\(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/4} \, dx\) [345]

Optimal result

Integrand size = 35, antiderivative size = 288 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4} \, dx=\frac {\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4}}{10 c^2 d^2 e}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/4}}{7 c d}+\frac {\left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/4}}{5 c^2 d^2 (d+e x)}-\frac {3 \left (c d^2-a e^2\right )^4 \sqrt [4]{-\frac {c d e (a e+c d x) (d+e x)}{\left (c d^2-a e^2\right )^2}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {a e^2+c d (d+2 e x)}{c d^2-a e^2}\right )\right |2\right )}{20 \sqrt {2} c^3 d^3 e^2 \sqrt [4]{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:

1/10*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/4)/c^2/d^2/e+2/ 
7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/4)/c/d+1/5*(-a*e^2+c*d^2)*(a*d*e+(a 
*e^2+c*d^2)*x+c*d*e*x^2)^(7/4)/c^2/d^2/(e*x+d)-3/40*(-a*e^2+c*d^2)^4*(-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^2/(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 = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.31 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4} \, dx=\frac {4 ((a e+c d x) (d+e x))^{7/4} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {7}{4},\frac {11}{4},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{7 c d \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{7/4}} \] 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))^(7/4)*Hypergeometric2F1[-7/4, 7/4, 11/4, (e*( 
a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(7*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. \(1081\) vs. \(2(288)=576\).

Time = 1.28 (sec) , antiderivative size = 1081, normalized size of antiderivative = 3.75, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1160, 1087, 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.

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/4}}{5 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt [4]{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{20 c d e}\right )}{2 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/4}}{7 c d}\)

\(\Big \downarrow \) 1094

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

\(\displaystyle \frac {2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/4}}{7 c d}+\frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (c d^2+2 c e x d+a e^2\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/4}}{5 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \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 )}{5 c d e \left (c d^2+2 c e x d+a e^2\right )}\right )}{2 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)^(7/4))/(7*c*d) + ((d^2 - (a*e^2 
)/c)*(((c*d^2 + a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2) 
^(3/4))/(5*c*d*e) - (3*(c*d^2 - a*e^2)^2*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*ArcTa 
n[(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)])))/(S 
qrt[c]*Sqrt[d]*Sqrt[e]) + ((c*d^2 - a*e^2)^(3/2)*(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)]*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*...
 

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 1087

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

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]

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

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]

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

Sympy [F]

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

Maxima [F]

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

Giac [F]

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

Mupad [F(-1)]

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

\[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/4} \, dx=\frac {-16 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} a^{2} d \,e^{3}+12 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} a^{2} e^{4} x +96 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} a c \,d^{3} e +80 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} a c \,d^{2} e^{2} x +40 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} a c d \,e^{3} x^{2}+68 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} c^{2} d^{4} x +40 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} c^{2} d^{3} e \,x^{2}-21 \left (\int \frac {\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} x}{a c d \,e^{3} x^{2}+c^{2} d^{3} e \,x^{2}+a^{2} e^{4} x +2 a c \,d^{2} e^{2} x +c^{2} d^{4} x +a^{2} d \,e^{3}+a c \,d^{3} e}d x \right ) a^{4} e^{8}+42 \left (\int \frac {\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} x}{a c d \,e^{3} x^{2}+c^{2} d^{3} e \,x^{2}+a^{2} e^{4} x +2 a c \,d^{2} e^{2} x +c^{2} d^{4} x +a^{2} d \,e^{3}+a c \,d^{3} e}d x \right ) a^{3} c \,d^{2} e^{6}-42 \left (\int \frac {\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} x}{a c d \,e^{3} x^{2}+c^{2} d^{3} e \,x^{2}+a^{2} e^{4} x +2 a c \,d^{2} e^{2} x +c^{2} d^{4} x +a^{2} d \,e^{3}+a c \,d^{3} e}d x \right ) a \,c^{3} d^{6} e^{2}+21 \left (\int \frac {\left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{4}} x}{a c d \,e^{3} x^{2}+c^{2} d^{3} e \,x^{2}+a^{2} e^{4} x +2 a c \,d^{2} e^{2} x +c^{2} d^{4} x +a^{2} d \,e^{3}+a c \,d^{3} e}d x \right ) c^{4} d^{8}}{140 c d \left (a \,e^{2}+c \,d^{2}\right )} \] Input:

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

Output:

( - 16*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*a**2*d*e**3 + 12* 
(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*a**2*e**4*x + 96*(a*d*e 
+ a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*a*c*d**3*e + 80*(a*d*e + a*e**2 
*x + c*d**2*x + c*d*e*x**2)**(3/4)*a*c*d**2*e**2*x + 40*(a*d*e + a*e**2*x 
+ c*d**2*x + c*d*e*x**2)**(3/4)*a*c*d*e**3*x**2 + 68*(a*d*e + a*e**2*x + c 
*d**2*x + c*d*e*x**2)**(3/4)*c**2*d**4*x + 40*(a*d*e + a*e**2*x + c*d**2*x 
 + c*d*e*x**2)**(3/4)*c**2*d**3*e*x**2 - 21*int(((a*d*e + a*e**2*x + c*d** 
2*x + c*d*e*x**2)**(3/4)*x)/(a**2*d*e**3 + a**2*e**4*x + a*c*d**3*e + 2*a* 
c*d**2*e**2*x + a*c*d*e**3*x**2 + c**2*d**4*x + c**2*d**3*e*x**2),x)*a**4* 
e**8 + 42*int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*x)/(a**2* 
d*e**3 + a**2*e**4*x + a*c*d**3*e + 2*a*c*d**2*e**2*x + a*c*d*e**3*x**2 + 
c**2*d**4*x + c**2*d**3*e*x**2),x)*a**3*c*d**2*e**6 - 42*int(((a*d*e + a*e 
**2*x + c*d**2*x + c*d*e*x**2)**(3/4)*x)/(a**2*d*e**3 + a**2*e**4*x + a*c* 
d**3*e + 2*a*c*d**2*e**2*x + a*c*d*e**3*x**2 + c**2*d**4*x + c**2*d**3*e*x 
**2),x)*a*c**3*d**6*e**2 + 21*int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x* 
*2)**(3/4)*x)/(a**2*d*e**3 + a**2*e**4*x + a*c*d**3*e + 2*a*c*d**2*e**2*x 
+ a*c*d*e**3*x**2 + c**2*d**4*x + c**2*d**3*e*x**2),x)*c**4*d**8)/(140*c*d 
*(a*e**2 + c*d**2))