\(\int \frac {(d+e x)^3}{x^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [110]

Optimal result

Integrand size = 40, antiderivative size = 340 \[ \int \frac {(d+e x)^3}{x^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)}{a^4 e^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a^2 e^2 x^3}+\frac {\left (11 c d^2-7 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^3 e^3 x^2}-\frac {\left (57 c^2 d^4-52 a c d^2 e^2+3 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^4 d e^4 x}+\frac {\left (c d^2-a e^2\right ) \left (35 c^2 d^4-10 a c d^2 e^2-a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 a^{9/2} d^{3/2} e^{9/2}} \] Output:

-2*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)/a^4/e^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2 
)^(1/2)-1/3*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/e^2/x^3+1/12*(-7 
*a*e^2+11*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^2-1/24* 
(3*a^2*e^4-52*a*c*d^2*e^2+57*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 
/2)/a^4/d/e^4/x+1/8*(-a*e^2+c*d^2)*(-a^2*e^4-10*a*c*d^2*e^2+35*c^2*d^4)*ar 
ctanh(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^(9/2)/d^(3/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 10.21 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x)^3}{x^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {a} \sqrt {d} \sqrt {e} (d+e x) \left (105 c^3 d^5 x^3+5 a c^2 d^3 e x^2 (7 d-20 e x)+a^2 c d e^2 x \left (-14 d^2-38 d e x+3 e^2 x^2\right )+a^3 e^3 \left (8 d^2+14 d e x+3 e^2 x^2\right )\right )+3 \left (35 c^3 d^6-45 a c^2 d^4 e^2+9 a^2 c d^2 e^4+a^3 e^6\right ) x^3 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{24 a^{9/2} d^{3/2} e^{9/2} x^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(d + e*x)^3/(x^4*(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)*(105*c^3*d^5*x^3 + 5*a*c^2*d^3*e*x^2* 
(7*d - 20*e*x) + a^2*c*d*e^2*x*(-14*d^2 - 38*d*e*x + 3*e^2*x^2) + a^3*e^3* 
(8*d^2 + 14*d*e*x + 3*e^2*x^2))) + 3*(35*c^3*d^6 - 45*a*c^2*d^4*e^2 + 9*a^ 
2*c*d^2*e^4 + a^3*e^6)*x^3*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])])/(24*a^(9/2)*d^(3/2) 
*e^(9/2)*x^3*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {1212, 25, 2181, 27, 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.

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

\(\Big \downarrow \) 1212

\(\displaystyle -c^2 d^2 e^4 \int -\frac {-\frac {\left (c d^2-a e^2\right )^2 x^3}{a^4 c d e^8}+\frac {\left (c d^2-a e^2\right )^2 x^2}{a^3 c^2 d^2 e^7}-\frac {\left (c d^2-2 a e^2\right ) x}{a^2 c^2 d e^6}+\frac {1}{a c^2 e^5}}{x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle c^2 d^2 e^4 \int \frac {-\frac {\left (c d^2-a e^2\right )^2 x^3}{a^4 c d e^8}+\frac {\left (c d^2-a e^2\right )^2 x^2}{a^3 c^2 d^2 e^7}-\frac {\left (c d^2-2 a e^2\right ) x}{a^2 c^2 d e^6}+\frac {1}{a c^2 e^5}}{x^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2181

\(\displaystyle c^2 d^2 e^4 \left (-\frac {\int \frac {\frac {6 \left (c d^2-a e^2\right )^2 x^2}{a^3 c e^7}-\frac {2 \left (3 c^2 d^4-8 a c e^2 d^2+3 a^2 e^4\right ) x}{a^2 c^2 d e^6}+\frac {11 c d^2-7 a e^2}{a c^2 e^5}}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 c^2 d e^6 x^3}\right )-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle c^2 d^2 e^4 \left (-\frac {\int \frac {\frac {6 \left (c d^2-a e^2\right )^2 x^2}{a^3 c e^7}-\frac {2 \left (3 c^2 d^4-8 a c e^2 d^2+3 a^2 e^4\right ) x}{a^2 c^2 d e^6}+\frac {11 c d^2-7 a e^2}{a c^2 e^5}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 c^2 d e^6 x^3}\right )-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2181

\(\displaystyle c^2 d^2 e^4 \left (-\frac {-\frac {\int \frac {\frac {57 d^4}{a}-\frac {52 e^2 d^2}{c}-2 \left (\frac {12 c d^4}{a^2 e^4}-\frac {35 d^2}{a e^2}+\frac {19}{c}\right ) e^3 x d+\frac {3 a e^4}{c^2}}{2 e^5 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d e^6 x^2}}{6 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 c^2 d e^6 x^3}\right )-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle c^2 d^2 e^4 \left (-\frac {-\frac {\int \frac {\frac {57 d^4}{a}-\frac {52 e^2 d^2}{c}-2 \left (\frac {12 c d^4}{a^2 e^4}-\frac {35 d^2}{a e^2}+\frac {19}{c}\right ) e^3 x d+\frac {3 a e^4}{c^2}}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e^6}-\frac {\left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d e^6 x^2}}{6 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 c^2 d e^6 x^3}\right )-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1228

