\(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx\) [222]

Optimal result

Integrand size = 35, antiderivative size = 486 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {5 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{1024 c^4 d^4 e^3}-\frac {5 \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1536 c^4 d^4 e^2 (d+e x)}+\frac {\left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{384 c^4 d^4 e (d+e x)^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{7 c d}+\frac {\left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{64 c^4 d^4 (d+e x)^3}+\frac {\left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{24 c^3 d^3 (d+e x)^2}+\frac {\left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{12 c^2 d^2 (d+e x)}-\frac {5 \left (c d^2-a e^2\right )^7 \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{1024 c^{9/2} d^{9/2} e^{7/2}} \] Output:

5/1024*(-a*e^2+c*d^2)^6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4/e^ 
3-5/1536*(-a*e^2+c*d^2)^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^4/ 
e^2/(e*x+d)+1/384*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2) 
/c^4/d^4/e/(e*x+d)^2+1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d+1/64* 
(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^4/d^4/(e*x+d)^3 
+1/24*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^3/d^3/(e* 
x+d)^2+1/12*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2 
/(e*x+d)-5/1024*(-a*e^2+c*d^2)^7*arctanh(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c* 
d*e*x^2)^(1/2)/c^(1/2)/d^(1/2)/(e*x+d))/c^(9/2)/d^(9/2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 1.31 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.90 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-105 a^6 e^{12}+70 a^5 c d e^{10} (10 d+e x)-7 a^4 c^2 d^2 e^8 \left (283 d^2+66 d e x+8 e^2 x^2\right )+4 a^3 c^3 d^3 e^6 \left (768 d^3+323 d^2 e x+92 d e^2 x^2+12 e^3 x^3\right )+a^2 c^4 d^4 e^4 \left (1981 d^4+17140 d^3 e x+27648 d^2 e^2 x^2+18800 d e^3 x^3+4736 e^4 x^4\right )+2 a c^5 d^5 e^2 \left (-350 d^5+231 d^4 e x+9032 d^3 e^2 x^2+18248 d^2 e^3 x^3+13824 d e^4 x^4+3712 e^5 x^5\right )+c^6 d^6 \left (105 d^6-70 d^5 e x+56 d^4 e^2 x^2+6096 d^3 e^3 x^3+13696 d^2 e^4 x^4+11008 d e^5 x^5+3072 e^6 x^6\right )\right )}{(a e+c d x)^2 (d+e x)^2}-\frac {105 \left (c d^2-a e^2\right )^7 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{21504 c^{9/2} d^{9/2} e^{7/2}} \] Input:

Integrate[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
 

Output:

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[e]*(-105*a^6*e^12 
+ 70*a^5*c*d*e^10*(10*d + e*x) - 7*a^4*c^2*d^2*e^8*(283*d^2 + 66*d*e*x + 8 
*e^2*x^2) + 4*a^3*c^3*d^3*e^6*(768*d^3 + 323*d^2*e*x + 92*d*e^2*x^2 + 12*e 
^3*x^3) + a^2*c^4*d^4*e^4*(1981*d^4 + 17140*d^3*e*x + 27648*d^2*e^2*x^2 + 
18800*d*e^3*x^3 + 4736*e^4*x^4) + 2*a*c^5*d^5*e^2*(-350*d^5 + 231*d^4*e*x 
+ 9032*d^3*e^2*x^2 + 18248*d^2*e^3*x^3 + 13824*d*e^4*x^4 + 3712*e^5*x^5) + 
 c^6*d^6*(105*d^6 - 70*d^5*e*x + 56*d^4*e^2*x^2 + 6096*d^3*e^3*x^3 + 13696 
*d^2*e^4*x^4 + 11008*d*e^5*x^5 + 3072*e^6*x^6)))/((a*e + c*d*x)^2*(d + e*x 
)^2) - (105*(c*d^2 - a*e^2)^7*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqr 
t[e]*Sqrt[a*e + c*d*x])])/((a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/(21504*c 
^(9/2)*d^(9/2)*e^(7/2))
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.81, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1160, 1087, 1087, 1087, 1092, 219}

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.

