\(\int \frac {(d+e x)^3}{x^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [109]

Optimal result

Integrand size = 40, antiderivative size = 250 \[ \int \frac {(d+e x)^3}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)}{a^3 e^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 a^2 e^2 x^2}+\frac {\left (7 c d^2-5 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 a^3 e^3 x}-\frac {3 \left (c d^2-a e^2\right ) \left (5 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 a^{7/2} \sqrt {d} e^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 10.15 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^3}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {a} \sqrt {d} \sqrt {e} (d+e x) \left (-15 c^2 d^3 x^2+a^2 e^2 (2 d+5 e x)+a c d e x (-5 d+13 e x)\right )-3 \left (5 c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) x^2 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{4 a^{7/2} \sqrt {d} e^{7/2} x^2 \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(-(Sqrt[a]*Sqrt[d]*Sqrt[e]*(d + e*x)*(-15*c^2*d^3*x^2 + a^2*e^2*(2*d + 5*e 
*x) + a*c*d*e*x*(-5*d + 13*e*x))) - 3*(5*c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 
)*x^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/ 
(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(4*a^(7/2)*Sqrt[d]*e^(7/2)*x^2*Sqrt[(a*e 
 + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1212, 25, 2181, 27, 1228, 1154, 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.

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

\(\Big \downarrow \) 1212

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(2*c*d*(c*d^2 - a*e^2)*(d + e*x))/(a^3*e^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x 
+ c*d*e*x^2]) + c*d*e^3*(-1/2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/ 
(a^2*c*e^5*x^2) - (-(((7*c*d^2 - 5*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2])/(a^2*c*e*x)) + (3*(c*d^2 - a*e^2)*(5*c*d^2 - a*e^2)*ArcTanh[( 
2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^ 
2 + a*e^2)*x + c*d*e*x^2])])/(2*a^(5/2)*c*Sqrt[d]*e^(3/2)))/(4*a*d*e^5))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1212

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

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) 
^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - 
b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1280\) vs. \(2(224)=448\).

Time = 2.34 (sec) , antiderivative size = 1281, normalized size of antiderivative = 5.12

method	result	size
default	\(\text {Expression too large to display}\)	\(1281\)

Input:

int((e*x+d)^3/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETURN 
VERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 1.94 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^3}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left ({\left (5 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x^{3} + {\left (5 \, a c^{2} d^{4} e - 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{2}\right )} \sqrt {a d e} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{3} d^{2} e^{3} - {\left (15 \, a c^{2} d^{4} e - 13 \, a^{2} c d^{2} e^{3}\right )} x^{2} - 5 \, {\left (a^{2} c d^{3} e^{2} - a^{3} d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (a^{4} c d^{2} e^{4} x^{3} + a^{5} d e^{5} x^{2}\right )}}, \frac {3 \, {\left ({\left (5 \, c^{3} d^{5} - 6 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x^{3} + {\left (5 \, a c^{2} d^{4} e - 6 \, a^{2} c d^{2} e^{3} + a^{3} e^{5}\right )} x^{2}\right )} \sqrt {-a d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{3} d^{2} e^{3} - {\left (15 \, a c^{2} d^{4} e - 13 \, a^{2} c d^{2} e^{3}\right )} x^{2} - 5 \, {\left (a^{2} c d^{3} e^{2} - a^{3} d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (a^{4} c d^{2} e^{4} x^{3} + a^{5} d e^{5} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/16*(3*((5*c^3*d^5 - 6*a*c^2*d^3*e^2 + a^2*c*d*e^4)*x^3 + (5*a*c^2*d^4*e 
 - 6*a^2*c*d^2*e^3 + a^3*e^5)*x^2)*sqrt(a*d*e)*log((8*a^2*d^2*e^2 + (c^2*d 
^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a* 
e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e 
^3)*x)/x^2) - 4*(2*a^3*d^2*e^3 - (15*a*c^2*d^4*e - 13*a^2*c*d^2*e^3)*x^2 - 
 5*(a^2*c*d^3*e^2 - a^3*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2) 
*x))/(a^4*c*d^2*e^4*x^3 + a^5*d*e^5*x^2), 1/8*(3*((5*c^3*d^5 - 6*a*c^2*d^3 
*e^2 + a^2*c*d*e^4)*x^3 + (5*a*c^2*d^4*e - 6*a^2*c*d^2*e^3 + a^3*e^5)*x^2) 
*sqrt(-a*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a* 
d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a* 
c*d^3*e + a^2*d*e^3)*x)) - 2*(2*a^3*d^2*e^3 - (15*a*c^2*d^4*e - 13*a^2*c*d 
^2*e^3)*x^2 - 5*(a^2*c*d^3*e^2 - a^3*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c 
*d^2 + a*e^2)*x))/(a^4*c*d^2*e^4*x^3 + a^5*d*e^5*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

integrate((e*x+d)^3/x^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorit 
hm="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%%%{%%%{1,[1,1,11]%%%},[2,7]%%%}+%%%{%%%{-4,[2,3,9]%%%},[2,6]% 
%%}+%%%{%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 1140, normalized size of antiderivative = 4.56 \[ \int \frac {(d+e x)^3}{x^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(9*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) 
 - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d 
+ e*x))*a**3*e**6*x**2 - 39*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log( 
sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) 
+ sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**2*c*d**2*e**4*x**2 - 45*sqrt(e)*sqrt(d 
)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c) 
*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*c**2*d* 
*4*e**2*x**2 + 75*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sq 
rt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)* 
sqrt(c)*sqrt(d + e*x))*c**3*d**6*x**2 + 9*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e 
 + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e 
**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**3*e**6*x**2 - 39*sqrt(e) 
*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2* 
sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a* 
*2*c*d**2*e**4*x**2 - 45*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqr 
t(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + s 
qrt(d)*sqrt(c)*sqrt(d + e*x))*a*c**2*d**4*e**2*x**2 + 75*sqrt(e)*sqrt(d)*s 
qrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(c)*sq 
rt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*c**3*d**6*x* 
*2 - 9*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(2*sqrt(e)*sqrt(d)*...