\(\int \frac {(e x)^{3/2} (c-d x^2)^{3/2}}{(a-b x^2)^2} \, dx\) [1132]

Optimal result

Integrand size = 30, antiderivative size = 381 \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=-\frac {7 d e \sqrt {e x} \sqrt {c-d x^2}}{6 b^2}+\frac {e \sqrt {e x} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {\sqrt [4]{c} d^{3/4} (17 b c-21 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 b^3 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-7 a d) (b c-a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b^3 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-7 a d) (b c-a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b^3 \sqrt [4]{d} \sqrt {c-d x^2}} \] Output:

-7/6*d*e*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^2+1/2*e*(e*x)^(1/2)*(-d*x^2+c)^(3/ 
2)/b/(-b*x^2+a)-1/6*c^(1/4)*d^(3/4)*(-21*a*d+17*b*c)*e^(3/2)*(1-d*x^2/c)^( 
1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/b^3/(-d*x^2+c)^(1/2) 
-1/4*c^(1/4)*(-7*a*d+b*c)*(-a*d+b*c)*e^(3/2)*(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/ 
b^3/d^(1/4)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-7*a*d+b*c)*(-a*d+b*c)*e^(3/2)*( 
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/b^3/d^(1/4)/(-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 = 11.18 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.51 \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {e \sqrt {e x} \left (5 a \left (c-d x^2\right ) \left (-3 b c+7 a d-4 b d x^2\right )-5 c (-3 b c+7 a d) \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+d (-17 b c+21 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a b^2 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

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

Output:

(e*Sqrt[e*x]*(5*a*(c - d*x^2)*(-3*b*c + 7*a*d - 4*b*d*x^2) - 5*c*(-3*b*c + 
 7*a*d)*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2) 
/c, (b*x^2)/a] + d*(-17*b*c + 21*a*d)*x^2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]* 
AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a*b^2*(-a + b*x^2)* 
Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {368, 27, 967, 27, 1025, 25, 1021, 765, 762, 925, 27, 1543, 1542}

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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 967

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1025

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}-\frac {\int -\frac {c (3 b c-7 a d) e^2-d (17 b c-21 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b}}{4 b e^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\int \frac {c (3 b c-7 a d) e^2-d (17 b c-21 a d) e^2 x^2}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{3 b}+\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\)

\(\Big \downarrow \) 1021

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 e^2 (b c-7 a d) (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {d (17 b c-21 a d) \int \frac {1}{\sqrt {c-d x^2}}d\sqrt {e x}}{b}}{3 b}+\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\)

\(\Big \downarrow \) 765

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 e^2 (b c-7 a d) (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {d \sqrt {1-\frac {d x^2}{c}} (17 b c-21 a d) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{b \sqrt {c-d x^2}}}{3 b}+\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\)

\(\Big \downarrow \) 762

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 e^2 (b c-7 a d) (b c-a d) \int \frac {1}{\sqrt {c-d x^2} \left (a e^2-b e^2 x^2\right )}d\sqrt {e x}}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (17 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}}{3 b}+\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\)

\(\Big \downarrow \) 925

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 e^2 (b c-7 a d) (b c-a d) \left (\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}+\frac {\int \frac {\sqrt {a} e}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 a e^2}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (17 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}}{3 b}+\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 e^2 (b c-7 a d) (b c-a d) \left (\frac {\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}+\frac {\int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {c-d x^2}}d\sqrt {e x}}{2 \sqrt {a} e}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (17 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}}{3 b}+\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\)

\(\Big \downarrow \) 1543

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {3 e^2 (b c-7 a d) (b c-a d) \left (\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {a} e-\sqrt {b} e x\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}+\frac {\sqrt {1-\frac {d x^2}{c}} \int \frac {1}{\left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {1-\frac {d x^2}{c}}}d\sqrt {e x}}{2 \sqrt {a} e \sqrt {c-d x^2}}\right )}{b}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (17 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}}{3 b}+\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\)

\(\Big \downarrow \) 1542

\(\displaystyle 2 e^3 \left (\frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{4 b \left (a e^2-b e^2 x^2\right )}-\frac {\frac {\frac {\sqrt [4]{c} d^{3/4} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (17 b c-21 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b \sqrt {c-d x^2}}+\frac {3 e^2 (b c-7 a d) (b c-a d) \left (\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2}}\right )}{b}}{3 b}+\frac {7 d \sqrt {e x} \sqrt {c-d x^2}}{3 b}}{4 b e^2}\right )\)

Input:

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

Output:

