\(\int \frac {(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [256]

Optimal result

Integrand size = 37, antiderivative size = 287 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^4}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 c^4 d^4}+\frac {35 e \left (c d^2-a e^2\right ) (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 c^3 d^3}+\frac {7 e (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 c^2 d^2}+\frac {35 \sqrt {e} \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}} \] Output:

-2*(e*x+d)^4/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/8*e*(-a*e^2+c* 
d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^4/d^4+35/12*e*(-a*e^2+c*d 
^2)*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3+7/3*e*(e*x+d)^ 
2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/d^2+35/8*e^(1/2)*(-a*e^2+c*d 
^2)^3*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(1/2))/c^(9/2)/d^(9/2)
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {c} \sqrt {d} (a e+c d x) (d+e x)^2 \left (105 a^3 e^6-35 a^2 c d e^4 (8 d-e x)+7 a c^2 d^2 e^2 \left (33 d^2-14 d e x-2 e^2 x^2\right )+c^3 d^3 \left (-48 d^3+87 d^2 e x+38 d e^2 x^2+8 e^3 x^3\right )\right )+105 \sqrt {e} \left (c d^2-a e^2\right )^3 (a e+c d x)^{3/2} (d+e x)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{24 c^{9/2} d^{9/2} ((a e+c d x) (d+e x))^{3/2}} \] Input:

Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
 

Output:

(Sqrt[c]*Sqrt[d]*(a*e + c*d*x)*(d + e*x)^2*(105*a^3*e^6 - 35*a^2*c*d*e^4*( 
8*d - e*x) + 7*a*c^2*d^2*e^2*(33*d^2 - 14*d*e*x - 2*e^2*x^2) + c^3*d^3*(-4 
8*d^3 + 87*d^2*e*x + 38*d*e^2*x^2 + 8*e^3*x^3)) + 105*Sqrt[e]*(c*d^2 - a*e 
^2)^3*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d 
+ e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(24*c^(9/2)*d^(9/2)*((a*e + c*d*x)*( 
d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {1124, 2192, 27, 2192, 27, 1160, 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.

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

\(\Big \downarrow \) 1124

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\int \frac {c^3 d^3 \left (19 c d^2-11 a e^2\right ) x^2 e^6+2 c^2 d^2 \left (18 c^2 d^4-14 a c e^2 d^2+3 a^2 e^4\right ) x e^5+6 c d \left (2 c d^2-a e^2\right ) \left (2 c^2 d^4-2 a c e^2 d^2+a^2 e^4\right ) e^4}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 c d e}+\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^2}-\frac {2 (d+e x) \left (c d^2-a e^2\right )^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {c^3 d^3 \left (19 c d^2-11 a e^2\right ) x^2 e^6+2 c^2 d^2 \left (18 c^2 d^4-14 a c e^2 d^2+3 a^2 e^4\right ) x e^5+6 c d \left (2 c d^2-a e^2\right ) \left (2 c^2 d^4-2 a c e^2 d^2+a^2 e^4\right ) e^4}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 c d e}+\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^2}-\frac {2 (d+e x) \left (c d^2-a e^2\right )^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {\frac {\int \frac {c^2 d^2 e^5 \left (2 \left (48 c^3 d^6-91 a c^2 e^2 d^4+59 a^2 c e^4 d^2-12 a^3 e^6\right )+c d e \left (87 c^2 d^4-136 a c e^2 d^2+57 a^2 e^4\right ) x\right )}{2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d e}+\frac {1}{2} c^2 d^2 e^5 x \left (19 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}+\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^2}-\frac {2 (d+e x) \left (c d^2-a e^2\right )^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{4} c d e^4 \int \frac {2 \left (48 c^3 d^6-91 a c^2 e^2 d^4+59 a^2 c e^4 d^2-12 a^3 e^6\right )+c d e \left (87 c^2 d^4-136 a c e^2 d^2+57 a^2 e^4\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\frac {1}{2} c^2 d^2 e^5 x \left (19 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}+\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^2}-\frac {2 (d+e x) \left (c d^2-a e^2\right )^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\frac {1}{4} c d e^4 \left (\frac {105}{2} \left (c d^2-a e^2\right )^3 \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx+\left (57 a^2 e^4-136 a c d^2 e^2+87 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} c^2 d^2 e^5 x \left (19 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}+\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^2}-\frac {2 (d+e x) \left (c d^2-a e^2\right )^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {1}{4} c d e^4 \left (105 \left (c d^2-a e^2\right )^3 \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}+\left (57 a^2 e^4-136 a c d^2 e^2+87 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}\right )+\frac {1}{2} c^2 d^2 e^5 x \left (19 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}+\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^2}-\frac {2 (d+e x) \left (c d^2-a e^2\right )^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {1}{4} c d e^4 \left (\left (57 a^2 e^4-136 a c d^2 e^2+87 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac {105 \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {e}}\right )+\frac {1}{2} c^2 d^2 e^5 x \left (19 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c d e}+\frac {1}{3} c^2 d^2 e^5 x^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^4 d^4 e^2}-\frac {2 (d+e x) \left (c d^2-a e^2\right )^3}{c^4 d^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
 

