2*c^2)*EllipticF(d^(1/4)*(e*x)^ (1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/a^2/c^(5/4)/(-a*d+b*c)^2/e^(3/2 )/(-d*x^2+c)^(1/2)-1/4*b^(3/2)*c^(1/4)*(-11*a*d+5*b*c)*EllipticPi(d^(1/4)* (e*x)^(1/2)/c^(1/4)/e^(1/2),-b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)*(1-d*x^2/c )^(1/2)/a^(5/2)/d^(1/4)/(-a*d+b*c)^2/e^(3/2)/(-d*x^2+c)^(1/2)+1/4*b^(3/2)* c^(1/4)*(-11*a*d+5*b*c)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),b^( 1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)*(1-d*x^2/c)^(1/2)/a^(5/2)/d^(1/4)/(-a*d+b* c)^2/e^(3/2)/(-d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.31 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {x \left (21 a \left (2 a^3 d^2 \left (2 c-3 d x^2\right )-5 b^3 c^2 x^2 \left (c-d x^2\right )+4 a b^2 c \left (c^2+c d x^2-2 d^2 x^4\right )+2 a^2 b d \left (-4 c^2+2 c d x^2+3 d^2 x^4\right )\right )+7 \left (-5 b^3 c^3+16 a b^2 c^2 d-8 a^2 b c d^2+6 a^3 d^3\right ) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) x^4 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 c^2 (b c-a d)^2 (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
(x*(21*a*(2*a^3*d^2*(2*c - 3*d*x^2) - 5*b^3*c^2*x^2*(c - d*x^2) + 4*a*b^2* c*(c^2 + c*d*x^2 - 2*d^2*x^4) + 2*a^2*b*d*(-4*c^2 + 2*c*d*x^2 + 3*d^2*x^4) ) + 7*(-5*b^3*c^3 + 16*a*b^2*c^2*d - 8*a^2*b*c*d^2 + 6*a^3*d^3)*x^2*(a - b *x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*x^4*(-a + b*x^2)*Sqrt[1 - (d* x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^3*c^2*(b *c - a*d)^2*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])
Time = 1.13 (sec) , antiderivative size = 606, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {368, 27, 972, 27, 1049, 27, 1053, 25, 1054, 2009}
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.
| \(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^3}{x \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {1}{e x \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(5 b c-4 a d) e^2-7 b d e^2 x^2}{e^3 x \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {(5 b c-4 a d) e^2-7 b d e^2 x^2}{e x \left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {d (2 a d+b c)}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 \left (\left (5 b^2 c^2-8 a b d c+6 a^2 d^2\right ) e^2-3 b d (b c+2 a d) e^2 x^2\right )}{e x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {\left (5 b^2 c^2-8 a b d c+6 a^2 d^2\right ) e^2-3 b d (b c+2 a d) e^2 x^2}{e x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {-\frac {\int -\frac {e x \left (b d \left (5 b^2 c^2-8 a b d c+6 a^2 d^2\right ) x^2 e^2+\left (5 b^3 c^3-16 a b^2 d c^2+8 a^2 b d^2 c-6 a^3 d^3\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {5 b^2 c}{a}+\frac {6 a d^2}{c}-8 b d\right )}{\sqrt {e x}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\int \frac {e x \left (b d \left (5 b^2 c^2-8 a b d c+6 a^2 d^2\right ) x^2 e^2+\left (5 b^3 c^3-16 a b^2 d c^2+8 a^2 b d^2 c-6 a^3 d^3\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {5 b^2 c}{a}+\frac {6 a d^2}{c}-8 b d\right )}{\sqrt {e x}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\int \left (\frac {e \left (5 b^3 c^3 e^2-11 a b^2 c^2 d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d \left (5 b^2 c^2-8 a b d c+6 a^2 d^2\right ) e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {5 b^2 c}{a}+\frac {6 a d^2}{c}-8 b d\right )}{\sqrt {e x}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \left (6 a^2 d^2-8 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \left (6 a^2 d^2-8 a b c d+5 b^2 c^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}-\frac {b^{3/2} c^{9/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-11 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {b^{3/2} c^{9/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-11 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}}{a c e^2}-\frac {\sqrt {c-d x^2} \left (\frac {5 b^2 c}{a}+\frac {6 a d^2}{c}-8 b d\right )}{\sqrt {e x}}}{c (b c-a d)}+\frac {d (2 a d+b c)}{c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b}{4 a e^2 \sqrt {e x} \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\) |
2*e^3*(b/(4*a*(b*c - a*d)*e^2*Sqrt[e*x]*Sqrt[c - d*x^2]*(a*e^2 - b*e^2*x^2 )) + ((d*(b*c + 2*a*d))/(c*(b*c - a*d)*Sqrt[e*x]*Sqrt[c - d*x^2]) + (-(((( 5*b^2*c)/a - 8*b*d + (6*a*d^2)/c)*Sqrt[c - d*x^2])/Sqrt[e*x]) + (-((c^(3/4 )*d^(1/4)*(5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*e^(3/2)*Sqrt[1 - (d*x^2)/c]* EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x ^2]) + (c^(3/4)*d^(1/4)*(5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1] )/Sqrt[c - d*x^2] - (b^(3/2)*c^(9/4)*(5*b*c - 11*a*d)*e^(3/2)*Sqrt[1 - (d* x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4) *Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*Sqrt[a]*d^(1/4)*Sqrt[c - d*x^2]) + (b^(3/2)*c^(9/4)*(5*b*c - 11*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[ (Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*S qrt[e])], -1])/(2*Sqrt[a]*d^(1/4)*Sqrt[c - d*x^2]))/(a*c*e^2))/(c*(b*c - a *d)))/(4*a*(b*c - a*d)*e^4))
3.10.24.3.1 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[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*2)/e^2))^p*(c + d*(x^(k*2)/e^2))^q, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[m ] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1348\) vs. \(2(506)=1012\).
Time = 3.09 (sec) , antiderivative size = 1349, normalized size of antiderivative = 2.15
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1349\) |
| default | \(\text {Expression too large to display}\) | \(3373\) |
((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-2*(-d*e*x^2+c*e)/e^2 /c^2/a^2/(x*(-d*e*x^2+c*e))^(1/2)+1/2*b^3/(a*d-b*c)^2/a^2/e^2*x*(-d*e*x^3+ c*e*x)^(1/2)/(-b*x^2+a)+d^3/e*x^2/c^2/(a*d-b*c)^2/(-(x^2-c/d)*d*e*x)^(1/2) +2/c*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1 /2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/e/a^2*EllipticE(((x+1/d*(c*d)^(1/2))*d/( c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/c*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d) ^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/e/a^2*Elli pticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/2*c*(d*x/(c *d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/( -d*e*x^3+c*e*x)^(1/2)*b^2/e/a^2/(a*d-b*c)^2*EllipticE(((x+1/d*(c*d)^(1/2)) *d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/4*c*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x /(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*b^2/ e/a^2/(a*d-b*c)^2*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2* 2^(1/2))+d^2/c*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d* x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/e/(a*d-b*c)^2*EllipticE(((x+1/ d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/2*d^2/c*(d*x/(c*d)^(1/2 )+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3 +c*e*x)^(1/2)/e/(a*d-b*c)^2*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^ (1/2),1/2*2^(1/2))+11/8*b/e/a/(a*d-b*c)^2*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^ (1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c...
Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]