\(\int \frac {x^3}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [124]

Optimal result

Integrand size = 40, antiderivative size = 240 \[ \int \frac {x^3}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 x^3}{\left (c d^2-a e^2\right ) (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {12 d x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right )^2 (d+e x)^3}+\frac {16 a d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right )^3 (d+e x)^2}-\frac {16 a d \left (c d^2-3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 \left (c d^2-a e^2\right )^4 (d+e x)} \] Output:

-2*x^3/(-a*e^2+c*d^2)/(e*x+d)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+12 
/5*d*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^2/(e*x+d)^ 
3+16/5*a*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^3/(e*x 
+d)^2-16/5*a*d*(-3*a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(- 
a*e^2+c*d^2)^4/(e*x+d)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.51 \[ \int \frac {x^3}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 (a e+c d x)^5 \left (d^3-\frac {5 a d^2 e (d+e x)}{a e+c d x}+\frac {15 a^2 d e^2 (d+e x)^2}{(a e+c d x)^2}+\frac {5 a^3 e^3 (d+e x)^3}{(a e+c d x)^3}\right )}{5 \left (c d^2-a e^2\right )^4 ((a e+c d x) (d+e x))^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.51, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1267, 27, 2169, 27, 1220, 1129, 1088}

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.

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

\(\Big \downarrow \) 1267

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2169

\(\displaystyle -\frac {-\frac {\int \frac {e^4 \left (d \left (c^2 d^4+12 a c e^2 d^2+3 a^2 e^4\right )+3 e \left (c d^2+a e^2\right ) \left (3 c d^2+a e^2\right ) x\right )}{2 (d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 c d e^3}-\frac {e \left (\frac {a e^2}{c d}+5 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^4}-\frac {1}{c d e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {e \int \frac {d \left (c^2 d^4+12 a c e^2 d^2+3 a^2 e^4\right )+3 e \left (c d^2+a e^2\right ) \left (3 c d^2+a e^2\right ) x}{(d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{4 c d}-\frac {e \left (\frac {a e^2}{c d}+5 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^4}-\frac {1}{c d e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1220

\(\displaystyle -\frac {-\frac {e \left (-\frac {3 \left (5 a^3 e^6+15 a^2 c d^2 e^4-5 a c^2 d^4 e^2+c^3 d^6\right ) \int \frac {1}{(d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{5 \left (c d^2-a e^2\right )}-\frac {16 c^2 d^5}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 c d}-\frac {e \left (\frac {a e^2}{c d}+5 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^4}-\frac {1}{c d e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle -\frac {-\frac {e \left (-\frac {3 \left (5 a^3 e^6+15 a^2 c d^2 e^4-5 a c^2 d^4 e^2+c^3 d^6\right ) \left (\frac {4 c d \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}-\frac {16 c^2 d^5}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 c d}-\frac {e \left (\frac {a e^2}{c d}+5 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^4}-\frac {1}{c d e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1088

\(\displaystyle -\frac {-\frac {e \left (-\frac {3 \left (5 a^3 e^6+15 a^2 c d^2 e^4-5 a c^2 d^4 e^2+c^3 d^6\right ) \left (\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}-\frac {16 c^2 d^5}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 c d}-\frac {e \left (\frac {a e^2}{c d}+5 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^4}-\frac {1}{c d e^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

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

Output:

-(1/(c*d*e^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])) - (-1/2*(e*(5*d 
 + (a*e^2)/(c*d)))/((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) 
 - (e*((-16*c^2*d^5)/(5*(c*d^2 - a*e^2)*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + 
a*e^2)*x + c*d*e*x^2]) - (3*(c^3*d^6 - 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 
+ 5*a^3*e^6)*(2/(3*(c*d^2 - a*e^2)*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)* 
x + c*d*e*x^2]) - (8*c*d*(c*d^2 + a*e^2 + 2*c*d*e*x))/(3*(c*d^2 - a*e^2)^3 
*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])))/(5*(c*d^2 - a*e^2))))/(4*c 
*d))/(2*c*d*e^4)
 