Output:

(-2*(c*d^2 - a*e^2)^3*(d + e*x))/(c^4*d^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2]) + ((c^2*d^2*e^5*x^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 
2])/3 + ((c^2*d^2*e^5*(19*c*d^2 - 11*a*e^2)*x*Sqrt[a*d*e + (c*d^2 + a*e^2) 
*x + c*d*e*x^2])/2 + (c*d*e^4*((87*c^2*d^4 - 136*a*c*d^2*e^2 + 57*a^2*e^4) 
*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] + (105*(c*d^2 - a*e^2)^3*ArcT 
anh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c 
*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sqrt[c]*Sqrt[d]*Sqrt[e])))/4)/(6*c*d*e) 
)/(c^4*d^4*e^2)
 

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 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 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 1124

Int[((d_.) + (e_.)*(x_))^(m_.)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[-2*e*(2*c*d - b*e)^(m - 2)*((d + e*x)/(c^(m - 1)*Sqrt[a + 
b*x + c*x^2])), x] + Simp[e^2/c^(m - 1)   Int[(1/Sqrt[a + b*x + c*x^2])*Exp 
andToSum[((2*c*d - b*e)^(m - 1) - c^(m - 1)*(d + e*x)^(m - 1))/(c*d - b*e - 
 c*e*x), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e 
^2, 0] && IGtQ[m, 0]
 

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 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3116\) vs. \(2(257)=514\).

Time = 1.71 (sec) , antiderivative size = 3117, normalized size of antiderivative = 10.86

method	result	size
default	\(\text {Expression too large to display}\)	\(3117\)

Input:

int((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURNVERB 
OSE)
 

Output:

2*d^5*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^ 
2+c*d^2)*x+c*d*x^2*e)^(1/2)+e^5*(1/3*x^4/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d* 
x^2*e)^(1/2)-7/6*(a*e^2+c*d^2)/d/e/c*(1/2*x^3/d/e/c/(a*d*e+(a*e^2+c*d^2)*x 
+c*d*x^2*e)^(1/2)-5/4*(a*e^2+c*d^2)/d/e/c*(x^2/d/e/c/(a*d*e+(a*e^2+c*d^2)* 
x+c*d*x^2*e)^(1/2)-3/2*(a*e^2+c*d^2)/d/e/c*(-x/d/e/c/(a*d*e+(a*e^2+c*d^2)* 
x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)* 
x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2* 
e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))+1/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))-2*a/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e) 
^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c 
*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)))-3/2*a/c*(-x/d/e/c/(a*d* 
e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c*(-1/d/e/c/(a*d* 
e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c* 
d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/ 
2))+1/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)))-4/3*a/c*(x^2/d/e/c/(a*d*e+(a*e^2 
+c*d^2)*x+c*d*x^2*e)^(1/2)-3/2*(a*e^2+c*d^2)/d/e/c*(-x/d/e/c/(a*d*e+(a*e^2 
+c*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c*(-1/d/e/c/(a*d*e+(a*e^2 
+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/...
 

Fricas [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.64 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [-\frac {105 \, {\left (a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} + {\left (c^{4} d^{7} - 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - a^{3} c d e^{6}\right )} x\right )} \sqrt {\frac {e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x - 4 \, {\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {e}{c d}}\right ) - 4 \, {\left (8 \, c^{3} d^{3} e^{3} x^{3} - 48 \, c^{3} d^{6} + 231 \, a c^{2} d^{4} e^{2} - 280 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \, {\left (19 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (87 \, c^{3} d^{5} e - 98 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, -\frac {105 \, {\left (a c^{3} d^{6} e - 3 \, a^{2} c^{2} d^{4} e^{3} + 3 \, a^{3} c d^{2} e^{5} - a^{4} e^{7} + {\left (c^{4} d^{7} - 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} - a^{3} c d e^{6}\right )} x\right )} \sqrt {-\frac {e}{c d}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {e}{c d}}}{2 \, {\left (c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) - 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{3} - 48 \, c^{3} d^{6} + 231 \, a c^{2} d^{4} e^{2} - 280 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \, {\left (19 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (87 \, c^{3} d^{5} e - 98 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}\right ] \] Input:

