\(\int \frac {\sqrt {c-d x^2}}{\sqrt {e x} (a-b x^2)^2} \, dx\) [1126]

Optimal result

Integrand size = 30, antiderivative size = 335 \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {e x} \left (a-b x^2\right )^2} \, dx=\frac {\sqrt {e x} \sqrt {c-d x^2}}{2 a e \left (a-b x^2\right )}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 a b \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (3 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^2 b \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (3 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^2 b \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}} \] Output:

1/2*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/a/e/(-b*x^2+a)+1/2*c^(1/4)*d^(3/4)*(1-d*x 
^2/c)^(1/2)*EllipticF(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)/a/b/e^(1/2)/( 
-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-a*d+3*b*c)*(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^2/b/ 
d^(1/4)/e^(1/2)/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(-a*d+3*b*c)*(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^2/b/d^(1/4)/e^(1/2)/(-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 = 11.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.48 \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {e x} \left (a-b x^2\right )^2} \, dx=\frac {5 a x \left (-c+d x^2\right )+15 c x \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+d x^3 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{10 a^2 \sqrt {e x} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \] Input:

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

Output:

(5*a*x*(-c + d*x^2) + 15*c*x*(-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*x^3*(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^2*Sqrt[e*x]*(-a 
 + b*x^2)*Sqrt[c - d*x^2])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {368, 27, 929, 25, 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.

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

\(\Big \downarrow \) 368

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 929

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

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

Input:

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

Output:

2*e^3*((Sqrt[e*x]*Sqrt[c - d*x^2])/(4*a*e^2*(a*e^2 - b*e^2*x^2)) + ((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]) + ((3*b*c - a*d)*e^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]) + (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*a*e^4))
 

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 929

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-x)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*n*(p + 1))), x] + Simp[1 
/(a*n*(p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(n*(p + 
 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, 
x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, 
 c, d, 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. \(743\) vs. \(2(253)=506\).

Time = 1.24 (sec) , antiderivative size = 744, normalized size of antiderivative = 2.22

method	result	size
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) e x}\, \left (\frac {\sqrt {-d e \,x^{3}+c e x}}{2 e a \left (-b \,x^{2}+a \right )}+\frac {\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 a b \sqrt {-d e \,x^{3}+c e x}}+\frac {\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 )}{8 b \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {3 \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 a \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 {\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 )}{8 b \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {3 \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 a \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 )}{\sqrt {e x}\, \sqrt {-x^{2} d +c}}\)	\(744\)
default	\(\text {Expression too large to display}\)	\(2239\)

Input:

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

Output:

((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(1/2/e/a*(-d*e*x^3+c*e 
*x)^(1/2)/(-b*x^2+a)+1/4/a/b*(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)*Ellip 
ticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/8/b/(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))-3/8/a/(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))*Elli 
pticPi(((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-1/8/b/(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))+3/8/a/(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. \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {e x} \left (a-b x^2\right )^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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