Integrand size = 30, antiderivative size = 444 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=-\frac {5 \sqrt {c-d x^2}}{2 a^2 e \sqrt {e x}}+\frac {\sqrt {c-d x^2}}{2 a e \sqrt {e x} \left (a-b x^2\right )}-\frac {5 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 e^{3/2} \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (5 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt {b} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \]
-5/2*(-d*x^2+c)^(1/2)/a^2/e/(e*x)^(1/2)+1/2*(-d*x^2+c)^(1/2)/a/e/(-b*x^2+a )/(e*x)^(1/2)-5/2*c^(3/4)*d^(1/4)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^ (1/2),I)*(1-d*x^2/c)^(1/2)/a^2/e^(3/2)/(-d*x^2+c)^(1/2)+5/2*c^(3/4)*d^(1/4 )*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/a^2/e ^(3/2)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-3*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)/e^(3/2)/b^(1/2)/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-3*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)/e^(3/2)/b^(1/2)/(-d*x^2+c) ^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\frac {x \left (21 a \left (4 a-5 b x^2\right ) \left (c-d x^2\right )+7 (-5 b c+8 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+15 b d x^4 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]
(x*(21*a*(4*a - 5*b*x^2)*(c - d*x^2) + 7*(-5*b*c + 8*a*d)*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] + 15* b*d*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*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])
Time = 0.84 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {368, 27, 969, 25, 27, 1053, 25, 27, 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.
| \(\displaystyle \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e^3 \sqrt {c-d x^2}}{x \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \int \frac {\sqrt {c-d x^2}}{e x \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\) |
| \(\Big \downarrow \) 969 |
| \(\displaystyle 2 e^3 \left (\frac {\sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}-\frac {\int -\frac {5 c e^2-3 d e^2 x^2}{e^3 x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {5 c e^2-3 d e^2 x^2}{e^3 x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^2}+\frac {\sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\int \frac {5 c e^2-3 d e^2 x^2}{e x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{4 a e^4}+\frac {\sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle 2 e^3 \left (\frac {-\frac {\int -\frac {c e x \left (5 b d x^2 e^2+(5 b c-8 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {5 \sqrt {c-d x^2}}{a \sqrt {e x}}}{4 a e^4}+\frac {\sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {c e x \left (5 b d x^2 e^2+(5 b c-8 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {5 \sqrt {c-d x^2}}{a \sqrt {e x}}}{4 a e^4}+\frac {\sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {e x \left (5 b d x^2 e^2+(5 b c-8 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a e^2}-\frac {5 \sqrt {c-d x^2}}{a \sqrt {e x}}}{4 a e^4}+\frac {\sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {\int \left (\frac {e \left (5 b c e^2-3 a d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {5 d e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{a e^2}-\frac {5 \sqrt {c-d x^2}}{a \sqrt {e x}}}{4 a e^4}+\frac {\sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e^3 \left (\frac {\frac {-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {5 c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {5 c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}}{a e^2}-\frac {5 \sqrt {c-d x^2}}{a \sqrt {e x}}}{4 a e^4}+\frac {\sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} \left (a e^2-b e^2 x^2\right )}\right )\) |
2*e^3*(Sqrt[c - d*x^2]/(4*a*e^2*Sqrt[e*x]*(a*e^2 - b*e^2*x^2)) + ((-5*Sqrt [c - d*x^2])/(a*Sqrt[e*x]) + ((-5*c^(3/4)*d^(1/4)*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] + (5*c^(3/4)*d^(1/4)*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] - (c^(1/4)*(5 *b*c - 3*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]*Sqrt[b]*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(5*b*c - 3*a*d)*e^(3/2 )*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcS in[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*Sqrt[a]*Sqrt[b]*d^(1/4) *Sqrt[c - d*x^2]))/(a*e^2))/(4*a*e^4))
3.10.1.3.1 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[(-(e*x)^(m + 1))*(a + b*x^n)^(p + 1)*((c + d*x^n )^q/(a*e*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[(e*x)^m*(a + b*x^n)^( p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 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[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. \(858\) vs. \(2(328)=656\).
Time = 3.09 (sec) , antiderivative size = 859, normalized size of antiderivative = 1.93
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {2 \left (-d e \,x^{2}+c e \right )}{e^{2} a^{2} \sqrt {x \left (-d e \,x^{2}+c e \right )}}+\frac {b x \sqrt {-d e \,x^{3}+c e x}}{2 a^{2} e^{2} \left (-b \,x^{2}+a \right )}+\frac {5 c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 e \,a^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {5 c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{4 e \,a^{2} \sqrt {-d e \,x^{3}+c e x}}+\frac {3 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{8 a e b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {5 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 a^{2} e d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{8 a e b \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {5 \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 a^{2} e d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\) | \(859\) |
| default | \(\text {Expression too large to display}\) | \(2556\) |
((-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 /a^2/(x*(-d*e*x^2+c*e))^(1/2)+1/2*b/a^2/e^2*x*(-d*e*x^3+c*e*x)^(1/2)/(-b*x ^2+a)+5/2/e/a^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)*EllipticE(((x+1/d*(c*d)^(1/ 2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-5/4/e/a^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)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+3/8/ a/e/b*(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*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b) ^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1 /2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))-5/8/a^2/e/d*(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*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*Elliptic Pi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^ (1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))*c+3/8/a/e/b*(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*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d) ^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^ (1/2)),1/2*2^(1/2))-5/8/a^2/e/d*(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*e*x)^(1/2...
Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int \frac {\sqrt {c - d x^{2}}}{\left (e x\right )^{\frac {3}{2}} \left (- a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx=\int \frac {\sqrt {c-d\,x^2}}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^2} \,d x \]