\(\int \frac {(e x)^{5/2}}{(a-b x^2)^2 (c-d x^2)^{5/2}} \, dx\) [1162]

Optimal result

Integrand size = 30, antiderivative size = 551 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt [4]{d} (4 b c+a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 \sqrt [4]{c} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (4 b c+a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 \sqrt [4]{c} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt {b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt {b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 \sqrt {a} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \] Output:

5/6*d*e*(e*x)^(3/2)/(-a*d+b*c)^2/(-d*x^2+c)^(3/2)+1/2*e*(e*x)^(3/2)/(-a*d+ 
b*c)/(-b*x^2+a)/(-d*x^2+c)^(3/2)+1/2*d*(a*d+4*b*c)*e*(e*x)^(3/2)/c/(-a*d+b 
*c)^3/(-d*x^2+c)^(1/2)-1/2*d^(1/4)*(a*d+4*b*c)*e^(5/2)*(1-d*x^2/c)^(1/2)*E 
llipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/c^(1/4)/(-a*d+b*c)^3/(-d*x 
^2+c)^(1/2)+1/2*d^(1/4)*(a*d+4*b*c)*e^(5/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^ 
(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/c^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2) 
+1/4*b^(1/2)*c^(1/4)*(7*a*d+3*b*c)*e^(5/2)*(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^(1 
/2)/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/4*b^(1/2)*c^(1/4)*(7*a*d+3*b*c 
)*e^(5/2)*(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^(1/2)/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c 
)^(1/2)
 

Mathematica [C] (verified)

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

Time = 11.32 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.50 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=-\frac {e (e x)^{3/2} \left (7 a \left (a^2 d^2 \left (c-3 d x^2\right )+a b d \left (11 c^2-10 c d x^2+3 d^2 x^4\right )+b^2 c \left (3 c^2-17 c d x^2+12 d^2 x^4\right )\right )+7 \left (3 b^2 c^2+11 a b c d+a^2 d^2\right ) \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b d (4 b c+a d) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a c (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \] Input:

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

Output:

-1/42*(e*(e*x)^(3/2)*(7*a*(a^2*d^2*(c - 3*d*x^2) + a*b*d*(11*c^2 - 10*c*d* 
x^2 + 3*d^2*x^4) + b^2*c*(3*c^2 - 17*c*d*x^2 + 12*d^2*x^4)) + 7*(3*b^2*c^2 
 + 11*a*b*c*d + a^2*d^2)*(-a + b*x^2)*(c - d*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*(4*b*c + a*d)*x^2*(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]))/(a*c*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {368, 27, 971, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1049

\(\displaystyle 2 e^3 \left (\frac {(e x)^{3/2}}{4 \left (c-d x^2\right )^{3/2} (b c-a d) \left (a e^2-b e^2 x^2\right )}-\frac {-\frac {\int -\frac {6 c e x \left (5 b d x^2 e^2+(3 b c+2 a d) e^2\right )}{\left (c-d x^2\right )^{3/2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{6 c (b c-a d)}-\frac {5 d (e x)^{3/2}}{3 \left (c-d x^2\right )^{3/2} (b c-a d)}}{4 e^2 (b c-a d)}\right )\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1049

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1054

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 971

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* 
((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Simp[e^n/(n*(b*c - a*d) 
*(p + 1))   Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - 
 n + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e 
, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n, m - n + 
 1] && GtQ[m - n + 1, 0] && 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. \(1392\) vs. \(2(429)=858\).

Time = 3.32 (sec) , antiderivative size = 1393, normalized size of antiderivative = 2.53

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1393\)
default	\(\text {Expression too large to display}\)	\(5066\)

Input:

int((e*x)^(5/2)/(-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^2*d*e^2/( 
a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*x*(-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d 
)+1/3*e^2/(a*d-b*c)^2/d*x*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2-1/2*d*e^3*x^2 
/c*(a*d+3*b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(-(x^2-c/d)*d*e*x)^(1 
/2)-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^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d 
-b*c)*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*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^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*Ell 
ipticF(((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^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*a*Elliptic 
E(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/4*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^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*a*EllipticF(((x 
+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-7/8*e^3/(a^2*d^2-2*a*b 
*c*d+b^2*c^2)/(a*d-b*c)*(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))*a-...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(e)*int((sqrt(x)*sqrt(c - d*x**2)*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)*e**2