\(\int \frac {1}{\sqrt {d+e x^2} (a+c x^4)} \, dx\) [386]

Optimal result

Integrand size = 21, antiderivative size = 370 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=-\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c} d \sqrt {c d^2+a e^2}}-\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} a^{3/4} \sqrt {c} d \sqrt {c d^2+a e^2}} \] Output:

-1/4*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(e-(a*e^2+c*d^2)^(1/2)/a^(1/2)) 
*arctan(2^(1/2)*a^(1/4)*c^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e 
*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a 
^(1/4)/c^(1/2)/d/(a*e^2+c*d^2)^(1/2)-1/4*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^ 
(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))*arctanh(2^(1/2)*a^(1/4)*c^(1/2)*(-a^ 
(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e-( 
a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a^(3/4)/c^(1/2)/d/(a*e^2+c*d^2)^(1/2 
)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.36 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx=-2 e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-c d^3+3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}-3 c d \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ] \] Input:

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

Output:

-2*e^(3/2)*RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 + 16*a*e^2*#1^2 - 4*c 
*d*#1^3 + c*#1^4 & , (Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]* 
#1)/(-(c*d^3) + 3*c*d^2*#1 + 8*a*e^2*#1 - 3*c*d*#1^2 + c*#1^3) & ]
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.41, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1489, 27, 291, 218, 221}

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.

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

\(\Big \downarrow \) 1489

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\)

Input:

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

Output:

-1/2*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x^2])] 
/((-a)^(3/4)*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]) - ArcTanh[(Sqrt[Sqrt[c]*d + Sqr 
t[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x^2])]/(2*(-a)^(3/4)*Sqrt[Sqrt[c]*d + S 
qrt[-a]*e])
 

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 291

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

rule 1489

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r 
= Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S 
imp[c/(2*r)   Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, 
q}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(672\) vs. \(2(290)=580\).

Time = 0.90 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.82

method	result	size
pseudoelliptic	\(-\frac {-\frac {\left (\left (-e \sqrt {a}-\sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+a e \sqrt {a \,e^{2}+c \,d^{2}}+a^{\frac {3}{2}} e^{2}\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )-\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{4}+\frac {\left (\left (-e \sqrt {a}-\sqrt {a \,e^{2}+c \,d^{2}}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+a e \sqrt {a \,e^{2}+c \,d^{2}}+a^{\frac {3}{2}} e^{2}\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{4}+d^{2} \left (\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right ) \left (a \sqrt {a \,e^{2}+c \,d^{2}}-a^{\frac {3}{2}} e \right ) c}{2 a^{\frac {3}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \sqrt {a \,e^{2}+c \,d^{2}}\, c \,d^{2}}\)	\(673\)
default	\(\frac {-\frac {\ln \left (\frac {\frac {2 \sqrt {-a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {\sqrt {-a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e +\frac {2 e \sqrt {\sqrt {-a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {-a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {2 \sqrt {-a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {\sqrt {-a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e -\frac {2 e \sqrt {\sqrt {-a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {-a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}}{2 \sqrt {-a}}-\frac {-\frac {\ln \left (\frac {\frac {-2 \sqrt {-a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e +\frac {2 e \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {-2 \sqrt {-a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e -\frac {2 e \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {-\sqrt {-a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {-a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {-a}\, \sqrt {c}\, e +c d}{c}}}}{2 \sqrt {-a}}\)	\(872\)

Input:

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

Output:

-1/2/a^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a* 
e)^(1/2)*(-1/4*((-e*a^(1/2)-(a*e^2+c*d^2)^(1/2))*(a*(a*e^2+c*d^2))^(1/2)+a 
*e*(a*e^2+c*d^2)^(1/2)+a^(3/2)*e^2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2 
)*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)*ln 
((a^(1/2)*(e*x^2+d)-(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2 
)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)+1/4*((-e*a^(1/2)-(a*e^2+c*d^2)^(1/2))*(a 
*(a*e^2+c*d^2))^(1/2)+a*e*(a*e^2+c*d^2)^(1/2)+a^(3/2)*e^2)*(2*(a*(a*e^2+c* 
d^2))^(1/2)+2*a*e)^(1/2)*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2) 
)^(1/2)-2*a*e)^(1/2)*ln((a^(1/2)*(e*x^2+d)+(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c* 
d^2))^(1/2)+2*a*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)+d^2*(arctan((2*a^ 
(1/2)*(e*x^2+d)^(1/2)+(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x)/x/(4*(a*e 
^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2))-arctan(((2 
*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x-2*a^(1/2)*(e*x^2+d)^(1/2))/x/(4*(a 
*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)))*(a*(a*e 
^2+c*d^2)^(1/2)-a^(3/2)*e)*c)/(a*e^2+c*d^2)^(1/2)/c/d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1217 vs. \(2 (292) = 584\).

Time = 0.29 (sec) , antiderivative size = 1217, normalized size of antiderivative = 3.29 \[ \int \frac {1}{\sqrt {d+e x^2} \left (a+c x^4\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*sqrt(-((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 
+ a^5*e^4)) - e)/(a*c*d^2 + a^2*e^2))*log(((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2 
/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4))*x^2 + 2*e*x^2 + 2*(a*e*x + (a^ 
2*c*d^2 + a^3*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4))* 
x)*sqrt(e*x^2 + d)*sqrt(-((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2 
*a^4*c*d^2*e^2 + a^5*e^4)) - e)/(a*c*d^2 + a^2*e^2)) + d)/x^2) - 1/8*sqrt( 
-((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4 
)) - e)/(a*c*d^2 + a^2*e^2))*log(((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2 
*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4))*x^2 + 2*e*x^2 - 2*(a*e*x + (a^2*c*d^2 + 
 a^3*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4))*x)*sqrt(e 
*x^2 + d)*sqrt(-((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^ 
2*e^2 + a^5*e^4)) - e)/(a*c*d^2 + a^2*e^2)) + d)/x^2) - 1/8*sqrt(((a*c*d^2 
 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4)) + e)/(a 
*c*d^2 + a^2*e^2))*log(-((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2* 
a^4*c*d^2*e^2 + a^5*e^4))*x^2 - 2*e*x^2 + 2*(a*e*x - (a^2*c*d^2 + a^3*e^2) 
*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4))*x)*sqrt(e*x^2 + d) 
*sqrt(((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^ 
5*e^4)) + e)/(a*c*d^2 + a^2*e^2)) - d)/x^2) + 1/8*sqrt(((a*c*d^2 + a^2*e^2 
)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d^2*e^2 + a^5*e^4)) + e)/(a*c*d^2 + a 
^2*e^2))*log(-((a*c*d^2 + a^2*e^2)*sqrt(-c*d^2/(a^3*c^2*d^4 + 2*a^4*c*d...
 

Sympy [F]

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

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

Output:

Integral(1/((a + c*x**4)*sqrt(d + e*x**2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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