\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^4 (d+e x)^4} \, dx\) [68]

Optimal result

Integrand size = 40, antiderivative size = 359 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=\frac {5 \left (c d^2-7 a e^2\right ) \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 a d^4 e (d+e x)}-\frac {5 \left (c d^2-7 a e^2\right ) \left (c-\frac {a e^2}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 a d e x (d+e x)^2}-\frac {\left (\frac {c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{12 x^2 (d+e x)^3}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3 a d e x^3 (d+e x)^4}-\frac {5 \left (c d^2-7 a e^2\right ) \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{8 \sqrt {a} d^{9/2} \sqrt {e}} \] Output:

5/8*(-7*a*e^2+c*d^2)*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 
/2)/a/d^4/e/(e*x+d)-5/24*(-7*a*e^2+c*d^2)*(c-a*e^2/d^2)*(a*d*e+(a*e^2+c*d^ 
2)*x+c*d*e*x^2)^(3/2)/a/d/e/x/(e*x+d)^2-1/12*(c/a/e-7*e/d^2)*(a*d*e+(a*e^2 
+c*d^2)*x+c*d*e*x^2)^(5/2)/x^2/(e*x+d)^3-1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e* 
x^2)^(7/2)/a/d/e/x^3/(e*x+d)^4-5/8*(-7*a*e^2+c*d^2)*(-a*e^2+c*d^2)^2*arcta 
nh(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^(1/2)/d^(9/2)/e^(1/2)
 

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (-\frac {\sqrt {d} \left (3 c^2 d^4 x^2 (11 d+27 e x)+2 a c d^2 e x \left (13 d^2-34 d e x-95 e^2 x^2\right )+a^2 e^2 \left (8 d^3-14 d^2 e x+35 d e^2 x^2+105 e^3 x^3\right )\right )}{x^3 (a e+c d x)^2 (d+e x)^3}+\frac {15 \left (c d^2-a e^2\right )^2 \left (-c d^2+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {e} (a e+c d x)^{5/2} (d+e x)^{5/2}}\right )}{24 d^{9/2}} \] Input:

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

Output:

(((a*e + c*d*x)*(d + e*x))^(5/2)*(-((Sqrt[d]*(3*c^2*d^4*x^2*(11*d + 27*e*x 
) + 2*a*c*d^2*e*x*(13*d^2 - 34*d*e*x - 95*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)))/(x^3*(a*e + c*d*x)^2*(d + e*x)^3)) 
+ (15*(c*d^2 - a*e^2)^2*(-(c*d^2) + 7*a*e^2)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c 
*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a]*Sqrt[e]*(a*e + c*d*x)^(5 
/2)*(d + e*x)^(5/2))))/(24*d^(9/2))
 

Rubi [A] (verified)

Time = 1.98 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.03, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2181

\(\displaystyle \frac {-\frac {\int -\frac {\frac {a^3 \left (13 c d^2-11 a e^2\right ) e^9}{d}+2 a^2 \left (\frac {3 a^2 e^4}{d^2}-11 a c e^2+9 c^2 d^2\right ) x e^8+\frac {6 a \left (c d^2-a e^2\right )^3 x^2 e^7}{d^3}}{2 x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{3 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^6}-\frac {2 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\frac {a^3 \left (13 c d^2-11 a e^2\right ) e^9}{d}+2 a^2 \left (\frac {3 a^2 e^4}{d^2}-11 a c e^2+9 c^2 d^2\right ) x e^8+\frac {6 a \left (c d^2-a e^2\right )^3 x^2 e^7}{d^3}}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^6}-\frac {2 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 2181

