\(\int \frac {x^4}{\sqrt {a+\frac {(c d^2+a e^2) x^2}{d e}+c x^4}} \, dx\) [709]

Optimal result

Integrand size = 36, antiderivative size = 331 \[ \int \frac {x^4}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=-\frac {2 \left (\frac {1}{e}+\frac {a e}{c d^2}\right ) x \left (a e+c d x^2\right )}{3 c \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}}+\frac {x \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}}{3 c}+\frac {2 a \left (c d^2+a e^2\right ) \sqrt {\frac {d \left (a e+c d x^2\right )}{a e \left (d+e x^2\right )}} \left (d+e x^2\right ) E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|1-\frac {c d^2}{a e^2}\right )}{3 c^2 d^{5/2} \sqrt {e} \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}}-\frac {a \sqrt {\frac {d \left (a e+c d x^2\right )}{a e \left (d+e x^2\right )}} \left (d+e x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),1-\frac {c d^2}{a e^2}\right )}{3 c \sqrt {d} \sqrt {e} \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}} \] Output:

-2/3*(1/e+a*e/c/d^2)*x*(c*d*x^2+a*e)/c/(a+(c*d/e+a*e/d)*x^2+c*x^4)^(1/2)+1 
/3*x*(a+(c*d/e+a*e/d)*x^2+c*x^4)^(1/2)/c+2/3*a*(a*e^2+c*d^2)*(d*(c*d*x^2+a 
*e)/a/e/(e*x^2+d))^(1/2)*(e*x^2+d)*EllipticE(e^(1/2)*x/d^(1/2)/(1+e*x^2/d) 
^(1/2),(1-c*d^2/a/e^2)^(1/2))/c^2/d^(5/2)/e^(1/2)/(a+(c*d/e+a*e/d)*x^2+c*x 
^4)^(1/2)-1/3*a*(d*(c*d*x^2+a*e)/a/e/(e*x^2+d))^(1/2)*(e*x^2+d)*InverseJac 
obiAM(arctan(e^(1/2)*x/d^(1/2)),(1-c*d^2/a/e^2)^(1/2))/c/d^(1/2)/e^(1/2)/( 
a+(c*d/e+a*e/d)*x^2+c*x^4)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.79 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.74 \[ \int \frac {x^4}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\frac {\sqrt {\frac {c d}{a e}} e x \left (a e+c d x^2\right ) \left (d+e x^2\right )+2 i d \left (c d^2+a e^2\right ) \sqrt {1+\frac {c d x^2}{a e}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {c d}{a e}} x\right )|\frac {a e^2}{c d^2}\right )-i d \left (2 c d^2+a e^2\right ) \sqrt {1+\frac {c d x^2}{a e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {c d}{a e}} x\right ),\frac {a e^2}{c d^2}\right )}{3 a \left (\frac {c d}{a e}\right )^{3/2} e^3 \sqrt {\frac {\left (a e+c d x^2\right ) \left (d+e x^2\right )}{d e}}} \] Input:

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

Output:

(Sqrt[(c*d)/(a*e)]*e*x*(a*e + c*d*x^2)*(d + e*x^2) + (2*I)*d*(c*d^2 + a*e^ 
2)*Sqrt[1 + (c*d*x^2)/(a*e)]*Sqrt[1 + (e*x^2)/d]*EllipticE[I*ArcSinh[Sqrt[ 
(c*d)/(a*e)]*x], (a*e^2)/(c*d^2)] - I*d*(2*c*d^2 + a*e^2)*Sqrt[1 + (c*d*x^ 
2)/(a*e)]*Sqrt[1 + (e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[(c*d)/(a*e)]*x], (a 
*e^2)/(c*d^2)])/(3*a*((c*d)/(a*e))^(3/2)*e^3*Sqrt[((a*e + c*d*x^2)*(d + e* 
x^2))/(d*e)])
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1442, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 1442

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

\(\displaystyle \frac {x \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}{3 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \left (\frac {2 \left (\frac {a e}{d}+\frac {c d}{e}\right )}{\sqrt {c}}+\sqrt {a}\right ) \sqrt {\frac {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {\frac {c d}{e}+\frac {a e}{d}}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {2 \left (\frac {a e}{d}+\frac {c d}{e}\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {\frac {c d}{e}+\frac {a e}{d}}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}-\frac {x \sqrt {x^2 \left (\frac {a e}{d}+\frac {c d}{e}\right )+a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{3 c}\)

Input:

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

Output:

(x*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4])/(3*c) - ((-2*((c*d)/e + (a*e 
)/d)*(-((x*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x 
^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + ((c*d)/e + (a*e)/d)*x^2 
+ c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4) 
], (2 - ((c*d)/e + (a*e)/d)/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + ((c*d 
)/e + (a*e)/d)*x^2 + c*x^4])))/Sqrt[c] + (a^(1/4)*(Sqrt[a] + (2*((c*d)/e + 
 (a*e)/d))/Sqrt[c])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + ((c*d)/e + (a*e)/d)* 
x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^( 
1/4)], (2 - ((c*d)/e + (a*e)/d)/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(1/4)*Sqrt[a + 
 ((c*d)/e + (a*e)/d)*x^2 + c*x^4]))/(3*c)
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1442

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), 
x] - Simp[d^4/(c*(m + 4*p + 1))   Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 
 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x 
] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* 
p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [A] (verified)