Defintions of rubi rules used

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 1088

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 
2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0]
 

rule 1129

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) 
))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d 
, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 
2], 0]
 

rule 1220

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x 
^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e 
*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1))   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[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0 
]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 
]
 

rule 1267

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d 
+ e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) 
 - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
 

rule 2169

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - 
b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 
, 0]
 

Maple [A] (verified)

Time = 2.61 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.95

method	result	size
gosper	\(\frac {2 \left (c d x +a e \right ) \left (5 a^{3} e^{6} x^{3}+15 a^{2} c \,d^{2} e^{4} x^{3}-5 a \,c^{2} d^{4} e^{2} x^{3}+c^{3} d^{6} x^{3}+30 a^{3} d \,e^{5} x^{2}+20 a^{2} c \,d^{3} e^{3} x^{2}-2 a \,c^{2} d^{5} e \,x^{2}+40 a^{3} d^{2} e^{4} x +8 a^{2} c \,d^{4} e^{2} x +16 a^{3} d^{3} e^{3}\right )}{5 \left (e x +d \right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(227\)
orering	\(\frac {2 \left (5 a^{3} e^{6} x^{3}+15 a^{2} c \,d^{2} e^{4} x^{3}-5 a \,c^{2} d^{4} e^{2} x^{3}+c^{3} d^{6} x^{3}+30 a^{3} d \,e^{5} x^{2}+20 a^{2} c \,d^{3} e^{3} x^{2}-2 a \,c^{2} d^{5} e \,x^{2}+40 a^{3} d^{2} e^{4} x +8 a^{2} c \,d^{4} e^{2} x +16 a^{3} d^{3} e^{3}\right ) \left (c d x +a e \right )}{5 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\)	\(228\)
trager	\(\frac {2 \left (5 a^{3} e^{6} x^{3}+15 a^{2} c \,d^{2} e^{4} x^{3}-5 a \,c^{2} d^{4} e^{2} x^{3}+c^{3} d^{6} x^{3}+30 a^{3} d \,e^{5} x^{2}+20 a^{2} c \,d^{3} e^{3} x^{2}-2 a \,c^{2} d^{5} e \,x^{2}+40 a^{3} d^{2} e^{4} x +8 a^{2} c \,d^{4} e^{2} x +16 a^{3} d^{3} e^{3}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{5 \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right ) \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \left (e x +d \right )^{3}}\)	\(230\)
default	\(\frac {-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}}{e^{2}}-\frac {4 d \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{e^{3} \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}+\frac {3 d^{2} \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{4}}-\frac {d^{3} \left (-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}-\frac {6 d e c \left (-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}+\frac {8 d e c \left (2 d e c \left (x +\frac {d}{e}\right )+a \,e^{2}-c \,d^{2}\right )}{3 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{5}}\)	\(596\)

Input:

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

Output:

2/5*(c*d*x+a*e)*(5*a^3*e^6*x^3+15*a^2*c*d^2*e^4*x^3-5*a*c^2*d^4*e^2*x^3+c^ 
3*d^6*x^3+30*a^3*d*e^5*x^2+20*a^2*c*d^3*e^3*x^2-2*a*c^2*d^5*e*x^2+40*a^3*d 
^2*e^4*x+8*a^2*c*d^4*e^2*x+16*a^3*d^3*e^3)/(e*x+d)/(a^4*e^8-4*a^3*c*d^2*e^ 
6+6*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a* 
d*e)^(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (226) = 452\).

