\(\int \frac {(d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/4}} \, dx\) [368]

Optimal result

Integrand size = 37, antiderivative size = 294 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}} \, dx=\frac {3 \left (c d^2-a e^2\right )^2 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (d+e x) \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c^2 d^2}+\frac {2 (d+e x)^2 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c d}-\frac {3 \sqrt {e} \left (c d^2-a e^2\right )^{5/2} (a e+c d x)^{3/2} \left (\frac {c d (d+e x)}{e (a e+c d x)}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {c d^2-a e^2}}{\sqrt {e} \sqrt {a e+c d x}}\right ),2\right )}{2 c^4 d^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}} \] Output:

3/2*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4)/c^3/d^3+3/5*( 
-a*e^2+c*d^2)*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4)/c^2/d^2+2/5* 
(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/4)/c/d-3/2*e^(1/2)*(-a*e^2+ 
c*d^2)^(5/2)*(c*d*x+a*e)^(3/2)*(c*d*(e*x+d)/e/(c*d*x+a*e))^(3/4)*InverseJa 
cobiAM(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^4/d^4/(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 = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.34 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}} \, dx=\frac {4 \left (c d^2-a e^2\right )^2 \sqrt [4]{(a e+c d x) (d+e x)} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {1}{4},\frac {5}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c^3 d^3 \sqrt [4]{\frac {c d (d+e x)}{c d^2-a e^2}}} \] Input:

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

Output:

(4*(c*d^2 - a*e^2)^2*((a*e + c*d*x)*(d + e*x))^(1/4)*Hypergeometric2F1[-9/ 
4, 1/4, 5/4, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c^3*d^3*((c*d*(d + e* 
x))/(c*d^2 - a*e^2))^(1/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.65 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {1166, 27, 1166, 27, 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.

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {2 \int \frac {9 e \left (c d^2-a e^2\right ) (d+e x)^2}{4 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}dx}{5 c d e}+\frac {2 (d+e x)^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 \left (c d^2-a e^2\right ) \int \frac {(d+e x)^2}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}dx}{10 c d}+\frac {2 (d+e x)^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\)

\(\Big \downarrow \) 1166

\(\displaystyle \frac {9 \left (c d^2-a e^2\right ) \left (\frac {2 \int \frac {5 e \left (c d^2-a e^2\right ) (d+e x)}{4 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}dx}{3 c d e}+\frac {2 (d+e x) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x)^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 \left (c d^2-a e^2\right ) \left (\frac {5 \left (c d^2-a e^2\right ) \int \frac {d+e x}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}dx}{6 c d}+\frac {2 (d+e x) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x)^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {9 \left (c d^2-a e^2\right ) \left (\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\left (d^2-\frac {a e^2}{c}\right ) \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}dx}{2 d}+\frac {2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )}{6 c d}+\frac {2 (d+e x) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x)^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {9 \left (c d^2-a e^2\right ) \left (\frac {5 \left (c d^2-a e^2\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \int \frac {1}{\sqrt {\left (c d^2-a e^2\right )^2+4 c d e \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )}}d\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{d \left (a e^2+c d^2+2 c d e x\right )}+\frac {2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )}{6 c d}+\frac {2 (d+e x) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x)^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {9 \left (c d^2-a e^2\right ) \left (\frac {5 \left (c d^2-a e^2\right ) \left (\frac {\sqrt {c d^2-a e^2} \left (d^2-\frac {a e^2}{c}\right ) \sqrt {\left (a e^2+c d^2+2 c d e x\right )^2} \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d^2-a e^2}+1\right ) \sqrt {\frac {4 c d e \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )+\left (c d^2-a e^2\right )^2}{\left (c d^2-a e^2\right )^2 \left (\frac {2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d^2-a e^2}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt [4]{e} \sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {c d^2-a e^2}}\right ),\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} d^{5/4} \sqrt [4]{e} \left (a e^2+c d^2+2 c d e x\right ) \sqrt {4 c d e \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )+\left (c d^2-a e^2\right )^2}}+\frac {2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d}\right )}{6 c d}+\frac {2 (d+e x) \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d}\right )}{10 c d}+\frac {2 (d+e x)^2 \sqrt [4]{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d}\)

Input:

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

Output:

(2*(d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/4))/(5*c*d) + (9 
*(c*d^2 - a*e^2)*((2*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1/ 
4))/(3*c*d) + (5*(c*d^2 - a*e^2)*((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)])))/(6*c*d)))/(10*c*d)
 

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 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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