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm=" 
fricas")
 

Output:

[-1/96*(105*(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 - a^4*e^7 + 
 (c^4*d^7 - 3*a*c^3*d^5*e^2 + 3*a^2*c^2*d^3*e^4 - a^3*c*d*e^6)*x)*sqrt(e/( 
c*d))*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 + 8*(c^2*d 
^3*e + a*c*d*e^3)*x - 4*(2*c^2*d^2*e*x + c^2*d^3 + a*c*d*e^2)*sqrt(c*d*e*x 
^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d))) - 4*(8*c^3*d^3*e^3*x^3 - 48 
*c^3*d^6 + 231*a*c^2*d^4*e^2 - 280*a^2*c*d^2*e^4 + 105*a^3*e^6 + 2*(19*c^3 
*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 + (87*c^3*d^5*e - 98*a*c^2*d^3*e^3 + 35*a^ 
2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*x + a* 
c^4*d^4*e), -1/48*(105*(a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 
- a^4*e^7 + (c^4*d^7 - 3*a*c^3*d^5*e^2 + 3*a^2*c^2*d^3*e^4 - a^3*c*d*e^6)* 
x)*sqrt(-e/(c*d))*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(-e/(c*d))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2* 
e + a*e^3)*x)) - 2*(8*c^3*d^3*e^3*x^3 - 48*c^3*d^6 + 231*a*c^2*d^4*e^2 - 2 
80*a^2*c*d^2*e^4 + 105*a^3*e^6 + 2*(19*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 
+ (87*c^3*d^5*e - 98*a*c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a 
*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*x + a*c^4*d^4*e)]
 

Sympy [F]

\[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{5}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
 

Output:

Integral((d + e*x)**5/((d + e*x)*(a*e + c*d*x))**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm=" 
maxima")
 

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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm=" 
giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[%%%{16,[5,5,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,2 
]%%%}+%%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^5}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:

int((d + e*x)^5/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2),x)
 

Output:

int((d + e*x)^5/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.98 \[ \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-840 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{3} e^{6}+2520 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} c \,d^{2} e^{4}-2520 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,c^{2} d^{4} e^{2}+840 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{3} d^{6}+525 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{3} e^{6}-1575 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{4}+1575 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a \,c^{2} d^{4} e^{2}-525 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{3} d^{6}+840 \sqrt {e x +d}\, a^{3} c d \,e^{6}-2240 \sqrt {e x +d}\, a^{2} c^{2} d^{3} e^{4}+280 \sqrt {e x +d}\, a^{2} c^{2} d^{2} e^{5} x +1848 \sqrt {e x +d}\, a \,c^{3} d^{5} e^{2}-784 \sqrt {e x +d}\, a \,c^{3} d^{4} e^{3} x -112 \sqrt {e x +d}\, a \,c^{3} d^{3} e^{4} x^{2}-384 \sqrt {e x +d}\, c^{4} d^{7}+696 \sqrt {e x +d}\, c^{4} d^{6} e x +304 \sqrt {e x +d}\, c^{4} d^{5} e^{2} x^{2}+64 \sqrt {e x +d}\, c^{4} d^{4} e^{3} x^{3}}{192 \sqrt {c d x +a e}\, c^{5} d^{5}} \] Input:

int((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

( - 840*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + 
c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**3*e**6 + 
 2520*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c* 
d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c*d**2*e 
**4 - 2520*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e 
 + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a*c**2*d 
**4*e**2 + 840*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt 
(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c**3 
*d**6 + 525*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*e**6 - 1575*sqr 
t(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**2*e**4 + 1575*sqrt(e)*sqr 
t(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**2 - 525*sqrt(e)*sqrt(d)*sqrt 
(c)*sqrt(a*e + c*d*x)*c**3*d**6 + 840*sqrt(d + e*x)*a**3*c*d*e**6 - 2240*s 
qrt(d + e*x)*a**2*c**2*d**3*e**4 + 280*sqrt(d + e*x)*a**2*c**2*d**2*e**5*x 
 + 1848*sqrt(d + e*x)*a*c**3*d**5*e**2 - 784*sqrt(d + e*x)*a*c**3*d**4*e** 
3*x - 112*sqrt(d + e*x)*a*c**3*d**3*e**4*x**2 - 384*sqrt(d + e*x)*c**4*d** 
7 + 696*sqrt(d + e*x)*c**4*d**6*e*x + 304*sqrt(d + e*x)*c**4*d**5*e**2*x** 
2 + 64*sqrt(d + e*x)*c**4*d**4*e**3*x**3)/(192*sqrt(a*e + c*d*x)*c**5*d**5 
)