Time = 3.85 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.06 \[ \int \frac {x^3}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (16 \, a^{3} d^{3} e^{3} + {\left (c^{3} d^{6} - 5 \, a c^{2} d^{4} e^{2} + 15 \, a^{2} c d^{2} e^{4} + 5 \, a^{3} e^{6}\right )} x^{3} - 2 \, {\left (a c^{2} d^{5} e - 10 \, a^{2} c d^{3} e^{3} - 15 \, a^{3} d e^{5}\right )} x^{2} + 8 \, {\left (a^{2} c d^{4} e^{2} + 5 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{5 \, {\left (a c^{4} d^{11} e - 4 \, a^{2} c^{3} d^{9} e^{3} + 6 \, a^{3} c^{2} d^{7} e^{5} - 4 \, a^{4} c d^{5} e^{7} + a^{5} d^{3} e^{9} + {\left (c^{5} d^{9} e^{3} - 4 \, a c^{4} d^{7} e^{5} + 6 \, a^{2} c^{3} d^{5} e^{7} - 4 \, a^{3} c^{2} d^{3} e^{9} + a^{4} c d e^{11}\right )} x^{4} + {\left (3 \, c^{5} d^{10} e^{2} - 11 \, a c^{4} d^{8} e^{4} + 14 \, a^{2} c^{3} d^{6} e^{6} - 6 \, a^{3} c^{2} d^{4} e^{8} - a^{4} c d^{2} e^{10} + a^{5} e^{12}\right )} x^{3} + 3 \, {\left (c^{5} d^{11} e - 3 \, a c^{4} d^{9} e^{3} + 2 \, a^{2} c^{3} d^{7} e^{5} + 2 \, a^{3} c^{2} d^{5} e^{7} - 3 \, a^{4} c d^{3} e^{9} + a^{5} d e^{11}\right )} x^{2} + {\left (c^{5} d^{12} - a c^{4} d^{10} e^{2} - 6 \, a^{2} c^{3} d^{8} e^{4} + 14 \, a^{3} c^{2} d^{6} e^{6} - 11 \, a^{4} c d^{4} e^{8} + 3 \, a^{5} d^{2} e^{10}\right )} x\right )}} \] Input:

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

Output:

2/5*(16*a^3*d^3*e^3 + (c^3*d^6 - 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 5*a^ 
3*e^6)*x^3 - 2*(a*c^2*d^5*e - 10*a^2*c*d^3*e^3 - 15*a^3*d*e^5)*x^2 + 8*(a^ 
2*c*d^4*e^2 + 5*a^3*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x 
)/(a*c^4*d^11*e - 4*a^2*c^3*d^9*e^3 + 6*a^3*c^2*d^7*e^5 - 4*a^4*c*d^5*e^7 
+ a^5*d^3*e^9 + (c^5*d^9*e^3 - 4*a*c^4*d^7*e^5 + 6*a^2*c^3*d^5*e^7 - 4*a^3 
*c^2*d^3*e^9 + a^4*c*d*e^11)*x^4 + (3*c^5*d^10*e^2 - 11*a*c^4*d^8*e^4 + 14 
*a^2*c^3*d^6*e^6 - 6*a^3*c^2*d^4*e^8 - a^4*c*d^2*e^10 + a^5*e^12)*x^3 + 3* 
(c^5*d^11*e - 3*a*c^4*d^9*e^3 + 2*a^2*c^3*d^7*e^5 + 2*a^3*c^2*d^5*e^7 - 3* 
a^4*c*d^3*e^9 + a^5*d*e^11)*x^2 + (c^5*d^12 - a*c^4*d^10*e^2 - 6*a^2*c^3*d 
^8*e^4 + 14*a^3*c^2*d^6*e^6 - 11*a^4*c*d^4*e^8 + 3*a^5*d^2*e^10)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4068 vs. \(2 (226) = 452\).

