\(\int \frac {1}{(e x)^{3/2} (a-b x^2)^2 \sqrt {c-d x^2}} \, dx\) [1146]

Optimal result

Integrand size = 30, antiderivative size = 535 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=-\frac {(5 b c-4 a d) \sqrt {c-d x^2}}{2 a^2 c (b c-a d) e \sqrt {e x}}+\frac {b \sqrt {c-d x^2}}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right )}-\frac {\sqrt [4]{d} (5 b c-4 a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (5 b c-4 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a^2 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (5 b c-7 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (5 b c-7 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}} \] Output:

-1/2*(-4*a*d+5*b*c)*(-d*x^2+c)^(1/2)/a^2/c/(-a*d+b*c)/e/(e*x)^(1/2)+1/2*b* 
(-d*x^2+c)^(1/2)/a/(-a*d+b*c)/e/(e*x)^(1/2)/(-b*x^2+a)-1/2*d^(1/4)*(-4*a*d 
+5*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^2/c^(1/4)/(-a*d+b*c)/e^(3/2)/(-d*x^2+c)^(1/2)+1/2*d^(1/4)*(-4*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)/(-a*d+b*c)/e^(3/2)/(-d*x^2+c)^(1/2)-1/4*b^(1/2)*c^(1/4)*(-7*a*d+5*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)/(-a*d+b*c)/e^(3/2)/(-d*x^2+ 
c)^(1/2)+1/4*b^(1/2)*c^(1/4)*(-7*a*d+5*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)/(-a*d+b*c)/e^(3/2)/(-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 = 11.21 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.44 \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {x \left (-21 a \left (c-d x^2\right ) \left (4 a^2 d+5 b^2 c x^2-4 a b \left (c+d x^2\right )\right )+7 \left (5 b^2 c^2-12 a b c d+4 a^2 d^2\right ) x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d (-5 b c+4 a d) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^3 c (b c-a d) (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

Integrate[1/((e*x)^(3/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]
 

Output:

(x*(-21*a*(c - d*x^2)*(4*a^2*d + 5*b^2*c*x^2 - 4*a*b*(c + d*x^2)) + 7*(5*b 
^2*c^2 - 12*a*b*c*d + 4*a^2*d^2)*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*Appe 
llF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*b*d*(-5*b*c + 4*a*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*c*(b*c - a*d)*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {368, 27, 972, 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.

\(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx\)

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \int \frac {1}{e x \sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )^2}d\sqrt {e x}\)

\(\Big \downarrow \) 972

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1053

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1054

\(\displaystyle 2 e^3 \left (\frac {\frac {\int \left (\frac {e \left (5 b^2 c^2 e^2-7 a b c d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d (5 b c-4 a d) e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{a c e^2}-\frac {\sqrt {c-d x^2} (5 b c-4 a d)}{a c \sqrt {e x}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-4 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-4 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}-\frac {\sqrt {b} c^{5/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {b} c^{5/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}}{a c e^2}-\frac {\sqrt {c-d x^2} (5 b c-4 a d)}{a c \sqrt {e x}}}{4 a e^4 (b c-a d)}+\frac {b \sqrt {c-d x^2}}{4 a e^2 \sqrt {e x} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

Input:

Int[1/((e*x)^(3/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]
 

Output:

2*e^3*((b*Sqrt[c - d*x^2])/(4*a*(b*c - a*d)*e^2*Sqrt[e*x]*(a*e^2 - b*e^2*x 
^2)) + (-(((5*b*c - 4*a*d)*Sqrt[c - d*x^2])/(a*c*Sqrt[e*x])) + (-((c^(3/4) 
*d^(1/4)*(5*b*c - 4*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 - 4*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*S 
qrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^2] - (Sqrt[b]*c^(5/4)*(5*b 
*c - 7*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sq 
rt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*Sq 
rt[a]*d^(1/4)*Sqrt[c - d*x^2]) + (Sqrt[b]*c^(5/4)*(5*b*c - 7*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))/(4*a*(b*c - a*d)*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 972

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]
 

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. \(1099\) vs. \(2(419)=838\).

Time = 1.90 (sec) , antiderivative size = 1100, normalized size of antiderivative = 2.06

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1100\)
default	\(\text {Expression too large to display}\)	\(2970\)

Input:

int(1/(e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(-1/2*b^2/a^2/(a*d-b*c 
)/e^2*x*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)-2*(-d*e*x^2+c*e)/e^2/c/a^2/(x*(- 
d*e*x^2+c*e))^(1/2)-1/2*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)/(a*d-b*c)*b/a^2/e*El 
lipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/4*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)/(a*d-b*c)*b/a^2/e*EllipticF(((x+1/d*(c*d)^(1/2))*d/ 
(c*d)^(1/2))^(1/2),1/2*2^(1/2))+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^2*Ellipt 
icE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(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)/e/a^2*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2 
*2^(1/2))-7/8/(a*d-b*c)/a/e*(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))+5 
/8/(a*d-b*c)/a^2/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*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-7...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {3}{2}} \left (- a + b x^{2}\right )^{2} \sqrt {c - d x^{2}}}\, dx \] Input:

integrate(1/(e*x)**(3/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)
 

Output:

Integral(1/((e*x)**(3/2)*(-a + b*x**2)**2*sqrt(c - d*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="maxima 
")
 

Output:

integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(3/2)), x)
 

Giac [F]

\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^2\,\sqrt {c-d\,x^2}} \,d x \] Input:

int(1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(1/2)),x)
 

Output:

int(1/((e*x)^(3/2)*(a - b*x^2)^2*(c - d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{(e x)^{3/2} \left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {-d \,x^{2}+c}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b d +3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}\, x^{2}}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} d \,x^{2}-\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a^{2} d +5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b c +\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) a b d \,x^{2}-5 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b^{2} d \,x^{6}+2 a b d \,x^{4}+b^{2} c \,x^{4}-a^{2} d \,x^{2}-2 a b c \,x^{2}+a^{2} c}d x \right ) b^{2} c \,x^{2}\right )}{\sqrt {x}\, a c \,e^{2} \left (-b \,x^{2}+a \right )} \] Input:

int(1/(e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x)
 

Output:

(sqrt(e)*( - 2*sqrt(c - d*x**2) - 3*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)* 
x**2)/(a**2*c - a**2*d*x**2 - 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 - 
b**2*d*x**6),x)*a*b*d + 3*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2)*x**2)/(a** 
2*c - a**2*d*x**2 - 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 - b**2*d*x** 
6),x)*b**2*d*x**2 - sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2))/(a**2*c - a**2* 
d*x**2 - 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 - b**2*d*x**6),x)*a**2* 
d + 5*sqrt(x)*int((sqrt(x)*sqrt(c - d*x**2))/(a**2*c - a**2*d*x**2 - 2*a*b 
*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 - b**2*d*x**6),x)*a*b*c + sqrt(x)*int 
((sqrt(x)*sqrt(c - d*x**2))/(a**2*c - a**2*d*x**2 - 2*a*b*c*x**2 + 2*a*b*d 
*x**4 + b**2*c*x**4 - b**2*d*x**6),x)*a*b*d*x**2 - 5*sqrt(x)*int((sqrt(x)* 
sqrt(c - d*x**2))/(a**2*c - a**2*d*x**2 - 2*a*b*c*x**2 + 2*a*b*d*x**4 + b* 
*2*c*x**4 - b**2*d*x**6),x)*b**2*c*x**2))/(sqrt(x)*a*c*e**2*(a - b*x**2))