\(\int \frac {x^4}{(d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [131]

Optimal result

Integrand size = 40, antiderivative size = 306 \[ \int \frac {x^4}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 x^4}{\left (c d^2-a e^2\right ) (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {16 d x^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 \left (c d^2-a e^2\right )^2 (d+e x)^4}-\frac {96 a d e x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^3 (d+e x)^3}-\frac {128 a^2 d^2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^4 (d+e x)^2}+\frac {128 a^2 d e \left (c d^2-3 a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 \left (c d^2-a e^2\right )^5 (d+e x)} \] Output:

-2*x^4/(-a*e^2+c*d^2)/(e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+16 
/7*d*x^3*(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)^ 
4-96/35*a*d*e*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)^3 
/(e*x+d)^3-128/35*a^2*d^2*e*(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)^2+128/35*a^2*d*e*(-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)^5/(e*x+d)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.52 \[ \int \frac {x^4}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {2 (a e+c d x)^5 \left (5 d^4-\frac {28 a d^3 e (d+e x)}{a e+c d x}+\frac {70 a^2 d^2 e^2 (d+e x)^2}{(a e+c d x)^2}-\frac {140 a^3 d e^3 (d+e x)^3}{(a e+c d x)^3}-\frac {35 a^4 e^4 (d+e x)^4}{(a e+c d x)^4}\right )}{35 \left (c d^2-a e^2\right )^5 (d+e x)^2 ((a e+c d x) (d+e x))^{3/2}} \] Input:

Integrate[x^4/((d + e*x)^3*(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*(5*d^4 - (28*a*d^3*e*(d + e*x))/(a*e + c*d*x) + (70*a^2 
*d^2*e^2*(d + e*x)^2)/(a*e + c*d*x)^2 - (140*a^3*d*e^3*(d + e*x)^3)/(a*e + 
 c*d*x)^3 - (35*a^4*e^4*(d + e*x)^4)/(a*e + c*d*x)^4))/(35*(c*d^2 - a*e^2) 
^5*(d + e*x)^2*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.73, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1267, 27, 2169, 27, 2169, 27, 1220, 1129, 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.

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

\(\Big \downarrow \) 1267

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2169

\(\displaystyle -\frac {-\frac {\int \frac {3 \left (3 c d^2+a e^2\right )^2 x^2 e^7+2 d \left (11 c^2 d^4+18 a c e^2 d^2+3 a^2 e^4\right ) x e^6+3 d^2 \left (c^2 d^4+6 a c e^2 d^2+a^2 e^4\right ) e^5}{2 (d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 c d e^4}-\frac {e \left (\frac {a e^2}{c d}+7 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^5}-\frac {1}{c d e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {3 \left (3 c d^2+a e^2\right )^2 x^2 e^7+2 d \left (11 c^2 d^4+18 a c e^2 d^2+3 a^2 e^4\right ) x e^6+3 d^2 \left (c^2 d^4+6 a c e^2 d^2+a^2 e^4\right ) e^5}{(d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{4 c d e^4}-\frac {e \left (\frac {a e^2}{c d}+7 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^5}-\frac {1}{c d e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 2169

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {3 e^8 \left (d \left (3 c^3 d^6+15 a c^2 e^2 d^4+25 a^2 c e^4 d^2+5 a^3 e^6\right )+e \left (19 c^3 d^6+15 a c^2 e^2 d^4+25 a^2 c e^4 d^2+5 a^3 e^6\right ) x\right )}{2 (d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 c d e^3}-\frac {e^5 \left (a e^2+3 c d^2\right )^2}{c d (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {e \left (\frac {a e^2}{c d}+7 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^5}-\frac {1}{c d e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {e^5 \int \frac {d \left (3 c^3 d^6+15 a c^2 e^2 d^4+25 a^2 c e^4 d^2+5 a^3 e^6\right )+e \left (19 c^3 d^6+15 a c^2 e^2 d^4+25 a^2 c e^4 d^2+5 a^3 e^6\right ) x}{(d+e x)^3 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{2 c d}-\frac {e^5 \left (a e^2+3 c d^2\right )^2}{c d (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {e \left (\frac {a e^2}{c d}+7 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^5}-\frac {1}{c d e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1220

