\(\int (a+b \sqrt [3]{x}+c x^{2/3})^p x \, dx\) [96]

Optimal result

Integrand size = 20, antiderivative size = 499 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p x \, dx=-\frac {3 \left (b^2 (2+p) (4+p) \left (2 a c (15+7 p)-b^2 \left (15+8 p+p^2\right )\right )-2 a c (3+2 p) \left (4 a c (5+2 p)-b^2 \left (20+9 p+p^2\right )\right )-2 b c (1+p) (4+p) \left (2 a c (15+7 p)-b^2 \left (15+8 p+p^2\right )\right ) \sqrt [3]{x}\right ) \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{1+p}}{8 c^5 (1+p) (2+p) (3+p) (3+2 p) (5+2 p)}-\frac {3 \left (4 a c (5+2 p)-b^2 \left (20+9 p+p^2\right )\right ) \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{1+p} x^{2/3}}{4 c^3 (2+p) (3+p) (5+2 p)}-\frac {3 b (5+p) \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{1+p} x}{2 c^2 (3+p) (5+2 p)}+\frac {3 \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{1+p} x^{4/3}}{2 c (3+p)}+\frac {3\ 2^{-2+p} b \left (60 a^2 c^2-20 a b^2 c (4+p)+b^4 \left (20+9 p+p^2\right )\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c \sqrt [3]{x}}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c \sqrt [3]{x}}{2 \sqrt {b^2-4 a c}}\right )}{c^5 \sqrt {b^2-4 a c} (1+p) (3+2 p) (5+2 p)} \] Output:

-3/8*(b^2*(2+p)*(4+p)*(2*a*c*(15+7*p)-b^2*(p^2+8*p+15))-2*a*c*(3+2*p)*(4*a 
*c*(5+2*p)-b^2*(p^2+9*p+20))-2*b*c*(p+1)*(4+p)*(2*a*c*(15+7*p)-b^2*(p^2+8* 
p+15))*x^(1/3))*(a+b*x^(1/3)+c*x^(2/3))^(p+1)/c^5/(p+1)/(2+p)/(3+p)/(3+2*p 
)/(5+2*p)-3/4*(4*a*c*(5+2*p)-b^2*(p^2+9*p+20))*(a+b*x^(1/3)+c*x^(2/3))^(p+ 
1)*x^(2/3)/c^3/(2+p)/(3+p)/(5+2*p)-3/2*b*(5+p)*(a+b*x^(1/3)+c*x^(2/3))^(p+ 
1)*x/c^2/(3+p)/(5+2*p)+3/2*(a+b*x^(1/3)+c*x^(2/3))^(p+1)*x^(4/3)/c/(3+p)+3 
*2^(-2+p)*b*(60*c^2*a^2-20*a*b^2*c*(4+p)+b^4*(p^2+9*p+20))*(-(b-(-4*a*c+b^ 
2)^(1/2)+2*c*x^(1/3))/(-4*a*c+b^2)^(1/2))^(-1-p)*(a+b*x^(1/3)+c*x^(2/3))^( 
p+1)*hypergeom([-p, p+1],[2+p],1/2*(b+(-4*a*c+b^2)^(1/2)+2*c*x^(1/3))/(-4* 
a*c+b^2)^(1/2))/c^5/(-4*a*c+b^2)^(1/2)/(p+1)/(3+2*p)/(5+2*p)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 1.22 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.35 \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p x \, dx=\frac {1}{2} \left (\frac {b-\sqrt {b^2-4 a c}+2 c \sqrt [3]{x}}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c \sqrt [3]{x}}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p x^2 \operatorname {AppellF1}\left (6,-p,-p,7,-\frac {2 c \sqrt [3]{x}}{b+\sqrt {b^2-4 a c}},\frac {2 c \sqrt [3]{x}}{-b+\sqrt {b^2-4 a c}}\right ) \] Input:

Integrate[(a + b*x^(1/3) + c*x^(2/3))^p*x,x]
 

Output:

((a + b*x^(1/3) + c*x^(2/3))^p*x^2*AppellF1[6, -p, -p, 7, (-2*c*x^(1/3))/( 
b + Sqrt[b^2 - 4*a*c]), (2*c*x^(1/3))/(-b + Sqrt[b^2 - 4*a*c])])/(2*((b - 
Sqrt[b^2 - 4*a*c] + 2*c*x^(1/3))/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 
 - 4*a*c] + 2*c*x^(1/3))/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1693, 1166, 25, 1236, 25, 1236, 1225, 1096}

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.

