\(\int \frac {x (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^3} \, dx\) [41]

Optimal result

Integrand size = 38, antiderivative size = 234 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=-\frac {\left (5 c d^2-a e^2\right ) \left (3 c d^2-5 a e^2-2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 e^3 \left (c d^2-a e^2\right )}-\frac {2 d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{e \left (c d^2-a e^2\right ) (d+e x)^3}+\frac {3 \left (c d^2-a e^2\right ) \left (5 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 \sqrt {c} \sqrt {d} e^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.89 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {e} \sqrt {a e+c d x} \left (a e^2 (13 d+5 e x)+c d \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )}{(d+e x)^2}-\frac {6 \left (5 c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {-\frac {c d^2}{e}+a e}-\sqrt {a e+c d x}\right )}\right )}{\sqrt {c} \sqrt {d} (d+e x)^{3/2}}\right )}{4 e^{7/2} (a e+c d x)^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1213, 25, 2192, 27, 1160, 1092, 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.

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

\(\Big \downarrow \) 1213

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2192

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*d*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(e^3*(d 
 + e*x)) + ((c*d*e^3*x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/2 + (e 
^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]) + ( 
3*(c*d^2 - a*e^2)*(5*c*d^2 - a*e^2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2 
*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2 
*Sqrt[c]*Sqrt[d]*Sqrt[e])))/4)/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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1213

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 - n + 2)   Int[Expan 
dToSum[((-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], x] /; FreeQ[{a, b, c, d, e} 
, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && IGtQ[n, 0] && EqQ[m 
+ p, -3/2]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(704\) vs. \(2(212)=424\).

Time = 2.85 (sec) , antiderivative size = 705, normalized size of antiderivative = 3.01

method	result	size
default	\(\frac {\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {6 d e c \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{2}\right )}{a \,e^{2}-c \,d^{2}}}{e^{3}}-\frac {d \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {4 d e c \left (\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {6 d e c \left (\frac {\left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \left (\frac {\left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 d e c}-\frac {\left (a \,e^{2}-c \,d^{2}\right )^{2} \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 d e c \sqrt {d e c}}\right )}{2}\right )}{a \,e^{2}-c \,d^{2}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{e^{4}}\)	\(705\)

Input:

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

Output:

1/e^3*(2/(a*e^2-c*d^2)/(x+d/e)^2*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^( 
5/2)-6*d*e*c/(a*e^2-c*d^2)*(1/3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3 
/2)+1/2*(a*e^2-c*d^2)*(1/4*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d 
/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2 
-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+ 
d/e))^(1/2))/(d*e*c)^(1/2))))-d/e^4*(-2/(a*e^2-c*d^2)/(x+d/e)^3*(d*e*c*(x+ 
d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+4*d*e*c/(a*e^2-c*d^2)*(2/(a*e^2-c*d^2) 
/(x+d/e)^2*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6*d*e*c/(a*e^2-c* 
d^2)*(1/3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)* 
(1/4*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x 
+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/ 
e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^( 
1/2)))))
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.35 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\left [\frac {3 \, {\left (5 \, c^{2} d^{5} - 6 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (5 \, c^{2} d^{4} e - 6 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (2 \, c^{2} d^{2} e^{3} x^{2} - 15 \, c^{2} d^{4} e + 13 \, a c d^{2} e^{3} - 5 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, {\left (c d e^{5} x + c d^{2} e^{4}\right )}}, -\frac {3 \, {\left (5 \, c^{2} d^{5} - 6 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (5 \, c^{2} d^{4} e - 6 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (2 \, c^{2} d^{2} e^{3} x^{2} - 15 \, c^{2} d^{4} e + 13 \, a c d^{2} e^{3} - 5 \, {\left (c^{2} d^{3} e^{2} - a c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, {\left (c d e^{5} x + c d^{2} e^{4}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^3,x, algorithm 
="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*(a*e^2-c*d^2)>0)', see `assume 
?` for mor
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.16 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {1}{4} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (\frac {2 \, c d x}{e^{2}} - \frac {7 \, c^{2} d^{3} e^{5} - 5 \, a c d e^{7}}{c d e^{8}}\right )} - \frac {3 \, {\left (5 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{8 \, \sqrt {c d e} e^{3}} - \frac {2 \, {\left (\sqrt {c d e} c^{2} d^{5} - 2 \, \sqrt {c d e} a c d^{3} e^{2} + \sqrt {c d e} a^{2} d e^{4}\right )}}{\sqrt {c d e} {\left ({\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} e + \sqrt {c d e} d\right )} e^{3}} \] Input:

integrate(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^3,x, algorithm 
="giac")
 

Output:

1/4*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*c*d*x/e^2 - (7*c^2*d^3* 
e^5 - 5*a*c*d*e^7)/(c*d*e^8)) - 3/8*(5*c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)* 
log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c 
*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*e^3) - 2*(sqrt(c*d*e)*c^2*d^5 - 
2*sqrt(c*d*e)*a*c*d^3*e^2 + sqrt(c*d*e)*a^2*d*e^4)/(sqrt(c*d*e)*((sqrt(c*d 
*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*e + sqrt(c*d*e)*d)*e^ 
3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.48 \[ \int \frac {x \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {13 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{3}+5 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a c d \,e^{4} x -15 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{4} e -5 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{3} e^{2} x +2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{3} x^{2}+3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} d \,e^{4}+3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{5} x -18 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{3} e^{2}-18 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{3} x +15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{5}+15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4} e x -\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a^{2} d \,e^{4}-\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a^{2} e^{5} x +11 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a c \,d^{3} e^{2}+11 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a c \,d^{2} e^{3} x -10 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{5}-10 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{4} e x}{4 c d \,e^{4} \left (e x +d \right )} \] Input:

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

Output:

(13*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d**2*e**3 + 5*sqrt(d + e*x)*sqrt(a 
*e + c*d*x)*a*c*d*e**4*x - 15*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**4*e 
- 5*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**3*e**2*x + 2*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*c**2*d**2*e**3*x**2 + 3*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e 
)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2) 
)*a**2*d*e**4 + 3*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + 
 sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*e**5*x - 18*sq 
rt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqr 
t(d + e*x))/sqrt(a*e**2 - c*d**2))*a*c*d**3*e**2 - 18*sqrt(e)*sqrt(d)*sqrt 
(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a 
*e**2 - c*d**2))*a*c*d**2*e**3*x + 15*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e) 
*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2)) 
*c**2*d**5 + 15*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + s 
qrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c**2*d**4*e*x - sqrt( 
e)*sqrt(d)*sqrt(c)*a**2*d*e**4 - sqrt(e)*sqrt(d)*sqrt(c)*a**2*e**5*x + 11* 
sqrt(e)*sqrt(d)*sqrt(c)*a*c*d**3*e**2 + 11*sqrt(e)*sqrt(d)*sqrt(c)*a*c*d** 
2*e**3*x - 10*sqrt(e)*sqrt(d)*sqrt(c)*c**2*d**5 - 10*sqrt(e)*sqrt(d)*sqrt( 
c)*c**2*d**4*e*x)/(4*c*d*e**4*(d + e*x))