\(\int \frac {(e x)^{3/2} \sqrt {c-d x^2}}{(a-b x^2)^2} \, dx\) [1125]

Optimal result

Integrand size = 30, antiderivative size = 328 \[ \int \frac {(e x)^{3/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {e \sqrt {e x} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right )}-\frac {3 \sqrt [4]{c} d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-3 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b^2 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-3 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a b^2 \sqrt [4]{d} \sqrt {c-d x^2}} \] Output:

1/2*e*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b/(-b*x^2+a)-3/2*c^(1/4)*d^(3/4)*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^2/( 
-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-3*a*d+b*c)*e^(3/2)*(1-d*x^2/c)^(1/2)*Ellipti 
cPi(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^2/d^(1/4)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-3*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^2/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 = 11.12 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.50 \[ \int \frac {(e x)^{3/2} \sqrt {c-d x^2}}{\left (a-b x^2\right )^2} \, dx=\frac {e \sqrt {e x} \left (-5 a \left (c-d x^2\right )+5 c \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{10 a b \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

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

Output:

(e*Sqrt[e*x]*(-5*a*(c - d*x^2) + 5*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*Appel 
lF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + 3*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]))/(10*a*b*(- 
a + b*x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {368, 27, 967, 27, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 967

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

2*e^3*((Sqrt[e*x]*Sqrt[c - d*x^2])/(4*b*(a*e^2 - b*e^2*x^2)) - ((3*c^(1/4) 
*d^(3/4)*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]) + ((b*c - 3*a*d)*e^2*((c^(1/4 
)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), A 
rcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*d^(1/4)*e^(3/2)*Sq 
rt[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)/(4*b*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 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 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. \(761\) vs. \(2(246)=492\).

Time = 0.55 (sec) , antiderivative size = 762, normalized size of antiderivative = 2.32

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {e \sqrt {-d e \,x^{3}+c e x}}{2 b \left (-b \,x^{2}+a \right )}-\frac {3 e^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{4 b^{2} \sqrt {-d e \,x^{3}+c e x}}-\frac {3 e^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) a}{8 b^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {e^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 b \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {3 e^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) a}{8 b^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}-\frac {e^{2} \sqrt {c d}\, \sqrt {1+\frac {x d}{\sqrt {c d}}}\, \sqrt {2-\frac {2 x d}{\sqrt {c d}}}\, \sqrt {-\frac {x d}{\sqrt {c d}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 b \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-x^{2} d +c}}\)	\(762\)
default	\(\text {Expression too large to display}\)	\(2243\)

Input:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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