\(\displaystyle \int (d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \, dx\)

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \int \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}dx}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \int \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 c d e}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (d^2-\frac {a e^2}{c}\right ) \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e}-\frac {5 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 c d e}-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\right )}{16 c d e}\right )}{24 c d e}\right )}{2 d}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{7 c d}\)

Input:

Int[(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]
 

Output:

(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(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)^(5/ 
2))/(12*c*d*e) - (5*(c*d^2 - a*e^2)^2*(((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)^(3/2))/(8*c*d*e) - (3*(c*d^2 - a*e^2)^2* 
(((c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) 
/(4*c*d*e) - ((c*d^2 - a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqr 
t[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(8*c^( 
3/2)*d^(3/2)*e^(3/2))))/(16*c*d*e)))/(24*c*d*e)))/(2*d)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 1092

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

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 [A] (verified)

Time = 1.50 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.53

method	result	size
default	\(d \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{24 d e c}\right )+e \left (\frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {7}{2}}}{7 d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}{12 c d e}+\frac {5 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{8 c d e}+\frac {3 \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{16 d e c}\right )}{24 d e c}\right )}{2 d e c}\right )\)	\(745\)

Input:

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_RETURNVERBOS 
E)
 

Output:

d*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/c/ 
d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/8*(2*c*d*e*x+a*e^2+c*d^2 
)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e^2-(a*e^2 
+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x 
^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+ 
1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/ 
(d*e*c)^(1/2))))+e*(1/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/d/e/c-1/2* 
(a*e^2+c*d^2)/d/e/c*(1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c 
*d*x^2*e)^(5/2)/c/d/e+5/24*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/8*(2*c 
*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4* 
a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a* 
e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/ 
e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+ 
c*d*x^2*e)^(1/2))/(d*e*c)^(1/2)))))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1270, normalized size of antiderivative = 2.61 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fr 
icas")
 

Output:

[-1/86016*(105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a^3 
*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^12 
 - a^7*e^14)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + 
 a^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^ 
2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - 4*(3072*c^7*d^7*e^ 
7*x^6 + 105*c^7*d^13*e - 700*a*c^6*d^11*e^3 + 1981*a^2*c^5*d^9*e^5 + 3072* 
a^3*c^4*d^7*e^7 - 1981*a^4*c^3*d^5*e^9 + 700*a^5*c^2*d^3*e^11 - 105*a^6*c* 
d*e^13 + 256*(43*c^7*d^8*e^6 + 29*a*c^6*d^6*e^8)*x^5 + 128*(107*c^7*d^9*e^ 
5 + 216*a*c^6*d^7*e^7 + 37*a^2*c^5*d^5*e^9)*x^4 + 16*(381*c^7*d^10*e^4 + 2 
281*a*c^6*d^8*e^6 + 1175*a^2*c^5*d^6*e^8 + 3*a^3*c^4*d^4*e^10)*x^3 + 8*(7* 
c^7*d^11*e^3 + 2258*a*c^6*d^9*e^5 + 3456*a^2*c^5*d^7*e^7 + 46*a^3*c^4*d^5* 
e^9 - 7*a^4*c^3*d^3*e^11)*x^2 - 2*(35*c^7*d^12*e^2 - 231*a*c^6*d^10*e^4 - 
8570*a^2*c^5*d^8*e^6 - 646*a^3*c^4*d^6*e^8 + 231*a^4*c^3*d^4*e^10 - 35*a^5 
*c^2*d^2*e^12)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^ 
4), 1/43008*(105*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21*a^2*c^5*d^10*e^4 - 35*a 
^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d^4*e^10 + 7*a^6*c*d^2*e^ 
12 - a^7*e^14)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a 
*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d 
^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(3072*c^7*d^7*e^7*x^6 + 105*c^7*d 
^13*e - 700*a*c^6*d^11*e^3 + 1981*a^2*c^5*d^9*e^5 + 3072*a^3*c^4*d^7*e^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5268 vs. \(2 (452) = 904\).