\(\displaystyle -\frac {-\frac {-\frac {e^5 \left (\frac {\left (-35 a^4 e^8-140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4-28 a c^3 d^6 e^2+5 c^4 d^8\right ) \int \frac {1}{(d+e x)^2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{7 \left (c d^2-a e^2\right )}-\frac {32 c^3 d^7}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c d}-\frac {e^5 \left (a e^2+3 c d^2\right )^2}{c d (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {e \left (\frac {a e^2}{c d}+7 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^5}-\frac {1}{c d e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle -\frac {-\frac {-\frac {e^5 \left (\frac {\left (-35 a^4 e^8-140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4-28 a c^3 d^6 e^2+5 c^4 d^8\right ) \left (\frac {6 c d \int \frac {1}{(d+e x) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {32 c^3 d^7}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c d}-\frac {e^5 \left (a e^2+3 c d^2\right )^2}{c d (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {e \left (\frac {a e^2}{c d}+7 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^5}-\frac {1}{c d e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle -\frac {-\frac {-\frac {e^5 \left (\frac {\left (-35 a^4 e^8-140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4-28 a c^3 d^6 e^2+5 c^4 d^8\right ) \left (\frac {6 c d \left (\frac {4 c d \int \frac {1}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )}+\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {32 c^3 d^7}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c d}-\frac {e^5 \left (a e^2+3 c d^2\right )^2}{c d (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {e \left (\frac {a e^2}{c d}+7 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^5}-\frac {1}{c d e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

\(\Big \downarrow \) 1088

\(\displaystyle -\frac {-\frac {-\frac {e^5 \left (\frac {\left (-35 a^4 e^8-140 a^3 c d^2 e^6+70 a^2 c^2 d^4 e^4-28 a c^3 d^6 e^2+5 c^4 d^8\right ) \left (\frac {6 c d \left (\frac {2}{3 (d+e x) \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 c d \left (a e^2+c d^2+2 c d e x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{5 \left (c d^2-a e^2\right )}+\frac {2}{5 (d+e x)^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{7 \left (c d^2-a e^2\right )}-\frac {32 c^3 d^7}{7 (d+e x)^3 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c d}-\frac {e^5 \left (a e^2+3 c d^2\right )^2}{c d (d+e x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{4 c d e^4}-\frac {e \left (\frac {a e^2}{c d}+7 d\right )}{2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{2 c d e^5}-\frac {1}{c d e^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\)

Input:

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

Output:

-(1/(c*d*e^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])) - (-1/2*(e*(7*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^5*(3*c*d^2 + a*e^2)^2)/(c*d*(d + e*x)^2*Sqrt[a*d*e + (c*d^2 + a*e 
^2)*x + c*d*e*x^2])) - (e^5*((-32*c^3*d^7)/(7*(c*d^2 - a*e^2)*(d + e*x)^3* 
Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + ((5*c^4*d^8 - 28*a*c^3*d^6* 
e^2 + 70*a^2*c^2*d^4*e^4 - 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*(2/(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]) + (6*c*d*( 
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))))/(7*(c*d^2 - a*e^2) 
)))/(2*c*d))/(4*c*d*e^4))/(2*c*d*e^5)
 

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 = 3.27 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.06

