\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^3 (d+e x)^4} \, dx\) [67]

Optimal result

Integrand size = 40, antiderivative size = 244 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\frac {15 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 d^3 (d+e x)}-\frac {5 \left (c-\frac {a e^2}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 x (d+e x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{2 d x^2 (d+e x)^3}-\frac {15 \sqrt {a} \sqrt {e} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{4 d^{7/2}} \] Output:

15/4*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/d^3/(e*x+d)- 
5/4*(c-a*e^2/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x/(e*x+d)^2-1/2* 
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/x^2/(e*x+d)^3-15/4*a^(1/2)*e^(1/ 
2)*(-a*e^2+c*d^2)^2*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))/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {d} \left (8 c^2 d^4 x^2-a c d^2 e x (9 d+25 e x)+a^2 e^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )\right )}{x^2 (a e+c d x)^2 (d+e x)^3}-\frac {15 \sqrt {a} \sqrt {e} \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{4 d^{7/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)^4), 
x]
 

Output:

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[d]*(8*c^2*d^4*x^2 - a*c*d^2*e*x*(9 
*d + 25*e*x) + a^2*e^2*(-2*d^2 + 5*d*e*x + 15*e^2*x^2)))/(x^2*(a*e + c*d*x 
)^2*(d + e*x)^3) - (15*Sqrt[a]*Sqrt[e]*(c*d^2 - a*e^2)^2*ArcTanh[(Sqrt[d]* 
Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/((a*e + c*d*x)^(5/2)* 
(d + e*x)^(5/2))))/(4*d^(7/2))
 

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1214, 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.

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

\(\Big \downarrow \) 1214

\(\displaystyle \frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}-\frac {\int -\frac {\frac {a^3 e^9}{d}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}+\frac {a \left (3 c^2 d^4-3 a c e^2 d^2+a^2 e^4\right ) x^2 e^7}{d^3}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\frac {a^3 e^9}{d}+\frac {a^2 \left (3 c d^2-a e^2\right ) x e^8}{d^2}+\frac {a \left (3 c^2 d^4-3 a c e^2 d^2+a^2 e^4\right ) x^2 e^7}{d^3}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^6}+\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {-\frac {\int -\frac {a^2 e^8 \left (a e \left (9 c d^2-7 a e^2\right )+2 d \left (\frac {2 a^2 e^4}{d^2}-7 a c e^2+6 c^2 d^2\right ) x\right )}{2 d x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^6}+\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {a e^7 \int \frac {a e \left (9 c d^2-7 a e^2\right )+2 d \left (\frac {2 a^2 e^4}{d^2}-7 a c e^2+6 c^2 d^2\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 d^2}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^6}+\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {a e^7 \left (\frac {15 \left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 d}-\frac {\left (9 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}\right )}{4 d^2}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^6}+\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {a e^7 \left (-\frac {15 \left (c d^2-a e^2\right )^2 \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{d}-\frac {\left (9 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}\right )}{4 d^2}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^6}+\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {a e^7 \left (-\frac {15 \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {a} d^{3/2} \sqrt {e}}-\frac {\left (9 c d^2-7 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}\right )}{4 d^2}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{e^6}+\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^3 (d+e x)}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)^4),x]
 

Output:

(2*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^3*(d 
+ e*x)) + (-1/2*(a^2*e^8*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d^2 
*x^2) + (a*e^7*(-(((9*c*d^2 - 7*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c* 
d*e*x^2])/(d*x)) - (15*(c*d^2 - a*e^2)^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*Sqrt[a]*d^(3/2)*Sqrt[e])))/(4*d^2))/e^6
 

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. \(5364\) vs. \(2(216)=432\).

