\(\int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} (a-b x^2)} \, dx\) [1098]

Optimal result

Integrand size = 30, antiderivative size = 457 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=-\frac {2 \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-2 a d) \sqrt {c-d x^2}}{5 a^2 c e^3 \sqrt {e x}}-\frac {2 \sqrt [4]{d} (5 b c-2 a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}+\frac {2 \sqrt [4]{d} (5 b c-2 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}} \] Output:

-2/5*(-d*x^2+c)^(1/2)/a/e/(e*x)^(5/2)-2/5*(-2*a*d+5*b*c)*(-d*x^2+c)^(1/2)/ 
a^2/c/e^3/(e*x)^(1/2)-2/5*d^(1/4)*(-2*a*d+5*b*c)*(1-d*x^2/c)^(1/2)*Ellipti 
cE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a^2/c^(1/4)/e^(7/2)/(-d*x^2+c)^( 
1/2)+2/5*d^(1/4)*(-2*a*d+5*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^2/c^(1/4)/e^(7/2)/(-d*x^2+c)^(1/2)-b^(1/2)*c^(1 
/4)*(-a*d+b*c)*(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^(5/2)/d^(1/4)/e^(7/2)/(-d*x^2+ 
c)^(1/2)+b^(1/2)*c^(1/4)*(-a*d+b*c)*(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^(5/2)/d^(1 
/4)/e^(7/2)/(-d*x^2+c)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.18 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\frac {x \left (14 \left (5 b^2 c^2-10 a b c d+2 a^2 d^2\right ) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-6 \left (7 a \left (c-d x^2\right ) \left (a c+5 b c x^2-2 a d x^2\right )+b d (-5 b c+2 a d) x^6 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )\right )}{105 a^3 c (e x)^{7/2} \sqrt {c-d x^2}} \] Input:

Integrate[Sqrt[c - d*x^2]/((e*x)^(7/2)*(a - b*x^2)),x]
 

Output:

(x*(14*(5*b^2*c^2 - 10*a*b*c*d + 2*a^2*d^2)*x^4*Sqrt[1 - (d*x^2)/c]*Appell 
F1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] - 6*(7*a*(c - d*x^2)*(a*c + 5*b 
*c*x^2 - 2*a*d*x^2) + b*d*(-5*b*c + 2*a*d)*x^6*Sqrt[1 - (d*x^2)/c]*AppellF 
1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])))/(105*a^3*c*(e*x)^(7/2)*Sqrt[ 
c - d*x^2])
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {368, 27, 975, 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.

\(\displaystyle \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx\)

\(\Big \downarrow \) 368

\(\displaystyle \frac {2 \int \frac {\sqrt {c-d x^2}}{e x^3 \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{e}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e \int \frac {\sqrt {c-d x^2}}{e^3 x^3 \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}\)

\(\Big \downarrow \) 975

\(\displaystyle 2 e \left (\frac {\int \frac {(5 b c-2 a d) e^2-3 b d e^2 x^2}{e^3 x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{5 a e^2}-\frac {\sqrt {c-d x^2}}{5 a e^2 (e x)^{5/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e \left (\frac {\int \frac {(5 b c-2 a d) e^2-3 b d e^2 x^2}{e x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{5 a e^4}-\frac {\sqrt {c-d x^2}}{5 a e^2 (e x)^{5/2}}\right )\)

\(\Big \downarrow \) 1053

\(\displaystyle 2 e \left (\frac {-\frac {\int -\frac {e x \left (b d (5 b c-2 a d) x^2 e^2+\left (5 b^2 c^2-10 a b d c+2 a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (5 b c-2 a d)}{a c \sqrt {e x}}}{5 a e^4}-\frac {\sqrt {c-d x^2}}{5 a e^2 (e x)^{5/2}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 e \left (\frac {\frac {\int \frac {e x \left (b d (5 b c-2 a d) x^2 e^2+\left (5 b^2 c^2-10 a b d c+2 a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (5 b c-2 a d)}{a c \sqrt {e x}}}{5 a e^4}-\frac {\sqrt {c-d x^2}}{5 a e^2 (e x)^{5/2}}\right )\)