method	result	size
gosper	\(\frac {2 \left (c d x +a e \right ) \left (35 a^{4} e^{8} x^{4}+140 a^{3} c \,d^{2} e^{6} x^{4}-70 a^{2} c^{2} d^{4} e^{4} x^{4}+28 a \,c^{3} d^{6} e^{2} x^{4}-5 c^{4} d^{8} x^{4}+280 a^{4} d \,e^{7} x^{3}+280 a^{3} c \,d^{3} e^{5} x^{3}-56 a^{2} c^{2} d^{5} e^{3} x^{3}+8 a \,c^{3} d^{7} e \,x^{3}+560 a^{4} d^{2} e^{6} x^{2}+224 a^{3} c \,d^{4} e^{4} x^{2}-16 a^{2} c^{2} d^{6} e^{2} x^{2}+448 a^{4} d^{3} e^{5} x +64 a^{3} c \,d^{5} e^{3} x +128 a^{4} d^{4} e^{4}\right )}{35 \left (e x +d \right )^{2} \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(323\)
trager	\(\frac {2 \left (35 a^{4} e^{8} x^{4}+140 a^{3} c \,d^{2} e^{6} x^{4}-70 a^{2} c^{2} d^{4} e^{4} x^{4}+28 a \,c^{3} d^{6} e^{2} x^{4}-5 c^{4} d^{8} x^{4}+280 a^{4} d \,e^{7} x^{3}+280 a^{3} c \,d^{3} e^{5} x^{3}-56 a^{2} c^{2} d^{5} e^{3} x^{3}+8 a \,c^{3} d^{7} e \,x^{3}+560 a^{4} d^{2} e^{6} x^{2}+224 a^{3} c \,d^{4} e^{4} x^{2}-16 a^{2} c^{2} d^{6} e^{2} x^{2}+448 a^{4} d^{3} e^{5} x +64 a^{3} c \,d^{5} e^{3} x +128 a^{4} d^{4} e^{4}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{35 \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right ) \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \left (e x +d \right )^{4}}\)	\(324\)
orering	\(\frac {2 \left (35 a^{4} e^{8} x^{4}+140 a^{3} c \,d^{2} e^{6} x^{4}-70 a^{2} c^{2} d^{4} e^{4} x^{4}+28 a \,c^{3} d^{6} e^{2} x^{4}-5 c^{4} d^{8} x^{4}+280 a^{4} d \,e^{7} x^{3}+280 a^{3} c \,d^{3} e^{5} x^{3}-56 a^{2} c^{2} d^{5} e^{3} x^{3}+8 a \,c^{3} d^{7} e \,x^{3}+560 a^{4} d^{2} e^{6} x^{2}+224 a^{3} c \,d^{4} e^{4} x^{2}-16 a^{2} c^{2} d^{6} e^{2} x^{2}+448 a^{4} d^{3} e^{5} x +64 a^{3} c \,d^{5} e^{3} x +128 a^{4} d^{4} e^{4}\right ) \left (c d x +a e \right )}{35 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (e x +d \right )^{2} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}\)	\(324\)
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^{3}}+\frac {d^{4} \left (-\frac {2}{7 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\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 )}}-\frac {8 d e c \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 )}{7 \left (a \,e^{2}-c \,d^{2}\right )}\right )}{e^{7}}-\frac {6 d \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{e^{4} \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 {6 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^{5}}-\frac {4 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^{6}}\)	\(906\)

Input:

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

Output:

2/35*(c*d*x+a*e)*(35*a^4*e^8*x^4+140*a^3*c*d^2*e^6*x^4-70*a^2*c^2*d^4*e^4* 
x^4+28*a*c^3*d^6*e^2*x^4-5*c^4*d^8*x^4+280*a^4*d*e^7*x^3+280*a^3*c*d^3*e^5 
*x^3-56*a^2*c^2*d^5*e^3*x^3+8*a*c^3*d^7*e*x^3+560*a^4*d^2*e^6*x^2+224*a^3* 
c*d^4*e^4*x^2-16*a^2*c^2*d^6*e^2*x^2+448*a^4*d^3*e^5*x+64*a^3*c*d^5*e^3*x+ 
128*a^4*d^4*e^4)/(e*x+d)^2/(a^5*e^10-5*a^4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10 
*a^2*c^3*d^6*e^4+5*a*c^4*d^8*e^2-c^5*d^10)/(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. 735 vs. \(2 (288) = 576\).

