\(\int \frac {\sqrt {e x}}{(a-b x^2)^2 (c-d x^2)^{5/2}} \, dx\) [1163]

Optimal result

Integrand size = 30, antiderivative size = 625 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac {b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt {c-d x^2}}-\frac {\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a c^{5/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a c^{5/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \] Output:

1/6*d*(2*a*d+3*b*c)*(e*x)^(3/2)/a/c/(-a*d+b*c)^2/e/(-d*x^2+c)^(3/2)+1/2*b* 
(e*x)^(3/2)/a/(-a*d+b*c)/e/(-b*x^2+a)/(-d*x^2+c)^(3/2)+1/2*d*(-a^2*d^2+5*a 
*b*c*d+b^2*c^2)*(e*x)^(3/2)/a/c^2/(-a*d+b*c)^3/e/(-d*x^2+c)^(1/2)-1/2*d^(1 
/4)*(-a^2*d^2+5*a*b*c*d+b^2*c^2)*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^(5/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)+ 
1/2*d^(1/4)*(-a^2*d^2+5*a*b*c*d+b^2*c^2)*e^(1/2)*(1-d*x^2/c)^(1/2)*Ellipti 
cF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/c^(5/4)/(-a*d+b*c)^3/(-d*x^2+c 
)^(1/2)-1/4*b^(3/2)*c^(1/4)*(-11*a*d+b*c)*e^(1/2)*(1-d*x^2/c)^(1/2)*Ellipt 
icPi(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)^3/(-d*x^2+c)^(1/2)+1/4*b^(3/2)*c^(1/4)*(-11* 
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)^3/(- 
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 = 11.38 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {\sqrt {e x} \left (7 a x \left (a b^2 c d^2 x^2 \left (17 c-15 d x^2\right )+a^3 d^3 \left (5 c-3 d x^2\right )-3 b^3 c^2 \left (c-d x^2\right )^2+a^2 b d^2 \left (-17 c^2+10 c d x^2+3 d^2 x^4\right )\right )+7 \left (b^3 c^3-12 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) x \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) x^3 \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a^2 c^2 (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \] Input:

Integrate[Sqrt[e*x]/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
 

Output:

(Sqrt[e*x]*(7*a*x*(a*b^2*c*d^2*x^2*(17*c - 15*d*x^2) + a^3*d^3*(5*c - 3*d* 
x^2) - 3*b^3*c^2*(c - d*x^2)^2 + a^2*b*d^2*(-17*c^2 + 10*c*d*x^2 + 3*d^2*x 
^4)) + 7*(b^3*c^3 - 12*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*x*(-a + b*x^ 
2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, ( 
b*x^2)/a] + 3*b*d*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*x^3*(-a + b*x^2)*(c - d* 
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^2*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {368, 27, 972, 27, 1049, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 972

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1049

\(\displaystyle 2 e^3 \left (\frac {\frac {d (e x)^{3/2} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {6 e x \left (\left (b^2 c^2-8 a b d c+2 a^2 d^2\right ) e^2-b d (3 b c+2 a d) e^2 x^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 c (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1049

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}-\frac {\int -\frac {2 e x \left (b d \left (b^2 c^2+5 a b d c-a^2 d^2\right ) x^2 e^2+\left (b^3 c^3-12 a b^2 d c^2-5 a^2 b d^2 c+a^3 d^3\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{2 c (b c-a d)}}{c (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\int \frac {e x \left (b d \left (b^2 c^2+5 a b d c-a^2 d^2\right ) x^2 e^2+\left (b^3 c^3-12 a b^2 d c^2-5 a^2 b d^2 c+a^3 d^3\right ) e^2\right )}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{c (b c-a d)}+\frac {d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 1054

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\int \left (\frac {e \left (b^3 c^3 e^2-11 a b^2 c^2 d e^2\right ) x}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}-\frac {d \left (b^2 c^2+5 a b d c-a^2 d^2\right ) e x}{\sqrt {c-d x^2}}\right )d\sqrt {e x}}{c (b c-a d)}+\frac {d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 e^3 \left (\frac {\frac {\frac {\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {c-d x^2}}-\frac {b^{3/2} c^{9/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-11 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {b^{3/2} c^{9/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-11 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt {a} \sqrt [4]{d} \sqrt {c-d x^2}}}{c (b c-a d)}+\frac {d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{c \sqrt {c-d x^2} (b c-a d)}}{c (b c-a d)}+\frac {d (e x)^{3/2} (2 a d+3 b c)}{3 c \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 a e^4 (b c-a d)}+\frac {b (e x)^{3/2}}{4 a e^2 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}\right )\)

