\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^4 (d+e x)^3} \, dx\) [46]

Optimal result

Integrand size = 40, antiderivative size = 362 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^4 (d+e x)^3} \, dx=-\frac {\left (3 c^2 d^4-100 a c d^2 e^2+105 a^2 e^4\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a d^4 (d+e x)}-\frac {a e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3 (d+e x)}-\frac {7 \left (c-\frac {a e^2}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2 (d+e x)}-\frac {\left (\frac {3 c^2 d^2}{a}-38 c e^2+\frac {35 a e^4}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 d e x (d+e x)}+\frac {\left (c d^2-a e^2\right ) \left (c^2 d^4+10 a c d^2 e^2-35 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{8 a^{3/2} d^{9/2} e^{3/2}} \] Output:

-1/24*(105*a^2*e^4-100*a*c*d^2*e^2+3*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(1/2)/a/d^4/(e*x+d)-1/3*a*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ 
d/x^3/(e*x+d)-7/12*(c-a*e^2/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x 
^2/(e*x+d)-1/24*(3*c^2*d^2/a-38*c*e^2+35*a*e^4/d^2)*(a*d*e+(a*e^2+c*d^2)*x 
+c*d*e*x^2)^(1/2)/d/e/x/(e*x+d)+1/8*(-a*e^2+c*d^2)*(-35*a^2*e^4+10*a*c*d^2 
*e^2+c^2*d^4)*arctanh(d^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^(1 
/2)/e^(1/2)/(e*x+d))/a^(3/2)/d^(9/2)/e^(3/2)
 

Mathematica [A] (verified)

Time = 10.18 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^4 (d+e x)^3} \, dx=\frac {\sqrt {a e+c d x} \left (-\sqrt {a} \sqrt {d} \sqrt {e} \sqrt {a e+c d x} \left (3 c^2 d^4 x^2 (d+e x)+2 a c d^2 e x \left (7 d^2-19 d e x-50 e^2 x^2\right )+a^2 e^2 \left (8 d^3-14 d^2 e x+35 d e^2 x^2+105 e^3 x^3\right )\right )+3 \left (c^3 d^6+9 a c^2 d^4 e^2-45 a^2 c d^2 e^4+35 a^3 e^6\right ) x^3 \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )\right )}{24 a^{3/2} d^{9/2} e^{3/2} x^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(Sqrt[a*e + c*d*x]*(-(Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*(3*c^2*d^4 
*x^2*(d + e*x) + 2*a*c*d^2*e*x*(7*d^2 - 19*d*e*x - 50*e^2*x^2) + a^2*e^2*( 
8*d^3 - 14*d^2*e*x + 35*d*e^2*x^2 + 105*e^3*x^3))) + 3*(c^3*d^6 + 9*a*c^2* 
d^4*e^2 - 45*a^2*c*d^2*e^4 + 35*a^3*e^6)*x^3*Sqrt[d + e*x]*ArcTanh[(Sqrt[d 
]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])]))/(24*a^(3/2)*d^(9/2 
)*e^(3/2)*x^3*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 1.94 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {1214, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2181

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {\frac {a e^5 \left (-\frac {3 \left (c d^2-a e^2\right ) \left (-35 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (\frac {3 c^2 d^4}{a}+57 a e^4-52 c d^2 e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d e x}\right )}{4 d^2}-\frac {a e^5 \left (7 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^4}+\frac {2 e^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {a e^5 \left (\frac {3 \left (c d^2-a e^2\right ) \left (-35 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a d e}-\frac {\left (\frac {3 c^2 d^4}{a}+57 a e^4-52 c d^2 e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d e x}\right )}{4 d^2}-\frac {a e^5 \left (7 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a e^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^4}+\frac {2 e^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

(2*e^2*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^4*( 
d + e*x)) + (-1/3*(a*e^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2 
*x^3) + (-1/2*(a*e^5*(7*c*d^2 - 11*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2])/(d^2*x^2) + (a*e^5*(-((((3*c^2*d^4)/a - 52*c*d^2*e^2 + 57*a*e 
^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d*e*x)) + (3*(c*d^2 - a* 
e^2)*(c^2*d^4 + 10*a*c*d^2*e^2 - 35*a^2*e^4)*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^(3/2)*d^(3/2)*e^(3/2))))/(4*d^2))/(6*a*d*e))/e^4
 

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 1214

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

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. \(5037\) vs. \(2(330)=660\).

Time = 4.74 (sec) , antiderivative size = 5038, normalized size of antiderivative = 13.92

method	result	size
default	\(\text {Expression too large to display}\)	\(5038\)

Input:

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

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 4.92 (sec) , antiderivative size = 768, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^4 (d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^4/(e*x+d)^3,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,[0,4,11]%%%},[2,5]%%%}+%%%{%%%{-5,[1,6,9]%%%},[2,4]% 
%%}+%%%{%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 1.92 (sec) , antiderivative size = 2498, normalized size of antiderivative = 6.90 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^4 (d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 112*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*d**4*e**5 + 196*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**4*d**3*e**6*x - 490*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**4*d**2*e**7*x**2 - 1470*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*d*e**8*x** 
3 - 80*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**6*e**3 - 56*sqrt(d + e*x) 
*sqrt(a*e + c*d*x)*a**3*c*d**5*e**4*x + 182*sqrt(d + e*x)*sqrt(a*e + c*d*x 
)*a**3*c*d**4*e**5*x**2 + 350*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**3* 
e**6*x**3 - 140*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**7*e**2*x + 33 
8*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**6*e**3*x**2 + 958*sqrt(d + 
e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**5*e**4*x**3 - 30*sqrt(d + e*x)*sqrt(a* 
e + c*d*x)*a*c**3*d**8*e*x**2 - 30*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3* 
d**7*e**2*x**3 - 735*sqrt(e)*sqrt(d)*sqrt(a)*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*d*e**8*x**3 - 735*sqrt(e)*sqrt(d)*sqrt(a)*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**9*x**4 + 420*sqrt(e)*sqrt(d)*sqrt(a)*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**3*e**6*x**3 + 420*sqrt(e)*sqrt(d)*sqrt 
(a)*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**7*x**4 + 486*sqrt( 
e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(...