\(\int \frac {d+e x}{\sqrt [4]{a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [363]

Optimal result

Integrand size = 35, antiderivative size = 171 \[ \int \frac {d+e x}{\sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{3 c d}+\frac {\left (c d^2-a e^2\right )^2 \sqrt [4]{-\frac {c d e (a e+c d x) (d+e x)}{\left (c d^2-a e^2\right )^2}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {a e^2+c d (d+2 e x)}{c d^2-a e^2}\right )\right |2\right )}{\sqrt {2} c^2 d^2 e \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:

2/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/c/d+1/2*(-a*e^2+c*d^2)^2*(-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^2/d^2/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 = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.51 \[ \int \frac {d+e x}{\sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {4 ((a e+c d x) (d+e x))^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{4},\frac {7}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{3 c d \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{3/4}} \] Input:

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

Output:

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

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(992\) vs. \(2(171)=342\).

Time = 0.68 (sec) , antiderivative size = 992, normalized size of antiderivative = 5.80, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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.

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1094

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

\(\displaystyle \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}\)

Input:

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

Output:

(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]*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)]*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]*Sqrt[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]*Sqr 
t[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])/(4*Sqrt[2]*c^(3/ 
4)*d^(3/4)*e^(3/4)*Sqrt[(c*d^2 - a*e^2)^2 + 4*c*d*e*(a*d*e + (c*d^2 + a...
 

Defintions of rubi rules used

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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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