\(\int \frac {\sqrt {e x}}{(a-b x^2)^2 (c-d x^2)^{3/2}} \, dx\) [1154]

Optimal result

Integrand size = 30, antiderivative size = 531 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {d (b c+2 a d) (e x)^{3/2}}{2 a c (b c-a d)^2 e \sqrt {c-d x^2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\sqrt [4]{d} (b c+2 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a \sqrt [4]{c} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (b c+2 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{c} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (b c-7 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (b c-7 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}} \] Output:

1/2*d*(2*a*d+b*c)*(e*x)^(3/2)/a/c/(-a*d+b*c)^2/e/(-d*x^2+c)^(1/2)+1/2*b*(e 
*x)^(3/2)/a/(-a*d+b*c)/e/(-b*x^2+a)/(-d*x^2+c)^(1/2)-1/2*d^(1/4)*(2*a*d+b* 
c)*e^(1/2)*(1-d*x^2/c)^(1/2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2) 
,I)/a/c^(1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)+1/2*d^(1/4)*(2*a*d+b*c)*e^(1/2 
)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/c^( 
1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2)-1/4*b^(1/2)*c^(1/4)*(-7*a*d+b*c)*e^(1/2 
)*(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)/d^(1/4)/(-a*d+b*c)^2/(-d*x^2+c)^(1/2) 
+1/4*b^(1/2)*c^(1/4)*(-7*a*d+b*c)*e^(1/2)*(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 
)/d^(1/4)/(-a*d+b*c)^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 = 11.23 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e x} \left (21 a x \left (-2 a^2 d^2+2 a b d^2 x^2+b^2 c \left (-c+d x^2\right )\right )+7 \left (-b^2 c^2+8 a b c d+2 a^2 d^2\right ) x \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d (b c+2 a d) x^3 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^2 c (b c-a d)^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

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

Output:

(Sqrt[e*x]*(21*a*x*(-2*a^2*d^2 + 2*a*b*d^2*x^2 + b^2*c*(-c + d*x^2)) + 7*( 
-(b^2*c^2) + 8*a*b*c*d + 2*a^2*d^2)*x*(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*(b*c + 2*a*d)*x^3*(-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^2*c*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 519, normalized size of antiderivative = 0.98, 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, 1049, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 972

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1049

\(\displaystyle 2 e^3 \left (\frac {\frac {d (e x)^{3/2} (2 a d+b c)}{c \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 e x \left (b d (b c+2 a d) x^2 e^2+\left (b^2 c^2-8 a b d c-2 a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\frac {\int \frac {e x \left (b d (b c+2 a d) x^2 e^2+\left (b^2 c^2-8 a b d c-2 a^2 d^2\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+b c)}{c \sqrt {c-d x^2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \sqrt {c-d x^2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

2*e^3*((b*(e*x)^(3/2))/(4*a*(b*c - a*d)*e^2*Sqrt[c - d*x^2]*(a*e^2 - b*e^2 
*x^2)) + ((d*(b*c + 2*a*d)*(e*x)^(3/2))/(c*(b*c - a*d)*Sqrt[c - d*x^2]) + 
(-((c^(3/4)*d^(1/4)*(b*c + 2*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[Ar 
cSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/Sqrt[c - d*x^2]) + (c^(3 
/4)*d^(1/4)*(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] - (Sqrt[b]*c^(5/ 
4)*(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]) + (Sqrt[b]*c^(5/4)*(b*c - 7*a*d)*e^(3/ 
2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), Arc 
Sin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*Sqrt[a]*d^(1/4)*Sqrt[c 
 - d*x^2]))/(c*(b*c - a*d)))/(4*a*(b*c - a*d)*e^4))
 

Defintions of rubi rules used

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 1049

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]
 

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. \(1096\) vs. \(2(415)=830\).

Time = 2.18 (sec) , antiderivative size = 1097, normalized size of antiderivative = 2.07

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1097\)
default	\(\text {Expression too large to display}\)	\(2938\)

Input:

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

Output:

1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*b^2/a/(a*d- 
b*c)^2*x*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)+d^2*e*x^2/c/(a*d-b*c)^2/(-(x^2- 
c/d)*d*e*x)^(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)*b*e/(a*d-b*c)^2/a*Elli 
pticE(((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)*b*e/(a*d-b*c)^2/a*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c 
*d)^(1/2))^(1/2),1/2*2^(1/2))+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/(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*(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*d-b*c)^2*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c 
*d)^(1/2))^(1/2),1/2*2^(1/2))+7/8*e/(a*d-b*c)^2*(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))-1/8*e/(a*d-b*c)^2/a/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)...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{3/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{b^{2} d^{2} x^{8}-2 a b \,d^{2} x^{6}-2 b^{2} c d \,x^{6}+a^{2} d^{2} x^{4}+4 a b c d \,x^{4}+b^{2} c^{2} x^{4}-2 a^{2} c d \,x^{2}-2 a b \,c^{2} x^{2}+a^{2} c^{2}}d x \right ) \] Input:

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

Output:

sqrt(e)*int((sqrt(x)*sqrt(c - d*x**2))/(a**2*c**2 - 2*a**2*c*d*x**2 + a**2 
*d**2*x**4 - 2*a*b*c**2*x**2 + 4*a*b*c*d*x**4 - 2*a*b*d**2*x**6 + b**2*c** 
2*x**4 - 2*b**2*c*d*x**6 + b**2*d**2*x**8),x)