\(\Big \downarrow \) 1054

\(\displaystyle 2 e \left (\frac {\frac {\int \left (\frac {5 e \left (b^2 c^2 e^2-a b c d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d (5 b c-2 a d) e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (5 b c-2 a d)}{a c \sqrt {e x}}}{5 a e^4}-\frac {\sqrt {c-d x^2}}{5 a e^2 (e x)^{5/2}}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 e \left (\frac {\frac {\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-2 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-2 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}-\frac {5 \sqrt {b} c^{5/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {5 \sqrt {b} c^{5/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}}{a c e^2}-\frac {\sqrt {c-d x^2} (5 b c-2 a d)}{a c \sqrt {e x}}}{5 a e^4}-\frac {\sqrt {c-d x^2}}{5 a e^2 (e x)^{5/2}}\right )\)

Input:

Int[Sqrt[c - d*x^2]/((e*x)^(7/2)*(a - b*x^2)),x]
 

Output:

2*e*(-1/5*Sqrt[c - d*x^2]/(a*e^2*(e*x)^(5/2)) + (-(((5*b*c - 2*a*d)*Sqrt[c 
 - d*x^2])/(a*c*Sqrt[e*x])) + (-((c^(3/4)*d^(1/4)*(5*b*c - 2*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])/Sqrt[c - d*x^2]) + (c^(3/4)*d^(1/4)*(5*b*c - 2*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] 
)/Sqrt[c - d*x^2] - (5*Sqrt[b]*c^(5/4)*(b*c - a*d)*e^(3/2)*Sqrt[1 - (d*x^2 
)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sq 
rt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*Sqrt[a]*d^(1/4)*Sqrt[c - d*x^2]) + (5 
*Sqrt[b]*c^(5/4)*(b*c - 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]))/(a*c*e^2))/(5*a*e^4))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 368

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]
 

rule 975

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*(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 - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m 
 + 1) + b*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && 
 NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomi 
alQ[a, b, c, d, e, m, n, p, q, x]
 

rule 1053

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]
 

rule 1054

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1029\) vs. \(2(345)=690\).

Time = 1.19 (sec) , antiderivative size = 1030, normalized size of antiderivative = 2.25

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1030\)
default	\(\text {Expression too large to display}\)	\(1542\)

Input:

int((-d*x^2+c)^(1/2)/(e*x)^(7/2)/(-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)*(-2/5/e^4/a*(-d*e*x^3+ 
c*e*x)^(1/2)/x^3+2/5*(-d*e*x^2+c*e)/e^4/c*(2*a*d-5*b*c)/a^2/(x*(-d*e*x^2+c 
*e))^(1/2)-4/5/a/e^3*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)*EllipticE(((x+1/d*(c*d) 
^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+2/a^2/e^3*c*(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)*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b 
+2/5/a/e^3*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)*EllipticF(((x+1/d*(c*d)^(1/2))*d/ 
(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/a^2/e^3*c*(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)*El 
lipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b+1/2/e^3/a 
*(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))-1/2/e^3/a^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))*b*c+1/2/e^3/a*(c*d)^(1/2)*(1+x*d/(c*d)^(...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((-d*x^2+c)^(1/2)/(e*x)^(7/2)/(-b*x^2+a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=- \int \frac {\sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {7}{2}} + b x^{2} \left (e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((-d*x**2+c)**(1/2)/(e*x)**(7/2)/(-b*x**2+a),x)
 

Output:

-Integral(sqrt(c - d*x**2)/(-a*(e*x)**(7/2) + b*x**2*(e*x)**(7/2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\int { -\frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((-d*x^2+c)^(1/2)/(e*x)^(7/2)/(-b*x^2+a),x, algorithm="maxima")
 

Output:

-integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(7/2)), x)
 

Giac [F]

\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\int { -\frac {\sqrt {-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((-d*x^2+c)^(1/2)/(e*x)^(7/2)/(-b*x^2+a),x, algorithm="giac")
 

Output:

integrate(-sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\int \frac {\sqrt {c-d\,x^2}}{{\left (e\,x\right )}^{7/2}\,\left (a-b\,x^2\right )} \,d x \] Input:

int((c - d*x^2)^(1/2)/((e*x)^(7/2)*(a - b*x^2)),x)
 

Output:

int((c - d*x^2)^(1/2)/((e*x)^(7/2)*(a - b*x^2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {c-d x^2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {-d \,x^{2}+c}}{\sqrt {x}\, a \,x^{3}-\sqrt {x}\, b \,x^{5}}d x \right )}{e^{4}} \] Input:

int((-d*x^2+c)^(1/2)/(e*x)^(7/2)/(-b*x^2+a),x)
 

Output:

(sqrt(e)*int(sqrt(c - d*x**2)/(sqrt(x)*a*x**3 - sqrt(x)*b*x**5),x))/e**4