Time = 4.52 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {x \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}{3 c}-\frac {a \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )}{3 c \sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}+\frac {2 \left (\frac {2 c d}{e}+\frac {2 a e}{d}\right ) a \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )\right )}{3 c \sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}\, \left (\frac {c d}{e}+\frac {a e}{d}+\frac {a \,e^{2}-c \,d^{2}}{d e}\right )}\)	\(353\)
elliptic	\(\frac {x \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}{3 c}-\frac {a \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )}{3 c \sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}}+\frac {2 \left (\frac {2 c d}{e}+\frac {2 a e}{d}\right ) a \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )\right )}{3 c \sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}\, \left (\frac {c d}{e}+\frac {a e}{d}+\frac {a \,e^{2}-c \,d^{2}}{d e}\right )}\)	\(353\)
risch	\(\frac {x \left (e \,x^{2}+d \right ) \left (c d \,x^{2}+a e \right )}{3 c d e \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c d \,x^{2}+a e \right )}{d e}}}-\frac {\left (-\frac {2 \left (2 a \,e^{2}+2 c \,d^{2}\right ) a \,e^{2} d^{2} \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {a d \,e^{3}+c \,d^{3} e}{e \,d^{3} c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {a d \,e^{3}+c \,d^{3} e}{e \,d^{3} c}}\right )\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {c \,d^{2} e^{2} x^{4}+a \,e^{3} x^{2} d +c e \,x^{2} d^{3}+a \,d^{2} e^{2}}\, \left (a d \,e^{3}+c \,d^{3} e +d e \left (a \,e^{2}-c \,d^{2}\right )\right )}+\frac {a d e \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {a d \,e^{3}+c \,d^{3} e}{e \,d^{3} c}}\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {c \,d^{2} e^{2} x^{4}+a \,e^{3} x^{2} d +c e \,x^{2} d^{3}+a \,d^{2} e^{2}}}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (c d \,x^{2}+a e \right ) d e}}{3 e c d \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c d \,x^{2}+a e \right )}{d e}}}\)	\(462\)

Input:

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

Output:

1/3/c*x*(a+x^2/d*e*a+x^2*d/e*c+c*x^4)^(1/2)-1/3/c*a/(-c*d/a/e)^(1/2)*(1+1/ 
a*x^2*d/e*c)^(1/2)*(1+e*x^2/d)^(1/2)/(a+x^2/d*e*a+x^2*d/e*c+c*x^4)^(1/2)*E 
llipticF(x*(-c*d/a/e)^(1/2),(-1+(c*d/e+1/d*a*e)*e/d/c)^(1/2))+2/3/c*(2*c*d 
/e+2/d*a*e)*a/(-c*d/a/e)^(1/2)*(1+1/a*x^2*d/e*c)^(1/2)*(1+e*x^2/d)^(1/2)/( 
a+x^2/d*e*a+x^2*d/e*c+c*x^4)^(1/2)/(c*d/e+1/d*a*e+(a*e^2-c*d^2)/d/e)*(Elli 
pticF(x*(-c*d/a/e)^(1/2),(-1+(c*d/e+1/d*a*e)*e/d/c)^(1/2))-EllipticE(x*(-c 
*d/a/e)^(1/2),(-1+(c*d/e+1/d*a*e)*e/d/c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.56 \[ \int \frac {x^4}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\frac {2 \, {\left (c d^{3} + a d e^{2}\right )} \sqrt {c} x \sqrt {-\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{x}\right )\,|\,\frac {a e^{2}}{c d^{2}}) - {\left (2 \, c d^{3} + 2 \, a d e^{2} + a e^{3}\right )} \sqrt {c} x \sqrt {-\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{x}\right )\,|\,\frac {a e^{2}}{c d^{2}}) + {\left (c d e^{2} x^{2} - 2 \, c d^{2} e - 2 \, a e^{3}\right )} \sqrt {\frac {c d e x^{4} + a d e + {\left (c d^{2} + a e^{2}\right )} x^{2}}{d e}}}{3 \, c^{2} d e^{2} x} \] Input:

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

Output:

1/3*(2*(c*d^3 + a*d*e^2)*sqrt(c)*x*sqrt(-d/e)*elliptic_e(arcsin(sqrt(-d/e) 
/x), a*e^2/(c*d^2)) - (2*c*d^3 + 2*a*d*e^2 + a*e^3)*sqrt(c)*x*sqrt(-d/e)*e 
lliptic_f(arcsin(sqrt(-d/e)/x), a*e^2/(c*d^2)) + (c*d*e^2*x^2 - 2*c*d^2*e 
- 2*a*e^3)*sqrt((c*d*e*x^4 + a*d*e + (c*d^2 + a*e^2)*x^2)/(d*e)))/(c^2*d*e 
^2*x)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \left (\sqrt {e \,x^{2}+d}\, \sqrt {c d \,x^{2}+a e}\, x -2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c d \,x^{2}+a e}\, x^{2}}{c d e \,x^{4}+a \,e^{2} x^{2}+c \,d^{2} x^{2}+a d e}d x \right ) a \,e^{2}-2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c d \,x^{2}+a e}\, x^{2}}{c d e \,x^{4}+a \,e^{2} x^{2}+c \,d^{2} x^{2}+a d e}d x \right ) c \,d^{2}-\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c d \,x^{2}+a e}}{c d e \,x^{4}+a \,e^{2} x^{2}+c \,d^{2} x^{2}+a d e}d x \right ) a d e \right )}{3 c d e} \] Input:

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

Output:

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