Time = 11.18 (sec) , antiderivative size = 735, normalized size of antiderivative = 2.40 \[ \int \frac {x^4}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (128 \, a^{4} d^{4} e^{4} - {\left (5 \, c^{4} d^{8} - 28 \, a c^{3} d^{6} e^{2} + 70 \, a^{2} c^{2} d^{4} e^{4} - 140 \, a^{3} c d^{2} e^{6} - 35 \, a^{4} e^{8}\right )} x^{4} + 8 \, {\left (a c^{3} d^{7} e - 7 \, a^{2} c^{2} d^{5} e^{3} + 35 \, a^{3} c d^{3} e^{5} + 35 \, a^{4} d e^{7}\right )} x^{3} - 16 \, {\left (a^{2} c^{2} d^{6} e^{2} - 14 \, a^{3} c d^{4} e^{4} - 35 \, a^{4} d^{2} e^{6}\right )} x^{2} + 64 \, {\left (a^{3} c d^{5} e^{3} + 7 \, a^{4} d^{3} e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{35 \, {\left (a c^{5} d^{14} e - 5 \, a^{2} c^{4} d^{12} e^{3} + 10 \, a^{3} c^{3} d^{10} e^{5} - 10 \, a^{4} c^{2} d^{8} e^{7} + 5 \, a^{5} c d^{6} e^{9} - a^{6} d^{4} e^{11} + {\left (c^{6} d^{11} e^{4} - 5 \, a c^{5} d^{9} e^{6} + 10 \, a^{2} c^{4} d^{7} e^{8} - 10 \, a^{3} c^{3} d^{5} e^{10} + 5 \, a^{4} c^{2} d^{3} e^{12} - a^{5} c d e^{14}\right )} x^{5} + {\left (4 \, c^{6} d^{12} e^{3} - 19 \, a c^{5} d^{10} e^{5} + 35 \, a^{2} c^{4} d^{8} e^{7} - 30 \, a^{3} c^{3} d^{6} e^{9} + 10 \, a^{4} c^{2} d^{4} e^{11} + a^{5} c d^{2} e^{13} - a^{6} e^{15}\right )} x^{4} + 2 \, {\left (3 \, c^{6} d^{13} e^{2} - 13 \, a c^{5} d^{11} e^{4} + 20 \, a^{2} c^{4} d^{9} e^{6} - 10 \, a^{3} c^{3} d^{7} e^{8} - 5 \, a^{4} c^{2} d^{5} e^{10} + 7 \, a^{5} c d^{3} e^{12} - 2 \, a^{6} d e^{14}\right )} x^{3} + 2 \, {\left (2 \, c^{6} d^{14} e - 7 \, a c^{5} d^{12} e^{3} + 5 \, a^{2} c^{4} d^{10} e^{5} + 10 \, a^{3} c^{3} d^{8} e^{7} - 20 \, a^{4} c^{2} d^{6} e^{9} + 13 \, a^{5} c d^{4} e^{11} - 3 \, a^{6} d^{2} e^{13}\right )} x^{2} + {\left (c^{6} d^{15} - a c^{5} d^{13} e^{2} - 10 \, a^{2} c^{4} d^{11} e^{4} + 30 \, a^{3} c^{3} d^{9} e^{6} - 35 \, a^{4} c^{2} d^{7} e^{8} + 19 \, a^{5} c d^{5} e^{10} - 4 \, a^{6} d^{3} e^{12}\right )} x\right )}} \] Input:

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

Output:

-2/35*(128*a^4*d^4*e^4 - (5*c^4*d^8 - 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^ 
4 - 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*x^4 + 8*(a*c^3*d^7*e - 7*a^2*c^2*d^5*e 
^3 + 35*a^3*c*d^3*e^5 + 35*a^4*d*e^7)*x^3 - 16*(a^2*c^2*d^6*e^2 - 14*a^3*c 
*d^4*e^4 - 35*a^4*d^2*e^6)*x^2 + 64*(a^3*c*d^5*e^3 + 7*a^4*d^3*e^5)*x)*sqr 
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^5*d^14*e - 5*a^2*c^4*d^12*e^ 
3 + 10*a^3*c^3*d^10*e^5 - 10*a^4*c^2*d^8*e^7 + 5*a^5*c*d^6*e^9 - a^6*d^4*e 
^11 + (c^6*d^11*e^4 - 5*a*c^5*d^9*e^6 + 10*a^2*c^4*d^7*e^8 - 10*a^3*c^3*d^ 
5*e^10 + 5*a^4*c^2*d^3*e^12 - a^5*c*d*e^14)*x^5 + (4*c^6*d^12*e^3 - 19*a*c 
^5*d^10*e^5 + 35*a^2*c^4*d^8*e^7 - 30*a^3*c^3*d^6*e^9 + 10*a^4*c^2*d^4*e^1 
1 + a^5*c*d^2*e^13 - a^6*e^15)*x^4 + 2*(3*c^6*d^13*e^2 - 13*a*c^5*d^11*e^4 
 + 20*a^2*c^4*d^9*e^6 - 10*a^3*c^3*d^7*e^8 - 5*a^4*c^2*d^5*e^10 + 7*a^5*c* 
d^3*e^12 - 2*a^6*d*e^14)*x^3 + 2*(2*c^6*d^14*e - 7*a*c^5*d^12*e^3 + 5*a^2* 
c^4*d^10*e^5 + 10*a^3*c^3*d^8*e^7 - 20*a^4*c^2*d^6*e^9 + 13*a^5*c*d^4*e^11 
 - 3*a^6*d^2*e^13)*x^2 + (c^6*d^15 - a*c^5*d^13*e^2 - 10*a^2*c^4*d^11*e^4 
+ 30*a^3*c^3*d^9*e^6 - 35*a^4*c^2*d^7*e^8 + 19*a^5*c*d^5*e^10 - 4*a^6*d^3* 
e^12)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^4/(e*x+d)^3/(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 [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 12.16 (sec) , antiderivative size = 13455, normalized size of antiderivative = 43.97 \[ \int \frac {x^4}{(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