Time = 4.51 (sec) , antiderivative size = 5268, normalized size of antiderivative = 10.84 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
 

Output:

Piecewise((sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(c**2*d**2*e**3* 
x**6/7 + x**5*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13* 
a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e) + x**4*(3*a**2*c*d*e**6 + 78*a*c**2*d 
**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c**2*d**2 
*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/( 
6*c*d*e))/(5*c*d*e) + x**3*(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d* 
*4*e**3 - 5*a*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13* 
a*e**2/2 + 13*c*d**2/2)/7)/(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/ 
2)*(3*a**2*c*d*e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e** 
2/2 + 11*c*d**2/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3 
*(13*a*e**2/2 + 13*c*d**2/2)/7)/(6*c*d*e))/(5*c*d*e))/(4*c*d*e) + x**2*(4* 
a**3*d*e**6 + 18*a**2*c*d**3*e**4 + 12*a*c**2*d**5*e**2 - 4*a*(3*a**2*c*d* 
e**6 + 78*a*c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2 
/2)*(3*a*c**2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 
 13*c*d**2/2)/7)/(6*c*d*e))/(5*c) + c**3*d**7 - (7*a*e**2/2 + 7*c*d**2/2)* 
(a**3*e**7 + 12*a**2*c*d**2*e**5 + 18*a*c**2*d**4*e**3 - 5*a*(3*a*c**2*d** 
2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**2/2)/7)/ 
(6*c) + 4*c**3*d**6*e - (9*a*e**2/2 + 9*c*d**2/2)*(3*a**2*c*d*e**6 + 78*a* 
c**2*d**3*e**4/7 + 6*c**3*d**5*e**2 - (11*a*e**2/2 + 11*c*d**2/2)*(3*a*c** 
2*d**2*e**5 + 4*c**3*d**4*e**3 - c**2*d**2*e**3*(13*a*e**2/2 + 13*c*d**...
 

Maxima [F(-2)]

Exception generated. \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="ma 
xima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.34 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\frac {1}{21504} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, c^{2} d^{2} e^{3} x + \frac {43 \, c^{8} d^{9} e^{8} + 29 \, a c^{7} d^{7} e^{10}}{c^{6} d^{6} e^{6}}\right )} x + \frac {107 \, c^{8} d^{10} e^{7} + 216 \, a c^{7} d^{8} e^{9} + 37 \, a^{2} c^{6} d^{6} e^{11}}{c^{6} d^{6} e^{6}}\right )} x + \frac {381 \, c^{8} d^{11} e^{6} + 2281 \, a c^{7} d^{9} e^{8} + 1175 \, a^{2} c^{6} d^{7} e^{10} + 3 \, a^{3} c^{5} d^{5} e^{12}}{c^{6} d^{6} e^{6}}\right )} x + \frac {7 \, c^{8} d^{12} e^{5} + 2258 \, a c^{7} d^{10} e^{7} + 3456 \, a^{2} c^{6} d^{8} e^{9} + 46 \, a^{3} c^{5} d^{6} e^{11} - 7 \, a^{4} c^{4} d^{4} e^{13}}{c^{6} d^{6} e^{6}}\right )} x - \frac {35 \, c^{8} d^{13} e^{4} - 231 \, a c^{7} d^{11} e^{6} - 8570 \, a^{2} c^{6} d^{9} e^{8} - 646 \, a^{3} c^{5} d^{7} e^{10} + 231 \, a^{4} c^{4} d^{5} e^{12} - 35 \, a^{5} c^{3} d^{3} e^{14}}{c^{6} d^{6} e^{6}}\right )} x + \frac {105 \, c^{8} d^{14} e^{3} - 700 \, a c^{7} d^{12} e^{5} + 1981 \, a^{2} c^{6} d^{10} e^{7} + 3072 \, a^{3} c^{5} d^{8} e^{9} - 1981 \, a^{4} c^{4} d^{6} e^{11} + 700 \, a^{5} c^{3} d^{4} e^{13} - 105 \, a^{6} c^{2} d^{2} e^{15}}{c^{6} d^{6} e^{6}}\right )} + \frac {5 \, {\left (c^{7} d^{14} - 7 \, a c^{6} d^{12} e^{2} + 21 \, a^{2} c^{5} d^{10} e^{4} - 35 \, a^{3} c^{4} d^{8} e^{6} + 35 \, a^{4} c^{3} d^{6} e^{8} - 21 \, a^{5} c^{2} d^{4} e^{10} + 7 \, a^{6} c d^{2} e^{12} - a^{7} e^{14}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{2048 \, \sqrt {c d e} c^{4} d^{4} e^{3}} \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="gi 
ac")
 