Time = 5.25 (sec) , antiderivative size = 5365, normalized size of antiderivative = 21.99

method	result	size
default	\(\text {Expression too large to display}\)	\(5365\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^3/(e*x+d)^4,x,method=_RETURN 
VERBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 1.57 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.43 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\left [\frac {15 \, {\left ({\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x^{3} + {\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4}\right )} x^{2}\right )} \sqrt {\frac {a e}{d}} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d^{2} e + {\left (c d^{3} + a d e^{2}\right )} x\right )} \sqrt {\frac {a e}{d}} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} d^{2} e^{2} - {\left (8 \, c^{2} d^{4} - 25 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4}\right )} x^{2} + {\left (9 \, a c d^{3} e - 5 \, a^{2} d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (d^{3} e x^{3} + d^{4} x^{2}\right )}}, \frac {15 \, {\left ({\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x^{3} + {\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4}\right )} x^{2}\right )} \sqrt {-\frac {a e}{d}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-\frac {a e}{d}}}{2 \, {\left (a c d e^{2} x^{2} + a^{2} d e^{2} + {\left (a c d^{2} e + a^{2} e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} d^{2} e^{2} - {\left (8 \, c^{2} d^{4} - 25 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4}\right )} x^{2} + {\left (9 \, a c d^{3} e - 5 \, a^{2} d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (d^{3} e x^{3} + d^{4} x^{2}\right )}}\right ] \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d)^4,x, algorit 
hm="fricas")
 

Output:

[1/16*(15*((c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x^3 + (c^2*d^5 - 2*a*c*d^ 
3*e^2 + a^2*d*e^4)*x^2)*sqrt(a*e/d)*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^2*e + (c*d^3 + a*d*e^2)*x)*sqrt(a*e/d) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/ 
x^2) - 4*(2*a^2*d^2*e^2 - (8*c^2*d^4 - 25*a*c*d^2*e^2 + 15*a^2*e^4)*x^2 + 
(9*a*c*d^3*e - 5*a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) 
)/(d^3*e*x^3 + d^4*x^2), 1/8*(15*((c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x^ 
3 + (c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*e^4)*x^2)*sqrt(-a*e/d)*arctan(1/2*sqr 
t(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)*sqr 
t(-a*e/d)/(a*c*d*e^2*x^2 + a^2*d*e^2 + (a*c*d^2*e + a^2*e^3)*x)) - 2*(2*a^ 
2*d^2*e^2 - (8*c^2*d^4 - 25*a*c*d^2*e^2 + 15*a^2*e^4)*x^2 + (9*a*c*d^3*e - 
 5*a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(d^3*e*x^3 + 
 d^4*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**3/(e*x+d)**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{3}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d)^4,x, algorit 
hm="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)^4*x^3), 
 x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d)^4,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,[0,3,9]%%%},[2,4]%%%}+%%%{%%%{-4,[1,5,7]%%%},[2,3]%% 
%}+%%%{%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^3\,{\left (d+e\,x\right )}^4} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)^4),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^3*(d + e*x)^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 1969, normalized size of antiderivative = 8.07 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^3 (d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^3/(e*x+d)^4,x)
 

Output:

( - 20*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*d**3*e**5 + 50*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a**3*d**2*e**6*x + 150*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a* 
*3*d*e**7*x**2 - 12*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d**5*e**3 - 60* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c*d**4*e**4*x - 160*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a**2*c*d**3*e**5*x**2 - 54*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a*c**2*d**6*e**2*x - 70*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**2*d**5*e**3*x 
**2 + 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**3*d**7*e*x**2 + 75*sqrt(e)*sqr 
t(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*d*e**7*x**2 + 75*sq 
rt(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*e**8*x**3 
- 105*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**2*c* 
d**3*e**5*x**2 - 105*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**2*c*d**2*e**6*x**3 - 15*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqr 
t(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*s 
qrt(c)*sqrt(d + e*x))*a*c**2*d**5*e**3*x**2 - 15*sqrt(e)*sqrt(d)*sqrt(a)*l 
og(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**4*x**3 + 45*sqrt(e)*...