Input:

Int[Sqrt[e*x]/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]
 

Output:

2*e^3*((b*(e*x)^(3/2))/(4*a*(b*c - a*d)*e^2*(c - d*x^2)^(3/2)*(a*e^2 - b*e 
^2*x^2)) + ((d*(3*b*c + 2*a*d)*(e*x)^(3/2))/(3*c*(b*c - a*d)*(c - d*x^2)^( 
3/2)) + ((d*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*(e*x)^(3/2))/(c*(b*c - a*d)*Sq 
rt[c - d*x^2]) + (-((c^(3/4)*d^(1/4)*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*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)*(b^2*c^2 + 5*a*b*c*d - a^2* 
d^2)*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] - (b^(3/2)*c^(9/4)*(b*c - 11*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)*S 
qrt[c - d*x^2]) + (b^(3/2)*c^(9/4)*(b*c - 11*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]))/(c*(b*c 
 - a*d)))/(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. \(1517\) vs. \(2(503)=1006\).

Time = 3.29 (sec) , antiderivative size = 1518, normalized size of antiderivative = 2.43

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1518\)
default	\(\text {Expression too large to display}\)	\(5677\)

Input:

int((e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),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^3*d/(a*d- 
b*c)/a/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+ 
1/3/(a*d-b*c)^2/c*x*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2+1/2*d^2*e*x^2/c^2*( 
a*d-5*b*c)/(a*d-b*c)^3/(-(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^2*e/(a*d-b*c)/a/(a^2*d^2-2*a*b*c*d+b^2*c^2)*EllipticE(((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^2*e/(a*d-b*c)/a/(a^2*d^2-2*a*b*c*d+b^2*c^2)*EllipticF(((x+1/d*(c*d) 
^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/2*d^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)*e/(a*d-b*c)^3*a*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2), 
1/2*2^(1/2))-1/4*d^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)*e/(a*d-b*c)^3*a*Ellipti 
cF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-5/2*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)^3*b*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^( 
1/2))^(1/2),1/2*2^(1/2))+5/4*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)^3*b 
*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-11/8*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((e*x)**(1/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)
 

Output:

Integral(sqrt(e*x)/((-a + b*x**2)**2*(c - d*x**2)**(5/2)), x)
 

Maxima [F]

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

integrate((e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(e*x)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)
 

Giac [F]

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

integrate((e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="giac")
 

Output:

integrate(sqrt(e*x)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)
 

Mupad [F(-1)]

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

int((e*x)^(1/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x)
 

Output:

int((e*x)^(1/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {-d \,x^{2}+c}}{-b^{2} d^{3} x^{10}+2 a b \,d^{3} x^{8}+3 b^{2} c \,d^{2} x^{8}-a^{2} d^{3} x^{6}-6 a b c \,d^{2} x^{6}-3 b^{2} c^{2} d \,x^{6}+3 a^{2} c \,d^{2} x^{4}+6 a b \,c^{2} d \,x^{4}+b^{2} c^{3} x^{4}-3 a^{2} c^{2} d \,x^{2}-2 a b \,c^{3} x^{2}+a^{2} c^{3}}d x \right ) \] Input:

int((e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)
 

Output:

sqrt(e)*int((sqrt(x)*sqrt(c - d*x**2))/(a**2*c**3 - 3*a**2*c**2*d*x**2 + 3 
*a**2*c*d**2*x**4 - a**2*d**3*x**6 - 2*a*b*c**3*x**2 + 6*a*b*c**2*d*x**4 - 
 6*a*b*c*d**2*x**6 + 2*a*b*d**3*x**8 + b**2*c**3*x**4 - 3*b**2*c**2*d*x**6 
 + 3*b**2*c*d**2*x**8 - b**2*d**3*x**10),x)