\(\displaystyle \int x \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle 3 \int \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^p x^{5/3}d\sqrt [3]{x}\)

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 25

\(\displaystyle 3 \left (\frac {x^{4/3} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+3)}-\frac {\int \left (4 a+b (p+5) \sqrt [3]{x}\right ) \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^p xd\sqrt [3]{x}}{2 c (p+3)}\right )\)

\(\Big \downarrow \) 1236

\(\displaystyle 3 \left (\frac {x^{4/3} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+3)}-\frac {\frac {\int -\left (\left (3 a b (p+5)-\left (4 a c (2 p+5)-b^2 \left (p^2+9 p+20\right )\right ) \sqrt [3]{x}\right ) \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^p x^{2/3}\right )d\sqrt [3]{x}}{c (2 p+5)}+\frac {b (p+5) x \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{c (2 p+5)}}{2 c (p+3)}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 3 \left (\frac {x^{4/3} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+3)}-\frac {\frac {b (p+5) x \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{c (2 p+5)}-\frac {\int \left (3 a b (p+5)-\left (4 a c (2 p+5)-b^2 \left (p^2+9 p+20\right )\right ) \sqrt [3]{x}\right ) \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^p x^{2/3}d\sqrt [3]{x}}{c (2 p+5)}}{2 c (p+3)}\right )\)

\(\Big \downarrow \) 1236

\(\displaystyle 3 \left (\frac {x^{4/3} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+3)}-\frac {\frac {b (p+5) x \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{c (2 p+5)}-\frac {\frac {\int \left (b (p+4) \sqrt [3]{x} \left (2 a c (7 p+15)-b^2 \left (p^2+8 p+15\right )\right )+2 a \left (4 a c (2 p+5)-b^2 \left (p^2+9 p+20\right )\right )\right ) \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^p \sqrt [3]{x}d\sqrt [3]{x}}{2 c (p+2)}-\frac {x^{2/3} \left (4 a c (2 p+5)-b^2 \left (p^2+9 p+20\right )\right ) \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+2)}}{c (2 p+5)}}{2 c (p+3)}\right )\)

\(\Big \downarrow \) 1225

\(\displaystyle 3 \left (\frac {x^{4/3} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+3)}-\frac {\frac {b (p+5) x \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{c (2 p+5)}-\frac {\frac {-\frac {b \left (p^2+5 p+6\right ) \left (60 a^2 c^2-20 a b^2 c (p+4)+b^4 \left (p^2+9 p+20\right )\right ) \int \left (a+c x^{2/3}+b \sqrt [3]{x}\right )^pd\sqrt [3]{x}}{2 c^2 (2 p+3)}-\frac {\left (-2 b c (p+1) (p+4) \sqrt [3]{x} \left (2 a c (7 p+15)-b^2 \left (p^2+8 p+15\right )\right )+b^2 (p+2) (p+4) \left (2 a c (7 p+15)-b^2 \left (p^2+8 p+15\right )\right )-2 a c (2 p+3) \left (4 a c (2 p+5)-b^2 \left (p^2+9 p+20\right )\right )\right ) \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}}{2 c (p+2)}-\frac {x^{2/3} \left (4 a c (2 p+5)-b^2 \left (p^2+9 p+20\right )\right ) \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+2)}}{c (2 p+5)}}{2 c (p+3)}\right )\)

\(\Big \downarrow \) 1096

\(\displaystyle 3 \left (\frac {x^{4/3} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+3)}-\frac {\frac {b (p+5) x \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{c (2 p+5)}-\frac {\frac {\frac {b 2^p \left (p^2+5 p+6\right ) \left (60 a^2 c^2-20 a b^2 c (p+4)+b^4 \left (p^2+9 p+20\right )\right ) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c \sqrt [3]{x}}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+\sqrt {b^2-4 a c}+2 c \sqrt [3]{x}}{2 \sqrt {b^2-4 a c}}\right )}{c^2 (p+1) (2 p+3) \sqrt {b^2-4 a c}}-\frac {\left (-2 b c (p+1) (p+4) \sqrt [3]{x} \left (2 a c (7 p+15)-b^2 \left (p^2+8 p+15\right )\right )+b^2 (p+2) (p+4) \left (2 a c (7 p+15)-b^2 \left (p^2+8 p+15\right )\right )-2 a c (2 p+3) \left (4 a c (2 p+5)-b^2 \left (p^2+9 p+20\right )\right )\right ) \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c^2 (p+1) (2 p+3)}}{2 c (p+2)}-\frac {x^{2/3} \left (4 a c (2 p+5)-b^2 \left (p^2+9 p+20\right )\right ) \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^{p+1}}{2 c (p+2)}}{c (2 p+5)}}{2 c (p+3)}\right )\)

Input:

Int[(a + b*x^(1/3) + c*x^(2/3))^p*x,x]
 

Output:

3*(((a + b*x^(1/3) + c*x^(2/3))^(1 + p)*x^(4/3))/(2*c*(3 + p)) - ((b*(5 + 
p)*(a + b*x^(1/3) + c*x^(2/3))^(1 + p)*x)/(c*(5 + 2*p)) - (-1/2*((4*a*c*(5 
 + 2*p) - b^2*(20 + 9*p + p^2))*(a + b*x^(1/3) + c*x^(2/3))^(1 + p)*x^(2/3 
))/(c*(2 + p)) + (-1/2*((b^2*(2 + p)*(4 + p)*(2*a*c*(15 + 7*p) - b^2*(15 + 
 8*p + p^2)) - 2*a*c*(3 + 2*p)*(4*a*c*(5 + 2*p) - b^2*(20 + 9*p + p^2)) - 
2*b*c*(1 + p)*(4 + p)*(2*a*c*(15 + 7*p) - b^2*(15 + 8*p + p^2))*x^(1/3))*( 
a + b*x^(1/3) + c*x^(2/3))^(1 + p))/(c^2*(1 + p)*(3 + 2*p)) + (2^p*b*(6 + 
5*p + p^2)*(60*a^2*c^2 - 20*a*b^2*c*(4 + p) + b^4*(20 + 9*p + p^2))*(-((b 
- Sqrt[b^2 - 4*a*c] + 2*c*x^(1/3))/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^( 
1/3) + c*x^(2/3))^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^ 
2 - 4*a*c] + 2*c*x^(1/3))/(2*Sqrt[b^2 - 4*a*c])])/(c^2*Sqrt[b^2 - 4*a*c]*( 
1 + p)*(3 + 2*p)))/(2*c*(2 + p)))/(c*(5 + 2*p)))/(2*c*(3 + p)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

rule 1236

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

rule 1693

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, 
x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ 
[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \left (a +b \,x^{\frac {1}{3}}+c \,x^{\frac {2}{3}}\right )^{p} x d x\]

Input:

int((a+b*x^(1/3)+c*x^(2/3))^p*x,x)
 

Output:

int((a+b*x^(1/3)+c*x^(2/3))^p*x,x)
 

Fricas [F(-1)]

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

integrate((a+b*x^(1/3)+c*x^(2/3))^p*x,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate((a+b*x**(1/3)+c*x**(2/3))**p*x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p x \, dx=\int { {\left (c x^{\frac {2}{3}} + b x^{\frac {1}{3}} + a\right )}^{p} x \,d x } \] Input:

integrate((a+b*x^(1/3)+c*x^(2/3))^p*x,x, algorithm="maxima")
 

Output:

integrate((c*x^(2/3) + b*x^(1/3) + a)^p*x, x)
 

Giac [F]

\[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p x \, dx=\int { {\left (c x^{\frac {2}{3}} + b x^{\frac {1}{3}} + a\right )}^{p} x \,d x } \] Input:

integrate((a+b*x^(1/3)+c*x^(2/3))^p*x,x, algorithm="giac")
 

Output:

integrate((c*x^(2/3) + b*x^(1/3) + a)^p*x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p x \, dx=\int x\,{\left (a+b\,x^{1/3}+c\,x^{2/3}\right )}^p \,d x \] Input:

int(x*(a + b*x^(1/3) + c*x^(2/3))^p,x)
 

Output:

int(x*(a + b*x^(1/3) + c*x^(2/3))^p, x)
 

Reduce [F]

\[ \int \left (a+b \sqrt [3]{x}+c x^{2/3}\right )^p x \, dx=\text {too large to display} \] Input:

int((a+b*x^(1/3)+c*x^(2/3))^p*x,x)
                                                                                    
                                                                                    
 

Output:

( - 192*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*a**2*c**3*p**4 - 864*x** 
(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*a**2*c**3*p**3 - 1104*x**(2/3)*(x** 
(2/3)*c + x**(1/3)*b + a)**p*a**2*c**3*p**2 - 360*x**(2/3)*(x**(2/3)*c + x 
**(1/3)*b + a)**p*a**2*c**3*p + 24*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)* 
*p*a*b**2*c**2*p**5 + 348*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*a*b**2 
*c**2*p**4 + 1488*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*a*b**2*c**2*p* 
*3 + 2100*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*a*b**2*c**2*p**2 + 720 
*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*a*b**2*c**2*p - 6*x**(2/3)*(x** 
(2/3)*c + x**(1/3)*b + a)**p*b**4*c*p**5 - 75*x**(2/3)*(x**(2/3)*c + x**(1 
/3)*b + a)**p*b**4*c*p**4 - 318*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p* 
b**4*c*p**3 - 501*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**4*c*p**2 - 
180*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b**4*c*p + 48*x**(2/3)*(x**( 
2/3)*c + x**(1/3)*b + a)**p*b*c**4*p**5*x + 240*x**(2/3)*(x**(2/3)*c + x** 
(1/3)*b + a)**p*b*c**4*p**4*x + 420*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a) 
**p*b*c**4*p**3*x + 300*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b*c**4*p 
**2*x + 72*x**(2/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*b*c**4*p*x + 168*x**( 
1/3)*(x**(2/3)*c + x**(1/3)*b + a)**p*a**2*b*c**2*p**4 + 1296*x**(1/3)*(x* 
*(2/3)*c + x**(1/3)*b + a)**p*a**2*b*c**2*p**3 + 2856*x**(1/3)*(x**(2/3)*c 
 + x**(1/3)*b + a)**p*a**2*b*c**2*p**2 + 1800*x**(1/3)*(x**(2/3)*c + x**(1 
/3)*b + a)**p*a**2*b*c**2*p - 12*x**(1/3)*(x**(2/3)*c + x**(1/3)*b + a)...