\(\displaystyle c^2 d^2 e^4 \left (-\frac {-\frac {-\frac {3 \left (c d^2-a e^2\right ) \left (-a^2 e^4-10 a c d^2 e^2+35 c^2 d^4\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a^2 c^2 d e}-\frac {\left (3 a^2 e^4-52 a c d^2 e^2+57 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^2 d e x}}{4 a d e^6}-\frac {\left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d e^6 x^2}}{6 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 c^2 d e^6 x^3}\right )-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle c^2 d^2 e^4 \left (-\frac {-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (-a^2 e^4-10 a c d^2 e^2+35 c^2 d^4\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a^2 c^2 d e}-\frac {\left (3 a^2 e^4-52 a c d^2 e^2+57 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^2 d e x}}{4 a d e^6}-\frac {\left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d e^6 x^2}}{6 a d e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 c^2 d e^6 x^3}\right )-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle c^2 d^2 e^4 \left (-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 c^2 d e^6 x^3}-\frac {-\frac {\left (11 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 a^2 c^2 d e^6 x^2}-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (-a^2 e^4-10 a c d^2 e^2+35 c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{5/2} c^2 d^{3/2} e^{3/2}}-\frac {\left (3 a^2 e^4-52 a c d^2 e^2+57 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a^2 c^2 d e x}}{4 a d e^6}}{6 a d e}\right )-\frac {2 c^2 d^2 (d+e x) \left (c d^2-a e^2\right )}{a^4 e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

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

Output:

(-2*c^2*d^2*(c*d^2 - a*e^2)*(d + e*x))/(a^4*e^4*Sqrt[a*d*e + (c*d^2 + a*e^ 
2)*x + c*d*e*x^2]) + c^2*d^2*e^4*(-1/3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2]/(a^2*c^2*d*e^6*x^3) - (-1/2*((11*c*d^2 - 7*a*e^2)*Sqrt[a*d*e + (c 
*d^2 + a*e^2)*x + c*d*e*x^2])/(a^2*c^2*d*e^6*x^2) - (-(((57*c^2*d^4 - 52*a 
*c*d^2*e^2 + 3*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(a^2* 
c^2*d*e*x)) + (3*(c*d^2 - a*e^2)*(35*c^2*d^4 - 10*a*c*d^2*e^2 - a^2*e^4)*A 
rcTanh[(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^2*d^(3/2)*e^(3/2)))/(4*a 
*d*e^6))/(6*a*d*e))
 

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. \(2255\) vs. \(2(310)=620\).

Time = 2.85 (sec) , antiderivative size = 2256, normalized size of antiderivative = 6.64

method	result	size
default	\(\text {Expression too large to display}\)	\(2256\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 6.83 (sec) , antiderivative size = 792, normalized size of antiderivative = 2.33 \[ \int \frac {(d+e x)^3}{x^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/96*(3*((35*c^4*d^7 - 45*a*c^3*d^5*e^2 + 9*a^2*c^2*d^3*e^4 + a^3*c*d*e^6 
)*x^4 + (35*a*c^3*d^6*e - 45*a^2*c^2*d^4*e^3 + 9*a^3*c*d^2*e^5 + a^4*e^7)* 
x^3)*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*(8*a^4*d^3*e^4 
 + (105*a*c^3*d^6*e - 100*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5)*x^3 + (35*a^2 
*c^2*d^5*e^2 - 38*a^3*c*d^3*e^4 + 3*a^4*d*e^6)*x^2 - 14*(a^3*c*d^4*e^3 - a 
^4*d^2*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^5*c*d^3*e^5 
*x^4 + a^6*d^2*e^6*x^3), -1/48*(3*((35*c^4*d^7 - 45*a*c^3*d^5*e^2 + 9*a^2* 
c^2*d^3*e^4 + a^3*c*d*e^6)*x^4 + (35*a*c^3*d^6*e - 45*a^2*c^2*d^4*e^3 + 9* 
a^3*c*d^2*e^5 + a^4*e^7)*x^3)*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*(8*a^4*d^3*e^4 + 
 (105*a*c^3*d^6*e - 100*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5)*x^3 + (35*a^2*c 
^2*d^5*e^2 - 38*a^3*c*d^3*e^4 + 3*a^4*d*e^6)*x^2 - 14*(a^3*c*d^4*e^3 - a^4 
*d^2*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^5*c*d^3*e^5*x 
^4 + a^6*d^2*e^6*x^3)]
 

Sympy [F]

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

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

Maxima [F(-2)]

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

integrate((e*x+d)^3/x^4/(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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

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

integrate((e*x+d)^3/x^4/(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,[1,1,14]%%%},[2,9]%%%}+%%%{%%%{-5,[2,3,12]%%%},[2,8] 
%%%}+%%%{
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 1509, normalized size of antiderivative = 4.44 \[ \int \frac {(d+e x)^3}{x^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 15*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)*sqr 
t(d + e*x))*a**4*e**8*x**3 - 156*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*c*d**2*e**6*x**3 + 486*sqrt(e)* 
sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*s 
qrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a** 
2*c**2*d**4*e**4*x**3 + 420*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**3*d**6*e**2*x**3 - 735*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))*c**4*d** 
8*x**3 - 15*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**4*e**8*x**3 - 156*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*c*d**2*e**6*x**3 + 486*sqr 
t(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d*x)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqr 
t(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x) 
)*a**2*c**2*d**4*e**4*x**3 + 420*sqrt(e)*sqrt(d)*sqrt(a)*sqrt(a*e + c*d...