Time = 0.25 (sec) , antiderivative size = 4068, normalized size of antiderivative = 16.95 \[ \int \frac {x^3}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2/5*(5*a^3*e^4*abs(e)/((c^4*d^8*sgn(1/(e*x + d))*sgn(e) - 4*a*c^3*d^6*e^2* 
sgn(1/(e*x + d))*sgn(e) + 6*a^2*c^2*d^4*e^4*sgn(1/(e*x + d))*sgn(e) - 4*a^ 
3*c*d^2*e^6*sgn(1/(e*x + d))*sgn(e) + a^4*e^8*sgn(1/(e*x + d))*sgn(e))*sqr 
t(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))) - (c^3*d^6*abs(e) - 5*a*c^ 
2*d^4*e^2*abs(e) + 15*a^2*c*d^2*e^4*abs(e) + 5*a^3*e^6*abs(e))*sgn(1/(e*x 
+ d))*sgn(e)/(sqrt(c*d*e)*c^4*d^8*e^2 - 4*sqrt(c*d*e)*a*c^3*d^6*e^4 + 6*sq 
rt(c*d*e)*a^2*c^2*d^4*e^6 - 4*sqrt(c*d*e)*a^3*c*d^2*e^8 + sqrt(c*d*e)*a^4* 
e^10) + (15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^2*c^16*d^3 
3*e^26*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 240*sqrt(c*d*e - c*d^2*e/(e*x 
+ d) + a*e^3/(e*x + d))*a^3*c^15*d^31*e^28*abs(e)*sgn(1/(e*x + d))^4*sgn(e 
)^4 + 1800*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^4*c^14*d^29 
*e^30*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 8400*sqrt(c*d*e - c*d^2*e/(e*x 
+ d) + a*e^3/(e*x + d))*a^5*c^13*d^27*e^32*abs(e)*sgn(1/(e*x + d))^4*sgn(e 
)^4 + 27300*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^6*c^12*d^2 
5*e^34*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 65520*sqrt(c*d*e - c*d^2*e/(e* 
x + d) + a*e^3/(e*x + d))*a^7*c^11*d^23*e^36*abs(e)*sgn(1/(e*x + d))^4*sgn 
(e)^4 + 120120*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^8*c^10* 
d^21*e^38*abs(e)*sgn(1/(e*x + d))^4*sgn(e)^4 - 171600*sqrt(c*d*e - c*d^2*e 
/(e*x + d) + a*e^3/(e*x + d))*a^9*c^9*d^19*e^40*abs(e)*sgn(1/(e*x + d))^4* 
sgn(e)^4 + 193050*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*a^1...
 

Mupad [B] (verification not implemented)

Time = 7.99 (sec) , antiderivative size = 3963, normalized size of antiderivative = 16.51 \[ \int \frac {x^3}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*d^3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(5*a^2*d^3*e^6 + 5*c 
^2*d^7*e^2 + 5*a^2*e^9*x^3 + 15*a^2*d^2*e^7*x + 15*a^2*d*e^8*x^2 + 15*c^2* 
d^6*e^3*x + 15*c^2*d^5*e^4*x^2 + 5*c^2*d^4*e^5*x^3 - 10*a*c*d^5*e^4 - 30*a 
*c*d^4*e^5*x - 30*a*c*d^3*e^6*x^2 - 10*a*c*d^2*e^7*x^3) - (12*d^2*(a*d*e + 
 a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(5*(3*a^2*d^2*e^6 + 3*c^2*d^6*e^2 + 
 3*a^2*e^8*x^2 + 6*c^2*d^5*e^3*x + 3*c^2*d^4*e^4*x^2 - 6*a*c*d^4*e^4 + 6*a 
^2*d*e^7*x - 12*a*c*d^3*e^5*x - 6*a*c*d^2*e^6*x^2)) - (6*c^2*d^6*(a*d*e + 
a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(5*(3*a^4*d^2*e^10 + 3*c^4*d^10*e^2 
+ 3*a^4*e^12*x^2 - 12*a*c^3*d^8*e^4 - 12*a^3*c*d^4*e^8 + 6*c^4*d^9*e^3*x + 
 18*a^2*c^2*d^6*e^6 + 3*c^4*d^8*e^4*x^2 + 6*a^4*d*e^11*x + 18*a^2*c^2*d^4* 
e^8*x^2 - 24*a*c^3*d^7*e^5*x - 24*a^3*c*d^3*e^9*x + 36*a^2*c^2*d^5*e^7*x - 
 12*a*c^3*d^6*e^6*x^2 - 12*a^3*c*d^2*e^10*x^2)) - (6*c*d^4*(a*d*e + a*e^2* 
x + c*d^2*x + c*d*e*x^2)^(1/2))/(5*(3*a^3*d^2*e^8 - 3*c^3*d^8*e^2 + 3*a^3* 
e^10*x^2 + 9*a*c^2*d^6*e^4 - 9*a^2*c*d^4*e^6 - 6*c^3*d^7*e^3*x - 3*c^3*d^6 
*e^4*x^2 + 6*a^3*d*e^9*x + 18*a*c^2*d^5*e^5*x - 18*a^2*c*d^3*e^7*x + 9*a*c 
^2*d^4*e^6*x^2 - 9*a^2*c*d^2*e^8*x^2)) + (16*c^3*d^7*(a*d*e + a*e^2*x + c* 
d^2*x + c*d*e*x^2)^(1/2))/(15*(a^5*d*e^12 + a^5*e^13*x - c^5*d^11*e^2 + 5* 
a*c^4*d^9*e^4 - 5*a^4*c*d^3*e^10 - c^5*d^10*e^3*x - 10*a^2*c^3*d^7*e^6 + 1 
0*a^3*c^2*d^5*e^8 + 5*a*c^4*d^8*e^5*x - 5*a^4*c*d^2*e^11*x - 10*a^2*c^3*d^ 
6*e^7*x + 10*a^3*c^2*d^4*e^9*x)) + (24*c*d^3*(a*d*e + a*e^2*x + c*d^2*x...
 