2*e^3*((Sqrt[e*x]*(c - d*x^2)^(3/2))/(4*b*(a*e^2 - b*e^2*x^2)) - ((7*d*Sqr 
t[e*x]*Sqrt[c - d*x^2])/(3*b) + ((c^(1/4)*d^(3/4)*(17*b*c - 21*a*d)*Sqrt[e 
]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e 
])], -1])/(b*Sqrt[c - d*x^2]) + (3*(b*c - 7*a*d)*(b*c - a*d)*e^2*((c^(1/4) 
*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*a*d^(1/4)*e^(3/2)*Sqr 
t[c - d*x^2]) + (c^(1/4)*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* 
a*d^(1/4)*e^(3/2)*Sqrt[c - d*x^2])))/b)/(3*b))/(4*b*e^2))
 

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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 925

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Simp[ 
1/(2*c)   Int[1/(Sqrt[a + b*x^4]*(1 - Rt[-d/c, 2]*x^2)), x], x] + Simp[1/(2 
*c)   Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a, b, 
 c, d}, x] && NeQ[b*c - a*d, 0]
 

rule 967

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

rule 1021

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x 
_)^(n_)]), x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* 
e - a*f)/b   Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, 
 d, e, f, n}, x]
 

rule 1025

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( 
f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( 
b*(n*(p + q + 1) + 1))), x] + Simp[1/(b*(n*(p + q + 1) + 1))   Int[(a + b*x 
^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e 
- a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 0]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1193\) vs. \(2(293)=586\).

Time = 3.03 (sec) , antiderivative size = 1194, normalized size of antiderivative = 3.13

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1194\)
risch	\(\text {Expression too large to display}\)	\(1291\)
default	\(\text {Expression too large to display}\)	\(3454\)

Input:

int((e*x)^(3/2)*(-d*x^2+c)^(3/2)/(-b*x^2+a)^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*e*(a*d-b*c 
)/b^2*(-d*e*x^3+c*e*x)^(1/2)/(-b*x^2+a)-2/3*d*e/b^2*(-d*e*x^3+c*e*x)^(1/2) 
+7/4*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)*EllipticF(((x+1/d*(c*d)^(1/ 
2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*e^2/b^3*a-17/12*(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)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/ 
2*2^(1/2))*e^2/b^2*c+7/8*e^2/b^3/(a*b)^(1/2)*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))*a^2-e^2/b^2/(a*b)^(1/2)*(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))*a*c+1/8*e^2/b/(a*b)^(1/2)/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))*c^2-7/8*e^2/b^3/(a*b)^(1/2)*d*(c*d)^(1/2)*(1+x*d/(c*d)^(1/2...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {too large to display} \] Input:

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

Output:

(sqrt(e)*e*(10*sqrt(x)*sqrt(c - d*x**2)*a*c*d + 6*sqrt(x)*sqrt(c - d*x**2) 
*a*d**2*x**2 - 6*sqrt(x)*sqrt(c - d*x**2)*b*c**2 - 6*sqrt(x)*sqrt(c - d*x* 
*2)*b*c*d*x**2 - 5*int(sqrt(c - d*x**2)/(sqrt(x)*a**3*c*d - sqrt(x)*a**3*d 
**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a**2*b*c*d*x**2 + 2*sqrt(x)*a**2* 
b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sqrt(x)*a*b**2*c*d*x**4 - sqrt( 
x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x**4 + sqrt(x)*b**3*c*d*x**6),x)*a 
**4*c**2*d**2 + 8*int(sqrt(c - d*x**2)/(sqrt(x)*a**3*c*d - sqrt(x)*a**3*d* 
*2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a**2*b*c*d*x**2 + 2*sqrt(x)*a**2*b 
*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sqrt(x)*a*b**2*c*d*x**4 - sqrt(x 
)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x**4 + sqrt(x)*b**3*c*d*x**6),x)*a* 
*3*b*c**3*d + 5*int(sqrt(c - d*x**2)/(sqrt(x)*a**3*c*d - sqrt(x)*a**3*d**2 
*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a**2*b*c*d*x**2 + 2*sqrt(x)*a**2*b*d 
**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sqrt(x)*a*b**2*c*d*x**4 - sqrt(x)* 
a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x**4 + sqrt(x)*b**3*c*d*x**6),x)*a**3 
*b*c**2*d**2*x**2 - 3*int(sqrt(c - d*x**2)/(sqrt(x)*a**3*c*d - sqrt(x)*a** 
3*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a**2*b*c*d*x**2 + 2*sqrt(x)*a* 
*2*b*d**2*x**4 + 2*sqrt(x)*a*b**2*c**2*x**2 - sqrt(x)*a*b**2*c*d*x**4 - sq 
rt(x)*a*b**2*d**2*x**6 - sqrt(x)*b**3*c**2*x**4 + sqrt(x)*b**3*c*d*x**6),x 
)*a**2*b**2*c**4 - 8*int(sqrt(c - d*x**2)/(sqrt(x)*a**3*c*d - sqrt(x)*a**3 
*d**2*x**2 - sqrt(x)*a**2*b*c**2 - sqrt(x)*a**2*b*c*d*x**2 + 2*sqrt(x)*...