Integrand size = 30, antiderivative size = 417 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=-\frac {2 c \sqrt {c-d x^2}}{a e \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{d} (b c+a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a b e^{3/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} (b c+a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{3/2} b^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}} \] Output:
-2*c*(-d*x^2+c)^(1/2)/a/e/(e*x)^(1/2)-2*c^(3/4)*d^(1/4)*(a*d+b*c)*(1-d*x^2 /c)^(1/2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/b/e^(3/2)/(-d *x^2+c)^(1/2)+2*c^(3/4)*d^(1/4)*(a*d+b*c)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1 /4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/b/e^(3/2)/(-d*x^2+c)^(1/2)-c^(1/4)*(- a*d+b*c)^2*(1-d*x^2/c)^(1/2)*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)/a^(3/2)/b^(3/2)/d^(1/4)/e^(3/2)/(-d* x^2+c)^(1/2)+c^(1/4)*(-a*d+b*c)^2*(1-d*x^2/c)^(1/2)*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)/a^(3/2)/b^(3/2 )/d^(1/4)/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 = 11.11 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.36 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\frac {x \left (-42 a c \left (c-d x^2\right )+14 c (b c-3 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+6 d (b c+a d) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{21 a^2 (e x)^{3/2} \sqrt {c-d x^2}} \] Input:
Integrate[(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)),x]
Output:
(x*(-42*a*c*(c - d*x^2) + 14*c*(b*c - 3*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*Appel lF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 6*d*(b*c + a*d)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(21*a^2*(e *x)^(3/2)*Sqrt[c - d*x^2])
Time = 0.76 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {368, 27, 974, 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.
| \(\displaystyle \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx\) |
| \(\Big \downarrow \) 368 |
| \(\displaystyle \frac {2 \int \frac {e \left (c-d x^2\right )^{3/2}}{x \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \int \frac {\left (c-d x^2\right )^{3/2}}{e x \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\) |
| \(\Big \downarrow \) 974 |
| \(\displaystyle 2 e \left (\frac {\int \frac {x \left (d (b c+a d) x^2 e^2+c (b c-3 a d) e^2\right )}{e \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a e^2}-\frac {c \sqrt {c-d x^2}}{a e^2 \sqrt {e x}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 e \left (\frac {\int \frac {e x \left (d (b c+a d) x^2 e^2+c (b c-3 a d) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a e^4}-\frac {c \sqrt {c-d x^2}}{a e^2 \sqrt {e x}}\right )\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle 2 e \left (\frac {\int \left (\frac {e \left (b^2 c^2 e^2+a^2 d^2 e^2-2 a b c d e^2\right ) x}{b \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d (b c+a d) e x}{b \sqrt {c-d x^2}}\right )d\sqrt {e x}}{a e^4}-\frac {c \sqrt {c-d x^2}}{a e^2 \sqrt {e x}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 e \left (\frac {-\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+b c) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt {c-d x^2}}}{a e^4}-\frac {c \sqrt {c-d x^2}}{a e^2 \sqrt {e x}}\right )\) |
Input:
Int[(c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)),x]
Output:
2*e*(-((c*Sqrt[c - d*x^2])/(a*e^2*Sqrt[e*x])) + (-((c^(3/4)*d^(1/4)*(b*c + a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^ (1/4)*Sqrt[e])], -1])/(b*Sqrt[c - d*x^2])) + (c^(3/4)*d^(1/4)*(b*c + a*d)* e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)* Sqrt[e])], -1])/(b*Sqrt[c - d*x^2]) - (c^(1/4)*(b*c - a*d)^2*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]*b^(3/2)*d^(1/4)*Sqr t[c - d*x^2]) + (c^(1/4)*(b*c - a*d)^2*e^(3/2)*Sqrt[1 - (d*x^2)/c]*Ellipti cPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/ 4)*Sqrt[e])], -1])/(2*Sqrt[a]*b^(3/2)*d^(1/4)*Sqrt[c - d*x^2]))/(a*e^4))
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[c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^ (q - 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1 ) + a*d*(q - 1)) + d*((c*b - a*d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q , 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
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. \(1291\) vs. \(2(317)=634\).
Time = 0.84 (sec) , antiderivative size = 1292, normalized size of antiderivative = 3.10
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1292\) |
| default | \(\text {Expression too large to display}\) | \(1743\) |
Input:
int((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)
Output:
((-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/a/(x*(-d*e*x^2+c*e))^(1/2)+2*c*d*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c* d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/e/b*Ellipt icE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-c*d*(1+x*d/(c*d )^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e* x^3+c*e*x)^(1/2)/e/b*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1 /2*2^(1/2))+2*c^2*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(- x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/e/a*EllipticE(((x+1/d*(c*d)^ (1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-c^2*(1+x*d/(c*d)^(1/2))^(1/2)*(2- 2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/e /a*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/2*a/ e/b^2*d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)* (-x*d/(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))+1/e/b*(c*d)^(1/2)*(1+ x*d/(c*d)^(1/2))^(1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(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))*c-1/2/a/e/d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2))^ (1/2)*(2-2*x*d/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*...
Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=- \int \frac {c \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {3}{2}} + b x^{2} \left (e x\right )^{\frac {3}{2}}}\, dx - \int \left (- \frac {d x^{2} \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {3}{2}} + b x^{2} \left (e x\right )^{\frac {3}{2}}}\right )\, dx \] Input:
integrate((-d*x**2+c)**(3/2)/(e*x)**(3/2)/(-b*x**2+a),x)
Output:
-Integral(c*sqrt(c - d*x**2)/(-a*(e*x)**(3/2) + b*x**2*(e*x)**(3/2)), x) - Integral(-d*x**2*sqrt(c - d*x**2)/(-a*(e*x)**(3/2) + b*x**2*(e*x)**(3/2)) , x)
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int { -\frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="maxima")
Output:
-integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)*(e*x)^(3/2)), x)
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int { -\frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a),x, algorithm="giac")
Output:
integrate(-(-d*x^2 + c)^(3/2)/((b*x^2 - a)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\int \frac {{\left (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{3/2}\,\left (a-b\,x^2\right )} \,d x \] Input:
int((c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)),x)
Output:
int((c - d*x^2)^(3/2)/((e*x)^(3/2)*(a - b*x^2)), x)
\[ \int \frac {\left (c-d x^2\right )^{3/2}}{(e x)^{3/2} \left (a-b x^2\right )} \, dx=\frac {\sqrt {e}\, \left (4 \sqrt {-d \,x^{2}+c}\, c d +\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{a b \,d^{2} x^{4}-b^{2} c d \,x^{4}-a^{2} d^{2} x^{2}+b^{2} c^{2} x^{2}+a^{2} c d -a b \,c^{2}}d x \right ) a^{2} d^{4}-4 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{a b \,d^{2} x^{4}-b^{2} c d \,x^{4}-a^{2} d^{2} x^{2}+b^{2} c^{2} x^{2}+a^{2} c d -a b \,c^{2}}d x \right ) a b c \,d^{3}+3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{a b \,d^{2} x^{4}-b^{2} c d \,x^{4}-a^{2} d^{2} x^{2}+b^{2} c^{2} x^{2}+a^{2} c d -a b \,c^{2}}d x \right ) b^{2} c^{2} d^{2}+3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{a b \,d^{2} x^{6}-b^{2} c d \,x^{6}-a^{2} d^{2} x^{4}+b^{2} c^{2} x^{4}+a^{2} c d \,x^{2}-a b \,c^{2} x^{2}}d x \right ) a^{2} c^{2} d^{2}-4 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{a b \,d^{2} x^{6}-b^{2} c d \,x^{6}-a^{2} d^{2} x^{4}+b^{2} c^{2} x^{4}+a^{2} c d \,x^{2}-a b \,c^{2} x^{2}}d x \right ) a b \,c^{3} d +\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{a b \,d^{2} x^{6}-b^{2} c d \,x^{6}-a^{2} d^{2} x^{4}+b^{2} c^{2} x^{4}+a^{2} c d \,x^{2}-a b \,c^{2} x^{2}}d x \right ) b^{2} c^{4}\right )}{\sqrt {x}\, e^{2} \left (a d -b c \right )} \] Input:
int((-d*x^2+c)^(3/2)/(e*x)^(3/2)/(-b*x^2+a),x)
Output:
(sqrt(e)*(4*sqrt(c - d*x**2)*c*d + sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x **2)/(a**2*c*d - a**2*d**2*x**2 - a*b*c**2 + a*b*d**2*x**4 + b**2*c**2*x** 2 - b**2*c*d*x**4),x)*a**2*d**4 - 4*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)* x**2)/(a**2*c*d - a**2*d**2*x**2 - a*b*c**2 + a*b*d**2*x**4 + b**2*c**2*x* *2 - b**2*c*d*x**4),x)*a*b*c*d**3 + 3*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2 )*x**2)/(a**2*c*d - a**2*d**2*x**2 - a*b*c**2 + a*b*d**2*x**4 + b**2*c**2* x**2 - b**2*c*d*x**4),x)*b**2*c**2*d**2 + 3*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2))/(a**2*c*d*x**2 - a**2*d**2*x**4 - a*b*c**2*x**2 + a*b*d**2*x**6 + b**2*c**2*x**4 - b**2*c*d*x**6),x)*a**2*c**2*d**2 - 4*sqrt(x)*int((sqrt(x )*sqrt(c - d*x**2))/(a**2*c*d*x**2 - a**2*d**2*x**4 - a*b*c**2*x**2 + a*b* d**2*x**6 + b**2*c**2*x**4 - b**2*c*d*x**6),x)*a*b*c**3*d + sqrt(x)*int((s qrt(x)*sqrt(c - d*x**2))/(a**2*c*d*x**2 - a**2*d**2*x**4 - a*b*c**2*x**2 + a*b*d**2*x**6 + b**2*c**2*x**4 - b**2*c*d*x**6),x)*b**2*c**4))/(sqrt(x)*e **2*(a*d - b*c))