\(\displaystyle \frac {\frac {-\frac {\int -\frac {a^2 e^8 \left (a e \left (33 c^2 d^4-94 a c e^2 d^2+57 a^2 e^4\right )+2 d \left (-\frac {12 a^3 e^6}{d^2}+47 a^2 c e^4-49 a c^2 d^2 e^2+12 c^3 d^4\right ) x\right )}{2 d x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {a^2 e^8 \left (13 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^6}-\frac {2 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {a e^7 \int \frac {a e \left (33 c^2 d^4-94 a c e^2 d^2+57 a^2 e^4\right )+2 d \left (-\frac {12 a^3 e^6}{d^2}+47 a^2 c e^4-49 a c^2 d^2 e^2+12 c^3 d^4\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 d^2}-\frac {a^2 e^8 \left (13 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^6}-\frac {2 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {\frac {a e^7 \left (\frac {15 \left (c d^2-7 a e^2\right ) \left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 d}-\frac {\left (57 a^2 e^4-94 a c d^2 e^2+33 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}\right )}{4 d^2}-\frac {a^2 e^8 \left (13 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^6}-\frac {2 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {a e^7 \left (-\frac {15 \left (c d^2-7 a e^2\right ) \left (c d^2-a e^2\right )^2 \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{d}-\frac {\left (57 a^2 e^4-94 a c d^2 e^2+33 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}\right )}{4 d^2}-\frac {a^2 e^8 \left (13 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^6}-\frac {2 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {a e^7 \left (-\frac {\left (57 a^2 e^4-94 a c d^2 e^2+33 c^2 d^4\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d x}-\frac {15 \left (c d^2-7 a e^2\right ) \left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {a} d^{3/2} \sqrt {e}}\right )}{4 d^2}-\frac {a^2 e^8 \left (13 c d^2-11 a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 d^2 x^2}}{6 a d e}-\frac {a^2 e^8 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d^2 x^3}}{e^6}-\frac {2 e \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d^4 (d+e x)}\)

Input:

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

Output:

(-2*e*(c*d^2 - a*e^2)^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^2*e^8*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/( 
d^2*x^3) + (-1/2*(a^2*e^8*(13*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^7*(-(((33*c^2*d^4 - 94*a*c*d^2*e^2 + 5 
7*a^2*e^4)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d*x)) - (15*(c*d^ 
2 - 7*a*e^2)*(c*d^2 - a*e^2)^2*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sq 
rt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sq 
rt[a]*d^(3/2)*Sqrt[e])))/(4*d^2))/(6*a*d*e))/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. \(8321\) vs. \(2(327)=654\).

Time = 6.66 (sec) , antiderivative size = 8322, normalized size of antiderivative = 23.18

method	result	size
default	\(\text {Expression too large to display}\)	\(8322\)

Input:

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

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 5.90 (sec) , antiderivative size = 758, normalized size of antiderivative = 2.11 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/96*(15*((c^3*d^6*e - 9*a*c^2*d^4*e^3 + 15*a^2*c*d^2*e^5 - 7*a^3*e^7)*x 
^4 + (c^3*d^7 - 9*a*c^2*d^5*e^2 + 15*a^2*c*d^3*e^4 - 7*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 + (81*a 
*c^2*d^5*e^2 - 190*a^2*c*d^3*e^4 + 105*a^3*d*e^6)*x^3 + (33*a*c^2*d^6*e - 
68*a^2*c*d^4*e^3 + 35*a^3*d^2*e^5)*x^2 + 2*(13*a^2*c*d^5*e^2 - 7*a^3*d^3*e 
^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*d^5*e^2*x^4 + a*d^6 
*e*x^3), 1/48*(15*((c^3*d^6*e - 9*a*c^2*d^4*e^3 + 15*a^2*c*d^2*e^5 - 7*a^3 
*e^7)*x^4 + (c^3*d^7 - 9*a*c^2*d^5*e^2 + 15*a^2*c*d^3*e^4 - 7*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 + (81*a*c^2*d^5*e^2 - 190*a 
^2*c*d^3*e^4 + 105*a^3*d*e^6)*x^3 + (33*a*c^2*d^6*e - 68*a^2*c*d^4*e^3 + 3 
5*a^3*d^2*e^5)*x^2 + 2*(13*a^2*c*d^5*e^2 - 7*a^3*d^3*e^4)*x)*sqrt(c*d*e*x^ 
2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*d^5*e^2*x^4 + a*d^6*e*x^3)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 1.78 (sec) , antiderivative size = 2498, normalized size of antiderivative = 6.96 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^4 (d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^4/(e*x+d)^4,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 - 224*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*a**3*c*d**5*e**4*x + 602*sqrt(d + e*x)*sqrt(a*e + c*d* 
x)*a**3*c*d**4*e**5*x**2 + 1610*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d** 
3*e**6*x**3 - 260*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**7*e**2*x + 
218*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**6*e**3*x**2 + 766*sqrt(d 
+ e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**5*e**4*x**3 - 330*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*a*c**3*d**8*e*x**2 - 810*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)*sqr 
t(d + e*x))*a**4*d*e**8*x**3 - 735*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**4*e**9*x**4 + 1050*sqrt(e)*sqrt(d)*sqrt(a)*log(sq 
rt(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 + 1050*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 + 180 
*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)...