\(\int \frac {(d+e x)^2}{\sqrt [4]{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [362]

Optimal result

Integrand size = 37, antiderivative size = 230 \[ \int \frac {(d+e x)^2}{\sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {7 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{15 c^2 d^2}+\frac {2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{5 c d}+\frac {7 \left (c d^2-a e^2\right )^3 \sqrt [4]{-\frac {c d e (a e+c d x) (d+e x)}{\left (c d^2-a e^2\right )^2}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {a e^2+c d (d+2 e x)}{c d^2-a e^2}\right )\right |2\right )}{10 \sqrt {2} c^3 d^3 e \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:

7/15*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/c^2/d^2+2/5*(e 
*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/c/d+7/20*(-a*e^2+c*d^2)^3*(- 
c*d*e*(c*d*x+a*e)*(e*x+d)/(-a*e^2+c*d^2)^2)^(1/4)*EllipticE(sin(1/2*arcsin 
((a*e^2+c*d*(2*e*x+d))/(-a*e^2+c*d^2))),2^(1/2))*2^(1/2)/c^3/d^3/e/(a*d*e+ 
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.40 \[ \int \frac {(d+e x)^2}{\sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {4 (d+e x) ((a e+c d x) (d+e x))^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {3}{4},\frac {7}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{3 c d \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{7/4}} \] Input:

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

Output:

(4*(d + e*x)*((a*e + c*d*x)*(d + e*x))^(3/4)*Hypergeometric2F1[-7/4, 3/4, 
7/4, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(3*c*d*((c*d*(d + e*x))/(c*d^2 
 - a*e^2))^(7/4))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1059\) vs. \(2(230)=460\).

Time = 0.77 (sec) , antiderivative size = 1059, normalized size of antiderivative = 4.60, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1166, 27, 1160, 1094, 834, 761, 1510}

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.

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \int \frac {7 e \left (c d^2-a e^2\right ) (d+e x)}{4 \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{5 c d e}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 \left (c d^2-a e^2\right ) \int \frac {d+e x}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {7 \left (c d^2-a e^2\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {7 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{d \left (a e^2+c d^2+2 c d e x\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {7 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \left (\frac {\left (c d^2-a e^2\right ) \int \frac {1}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}-\frac {\left (c d^2-a e^2\right ) \int \frac {1-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{d \left (a e^2+c d^2+2 c d e x\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {7 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \left (\frac {\left (c d^2-a e^2\right )^{3/2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d^2-a e^2}+1\right ) \sqrt {\frac {4 c d e \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )+\left (c d^2-a e^2\right )^2}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d^2-a e^2}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} d^{3/4} e^{3/4} \sqrt {4 c d e \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )+\left (c d^2-a e^2\right )^2}}-\frac {\left (c d^2-a e^2\right ) \int \frac {1-\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{d \left (a e^2+c d^2+2 c d e x\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4} (d+e x)}{5 c d}+\frac {7 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (c d^2+2 c e x d+a e^2\right )^2} \left (\frac {\left (c d^2-a e^2\right )^{3/2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} d^{3/4} e^{3/4} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}-\frac {\left (c d^2-a e^2\right ) \left (\frac {\sqrt {c d^2-a e^2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}-\frac {\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e} \sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{c d^2-a e^2}+1\right )}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )}{d \left (c d^2+2 c e x d+a e^2\right )}+\frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}{3 c d}\right )}{10 c d}\)

Input:

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

Output:

(2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4))/(5*c*d) + (7*( 
c*d^2 - a*e^2)*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/4))/(3*c*d) 
+ (2*(d^2 - (a*e^2)/c)*Sqrt[(c*d^2 + a*e^2 + 2*c*d*e*x)^2]*(-1/2*((c*d^2 - 
 a*e^2)*(-(((a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4)*Sqrt[(c*d^2 - a* 
e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)])/((c*d^2 - a*e^2 
)^2*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e 
*x^2])/(c*d^2 - a*e^2)))) + (Sqrt[c*d^2 - a*e^2]*(1 + (2*Sqrt[c]*Sqrt[d]*S 
qrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 - a*e^2))*Sqrt[ 
((c*d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c* 
d^2 - a*e^2)^2*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2])/(c*d^2 - a*e^2))^2)]*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4) 
*d^(1/4)*e^(1/4)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4))/Sqrt[c*d^2 
 - a*e^2]], 1/2])/(Sqrt[2]*c^(1/4)*d^(1/4)*e^(1/4)*Sqrt[(c*d^2 - a*e^2)^2 
+ 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)])))/(Sqrt[c]*Sqrt[d]*Sqr 
t[e]) + ((c*d^2 - a*e^2)^(3/2)*(1 + (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e 
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d^2 - a*e^2))*Sqrt[((c*d^2 - a*e^2)^2 
 + 4*c*d*e*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2))/((c*d^2 - a*e^2)^2*(1 
+ (2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/ 
(c*d^2 - a*e^2))^2)]*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*d^(1/4)*e^(1/4)*( 
a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4))/Sqrt[c*d^2 - a*e^2]], 1/2...
 

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 761

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

rule 834

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

rule 1094

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[4*p]
 

rule 1160

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

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1510

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/4)*(e*x + d)/(c*d*x + 
a*e), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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