(32*d^3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(7*(5*a^2*d^3*e^7 + 
 5*c^2*d^7*e^3 + 5*a^2*e^10*x^3 + 15*a^2*d^2*e^8*x + 15*a^2*d*e^9*x^2 + 15 
*c^2*d^6*e^4*x + 15*c^2*d^5*e^5*x^2 + 5*c^2*d^4*e^6*x^3 - 10*a*c*d^5*e^5 - 
 30*a*c*d^4*e^6*x - 30*a*c*d^3*e^7*x^2 - 10*a*c*d^2*e^8*x^3)) - (2*d^4*(a* 
d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(7*a^2*d^4*e^7 + 7*c^2*d^8*e^3 
 + 7*a^2*e^11*x^4 + 28*a^2*d^3*e^8*x + 28*a^2*d*e^10*x^3 + 28*c^2*d^7*e^4* 
x + 42*a^2*d^2*e^9*x^2 + 42*c^2*d^6*e^5*x^2 + 28*c^2*d^5*e^6*x^3 + 7*c^2*d 
^4*e^7*x^4 - 14*a*c*d^6*e^5 - 56*a*c*d^5*e^6*x - 84*a*c*d^4*e^7*x^2 - 56*a 
*c*d^3*e^8*x^3 - 14*a*c*d^2*e^9*x^4) + (626*c^5*d^11*(a*d*e + a*e^2*x + c* 
d^2*x + c*d*e*x^2)^(1/2))/(105*(a^7*d*e^17 + a^7*e^18*x - c^7*d^15*e^3 + 7 
*a*c^6*d^13*e^5 - 7*a^6*c*d^3*e^15 - c^7*d^14*e^4*x - 21*a^2*c^5*d^11*e^7 
+ 35*a^3*c^4*d^9*e^9 - 35*a^4*c^3*d^7*e^11 + 21*a^5*c^2*d^5*e^13 + 7*a*c^6 
*d^12*e^6*x - 7*a^6*c*d^2*e^16*x - 21*a^2*c^5*d^10*e^8*x + 35*a^3*c^4*d^8* 
e^10*x - 35*a^4*c^3*d^6*e^12*x + 21*a^5*c^2*d^4*e^14*x)) - (494*c^2*d^5*(a 
*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(105*(a^4*d*e^11 + a^4*e^12*x 
 + c^4*d^9*e^3 - 4*a*c^3*d^7*e^5 - 4*a^3*c*d^3*e^9 + c^4*d^8*e^4*x + 6*a^2 
*c^2*d^5*e^7 - 4*a*c^3*d^6*e^6*x - 4*a^3*c*d^2*e^10*x + 6*a^2*c^2*d^4*e^8* 
x)) + (6*c^2*d^7*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(7*(5*a^4* 
d^3*e^11 + 5*c^4*d^11*e^3 + 5*a^4*e^14*x^3 - 20*a*c^3*d^9*e^5 - 20*a^3*c*d 
^5*e^9 + 15*a^4*d^2*e^12*x + 15*a^4*d*e^13*x^2 + 15*c^4*d^10*e^4*x + 30...
 

Reduce [F]

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

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

Output:

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