Output:

1/21504*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*(4*(2*(8*(2*(12*c^2 
*d^2*e^3*x + (43*c^8*d^9*e^8 + 29*a*c^7*d^7*e^10)/(c^6*d^6*e^6))*x + (107* 
c^8*d^10*e^7 + 216*a*c^7*d^8*e^9 + 37*a^2*c^6*d^6*e^11)/(c^6*d^6*e^6))*x + 
 (381*c^8*d^11*e^6 + 2281*a*c^7*d^9*e^8 + 1175*a^2*c^6*d^7*e^10 + 3*a^3*c^ 
5*d^5*e^12)/(c^6*d^6*e^6))*x + (7*c^8*d^12*e^5 + 2258*a*c^7*d^10*e^7 + 345 
6*a^2*c^6*d^8*e^9 + 46*a^3*c^5*d^6*e^11 - 7*a^4*c^4*d^4*e^13)/(c^6*d^6*e^6 
))*x - (35*c^8*d^13*e^4 - 231*a*c^7*d^11*e^6 - 8570*a^2*c^6*d^9*e^8 - 646* 
a^3*c^5*d^7*e^10 + 231*a^4*c^4*d^5*e^12 - 35*a^5*c^3*d^3*e^14)/(c^6*d^6*e^ 
6))*x + (105*c^8*d^14*e^3 - 700*a*c^7*d^12*e^5 + 1981*a^2*c^6*d^10*e^7 + 3 
072*a^3*c^5*d^8*e^9 - 1981*a^4*c^4*d^6*e^11 + 700*a^5*c^3*d^4*e^13 - 105*a 
^6*c^2*d^2*e^15)/(c^6*d^6*e^6)) + 5/2048*(c^7*d^14 - 7*a*c^6*d^12*e^2 + 21 
*a^2*c^5*d^10*e^4 - 35*a^3*c^4*d^8*e^6 + 35*a^4*c^3*d^6*e^8 - 21*a^5*c^2*d 
^4*e^10 + 7*a^6*c*d^2*e^12 - a^7*e^14)*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d 
*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt( 
c*d*e)*c^4*d^4*e^3)
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx=\int \left (d+e\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2} \,d x \] Input:

int((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)
 

Output:

int((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 1308, normalized size of antiderivative = 2.69 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
 

Output:

( - 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*c*d*e**13 + 700*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**5*c**2*d**3*e**11 + 70*sqrt(d + e*x)*sqrt(a*e + c*d* 
x)*a**5*c**2*d**2*e**12*x - 1981*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**3 
*d**5*e**9 - 462*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**3*d**4*e**10*x - 
56*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**3*d**3*e**11*x**2 + 3072*sqrt(d 
 + e*x)*sqrt(a*e + c*d*x)*a**3*c**4*d**7*e**7 + 1292*sqrt(d + e*x)*sqrt(a* 
e + c*d*x)*a**3*c**4*d**6*e**8*x + 368*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a** 
3*c**4*d**5*e**9*x**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**4*d**4* 
e**10*x**3 + 1981*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**5*d**9*e**5 + 17 
140*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**5*d**8*e**6*x + 27648*sqrt(d + 
 e*x)*sqrt(a*e + c*d*x)*a**2*c**5*d**7*e**7*x**2 + 18800*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a**2*c**5*d**6*e**8*x**3 + 4736*sqrt(d + e*x)*sqrt(a*e + c* 
d*x)*a**2*c**5*d**5*e**9*x**4 - 700*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**6 
*d**11*e**3 + 462*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**6*d**10*e**4*x + 18 
064*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**6*d**9*e**5*x**2 + 36496*sqrt(d + 
 e*x)*sqrt(a*e + c*d*x)*a*c**6*d**8*e**6*x**3 + 27648*sqrt(d + e*x)*sqrt(a 
*e + c*d*x)*a*c**6*d**7*e**7*x**4 + 7424*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a 
*c**6*d**6*e**8*x**5 + 105*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**7*d**13*e - 
70*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**7*d**12*e**2*x + 56*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*c**7*d**11*e**3*x**2 + 6096*sqrt(d + e*x)*sqrt(a*e + c*...