Reduce [B] (verification not implemented)

Time = 30.43 (sec) , antiderivative size = 935, normalized size of antiderivative = 3.90 \[ \int \frac {x^3}{(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x^3/(e*x+d)^2/(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(c)*sqrt(a*e + c*d*x)*a**3*d**3*e**6 - 45*sqrt( 
e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d**2*e**7*x - 45*sqrt(e)*sqrt(d) 
*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d*e**8*x**2 - 15*sqrt(e)*sqrt(d)*sqrt(c)*s 
qrt(a*e + c*d*x)*a**3*e**9*x**3 - 15*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c* 
d*x)*a**2*c*d**5*e**4 - 45*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2* 
c*d**4*e**5*x - 45*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**3*e 
**6*x**2 - 15*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*c*d**2*e**7*x 
**3 - 5*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**7*e**2 - 15*sq 
rt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**6*e**3*x - 15*sqrt(e)*sq 
rt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**5*e**4*x**2 - 5*sqrt(e)*sqrt(d)* 
sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**5*x**3 + 3*sqrt(e)*sqrt(d)*sqrt(c 
)*sqrt(a*e + c*d*x)*c**3*d**9 + 9*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x 
)*c**3*d**8*e*x + 9*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**7*e* 
*2*x**2 + 3*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**6*e**3*x**3 
+ 32*sqrt(d + e*x)*a**3*c*d**4*e**6 + 80*sqrt(d + e*x)*a**3*c*d**3*e**7*x 
+ 60*sqrt(d + e*x)*a**3*c*d**2*e**8*x**2 + 10*sqrt(d + e*x)*a**3*c*d*e**9* 
x**3 + 16*sqrt(d + e*x)*a**2*c**2*d**5*e**5*x + 40*sqrt(d + e*x)*a**2*c**2 
*d**4*e**6*x**2 + 30*sqrt(d + e*x)*a**2*c**2*d**3*e**7*x**3 - 4*sqrt(d + e 
*x)*a*c**3*d**6*e**4*x**2 - 10*sqrt(d + e*x)*a*c**3*d**5*e**5*x**3 + 2*sqr 
t(d + e*x)*c**4*d**7*e**3*x**3)/(5*sqrt(a*e + c*d*x)*c*d*e**3*(a**4*d**...