\(\int \frac {d+e x}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/4}} \, dx\) [370]

Optimal result

Integrand size = 35, antiderivative size = 174 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d}-\frac {2 \sqrt {e} \sqrt {c d^2-a e^2} (a e+c d x)^{3/2} \left (\frac {c d (d+e x)}{e (a e+c d x)}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {c d^2-a e^2}}{\sqrt {e} \sqrt {a e+c d x}}\right ),2\right )}{c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.49 \[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}} \, dx=\frac {4 \sqrt [4]{(a e+c d x) (d+e x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{4},\frac {5}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d \sqrt [4]{\frac {c d (d+e x)}{c d^2-a e^2}}} \] Input:

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

Output:

(4*((a*e + c*d*x)*(d + e*x))^(1/4)*Hypergeometric2F1[-1/4, 1/4, 5/4, (e*(a 
*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d*((c*d*(d + e*x))/(c*d^2 - a*e^2))^( 
1/4))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(442\) vs. \(2(174)=348\).

Time = 0.41 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.54, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1160, 1094, 761}

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.

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1094

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

\(\Big \downarrow \) 761

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

Input:

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

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4))/(c*d) + (Sqrt[c*d^2 - a* 
e^2]*(d^2 - (a*e^2)/c)*Sqrt[(c*d^2 + a*e^2 + 2*c*d*e*x)^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])/(Sqrt[2]*c^(1/4)*d^(5/4)*e^(1/4)*(c*d^2 + a*e^ 
2 + 2*c*d*e*x)*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)])
 

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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F]

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

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

Output:

Integral((d + e*x)/((d + e*x)*(a*e + c*d*x))**(3/4), x)
 

Maxima [F]

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

integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4),x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4),x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}} \, dx=\left (\int \frac {x}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}}}d x \right ) e +\left (\int \frac {1}{\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}}}d x \right ) d \] Input:

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

Output:

int(x/(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4),x)*e + int(1/(a*d* 
e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4),x)*d