\(\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{(d+e x^2) \sqrt {a-c x^4}} \, dx\) [61]

Optimal result

Integrand size = 39, antiderivative size = 174 \[ \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {\sqrt [4]{c} \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} e \sqrt {a-c x^4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}} \sqrt {1+\frac {\sqrt {c} x^2}{\sqrt {a}}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{a} \sqrt [4]{c} d e \sqrt {a-c x^4}} \] Output:

-c^(1/4)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(1/4)/e/(-c*x^ 
4+a)^(1/2)+(c^(1/2)*d+a^(1/2)*e)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)* 
x^2/a^(1/2))^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1/2)/d,I)/a^ 
(1/4)/c^(1/4)/d/e/(-c*x^4+a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.57 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.86 \[ \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {i \sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \sqrt {1-\frac {c x^4}{a}} \left (-\sqrt {c} d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\left (\sqrt {c} d+\sqrt {a} e\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {c} d e \sqrt {a-c x^4}} \] Input:

Integrate[(1 - (Sqrt[c]*x^2)/Sqrt[a])/((d + e*x^2)*Sqrt[a - c*x^4]),x]
 

Output:

(I*Sqrt[-(Sqrt[c]/Sqrt[a])]*Sqrt[1 - (c*x^4)/a]*(-(Sqrt[c]*d*EllipticF[I*A 
rcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1]) + (Sqrt[c]*d + Sqrt[a]*e)*Ellipti 
cPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] 
))/(Sqrt[c]*d*e*Sqrt[a - c*x^4])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.40, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1787, 415, 289, 413, 412, 765, 762}

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.

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

\(\Big \downarrow \) 1787

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

\(\Big \downarrow \) 415

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

\(\Big \downarrow \) 289

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

Input:

Int[(1 - (Sqrt[c]*x^2)/Sqrt[a])/((d + e*x^2)*Sqrt[a - c*x^4]),x]
 

Output:

(Sqrt[1 - (Sqrt[c]*x^2)/Sqrt[a]]*Sqrt[a + Sqrt[a]*Sqrt[c]*x^2]*(-((c^(1/4) 
*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(a^(1/4)* 
e*Sqrt[1 - (Sqrt[c]*x^2)/Sqrt[a]]*Sqrt[a + Sqrt[a]*Sqrt[c]*x^2])) + (a^(1/ 
4)*((Sqrt[c]*d)/Sqrt[a] + e)*Sqrt[1 + (Sqrt[c]*x^2)/Sqrt[a]]*EllipticPi[-( 
(Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*d*e*S 
qrt[a + Sqrt[a]*Sqrt[c]*x^2])))/Sqrt[a - c*x^4]
 

Defintions of rubi rules used

rule 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

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

rule 415

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) 
^2]), x_Symbol] :> Simp[d/b   Int[1/(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], 
x] + Simp[(b*c - a*d)/b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2 
]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]
 

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 1787

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + ( 
c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + 
e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p])   Int[(d + e*x^n)^(p + q) 
*(f + g*x^n)^r*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p 
, q, r}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.07

method	result	size
default	\(-\frac {\frac {\sqrt {c}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\left (\sqrt {c}\, d +\sqrt {a}\, e \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{e d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{\sqrt {a}}\)	\(187\)
elliptic	\(\frac {\sqrt {\left (-c \,x^{4}+a \right ) a c}\, \left (-\sqrt {a}+\sqrt {c}\, x^{2}\right ) \left (\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-a \,c^{2} x^{4}+a^{2} c}}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, e}{\sqrt {c}\, d}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{e \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-a \,c^{2} x^{4}+a^{2} c}}\right )}{\left (c \,x^{2} \sqrt {-c \,x^{4}+a}-\sqrt {\left (-c \,x^{4}+a \right ) a c}\right ) \sqrt {a}}\)	\(340\)

Input:

int((1-c^(1/2)*x^2/a^(1/2))/(e*x^2+d)/(-c*x^4+a)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/a^(1/2)*(c^(1/2)/e/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2 
)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1 
/2))^(1/2),I)-(c^(1/2)*d+a^(1/2)*e)/e/d/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2) 
*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*Ellipti 
cPi(x*(c^(1/2)/a^(1/2))^(1/2),-a^(1/2)*e/c^(1/2)/d,(-c^(1/2)/a^(1/2))^(1/2 
)/(c^(1/2)/a^(1/2))^(1/2)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\text {Timed out} \] Input:

integrate((1-c^(1/2)*x^2/a^(1/2))/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm= 
"fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=- \frac {\int \left (- \frac {\sqrt {a}}{d \sqrt {a - c x^{4}} + e x^{2} \sqrt {a - c x^{4}}}\right )\, dx + \int \frac {\sqrt {c} x^{2}}{d \sqrt {a - c x^{4}} + e x^{2} \sqrt {a - c x^{4}}}\, dx}{\sqrt {a}} \] Input:

integrate((1-c**(1/2)*x**2/a**(1/2))/(e*x**2+d)/(-c*x**4+a)**(1/2),x)
 

Output:

-(Integral(-sqrt(a)/(d*sqrt(a - c*x**4) + e*x**2*sqrt(a - c*x**4)), x) + I 
ntegral(sqrt(c)*x**2/(d*sqrt(a - c*x**4) + e*x**2*sqrt(a - c*x**4)), x))/s 
qrt(a)
 

Maxima [F]

\[ \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\int { -\frac {\frac {\sqrt {c} x^{2}}{\sqrt {a}} - 1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}} \,d x } \] Input:

integrate((1-c^(1/2)*x^2/a^(1/2))/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm= 
"maxima")
                                                                                    
                                                                                    
 

Output:

-integrate((sqrt(c)*x^2/sqrt(a) - 1)/(sqrt(-c*x^4 + a)*(e*x^2 + d)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((1-c^(1/2)*x^2/a^(1/2))/(e*x^2+d)/(-c*x^4+a)^(1/2),x, algorithm= 
"giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Degree mismatch inside factorisatio 
n over extensionDegree mismatch inside factorisation over extensionDegree 
mismatch
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=-\int \frac {\frac {\sqrt {c}\,x^2}{\sqrt {a}}-1}{\sqrt {a-c\,x^4}\,\left (e\,x^2+d\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (d+e x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {-\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right )+\left (\int \frac {\sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a}{a} \] Input:

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

Output:

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