\(\int \frac {\sqrt {d+e x^2}}{a+c x^4} \, dx\) [385]

Optimal result

Integrand size = 21, antiderivative size = 287 \[ \int \frac {\sqrt {d+e x^2}}{a+c x^4} \, dx=\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} a^{3/4} \sqrt {c}}-\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} a^{3/4} \sqrt {c}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.12 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {d+e x^2}}{a+c x^4} \, dx=\frac {1}{2} e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+2 d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ] \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1489, 27, 301, 224, 219, 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.

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

\(\Big \downarrow \) 1489

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(((d - (Sqrt[-a]*e)/Sqrt[c])*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/ 
((-a)^(1/4)*Sqrt[d + e*x^2])])/((-a)^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]) + 
 (Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c])/Sqrt[-a] - (-((Sq 
rt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c]) + ((d + (Sqrt[-a]*e)/ 
Sqrt[c])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x 
^2])])/((-a)^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]))/(2*Sqrt[-a])
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 224

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

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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*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. \(550\) vs. \(2(219)=438\).

Time = 1.11 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.92

method	result	size
pseudoelliptic	\(-\frac {\left (a e -\sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}\right ) \left (-\frac {\ln \left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}-\sqrt {a}\, \left (e \,x^{2}+d \right )}{x^{2}}\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}{4}+\frac {\ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}{4}+\left (a e +\sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}\right ) \left (\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right )\right )}{2 a^{\frac {3}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, d c}\)	\(551\)
default	\(\text {Expression too large to display}\)	\(1842\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (221) = 442\).

Time = 0.21 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {d+e x^2}}{a+c x^4} \, dx=\frac {1}{8} \, \sqrt {-\frac {a c \sqrt {-\frac {d^{2}}{a^{3} c}} + e}{a c}} \log \left (\frac {a c d x^{2} \sqrt {-\frac {d^{2}}{a^{3} c}} + 2 \, \sqrt {e x^{2} + d} a^{2} c x \sqrt {-\frac {a c \sqrt {-\frac {d^{2}}{a^{3} c}} + e}{a c}} \sqrt {-\frac {d^{2}}{a^{3} c}} + 2 \, d e x^{2} + d^{2}}{x^{2}}\right ) - \frac {1}{8} \, \sqrt {-\frac {a c \sqrt {-\frac {d^{2}}{a^{3} c}} + e}{a c}} \log \left (\frac {a c d x^{2} \sqrt {-\frac {d^{2}}{a^{3} c}} - 2 \, \sqrt {e x^{2} + d} a^{2} c x \sqrt {-\frac {a c \sqrt {-\frac {d^{2}}{a^{3} c}} + e}{a c}} \sqrt {-\frac {d^{2}}{a^{3} c}} + 2 \, d e x^{2} + d^{2}}{x^{2}}\right ) + \frac {1}{8} \, \sqrt {\frac {a c \sqrt {-\frac {d^{2}}{a^{3} c}} - e}{a c}} \log \left (-\frac {a c d x^{2} \sqrt {-\frac {d^{2}}{a^{3} c}} + 2 \, \sqrt {e x^{2} + d} a^{2} c x \sqrt {\frac {a c \sqrt {-\frac {d^{2}}{a^{3} c}} - e}{a c}} \sqrt {-\frac {d^{2}}{a^{3} c}} - 2 \, d e x^{2} - d^{2}}{x^{2}}\right ) - \frac {1}{8} \, \sqrt {\frac {a c \sqrt {-\frac {d^{2}}{a^{3} c}} - e}{a c}} \log \left (-\frac {a c d x^{2} \sqrt {-\frac {d^{2}}{a^{3} c}} - 2 \, \sqrt {e x^{2} + d} a^{2} c x \sqrt {\frac {a c \sqrt {-\frac {d^{2}}{a^{3} c}} - e}{a c}} \sqrt {-\frac {d^{2}}{a^{3} c}} - 2 \, d e x^{2} - d^{2}}{x^{2}}\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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