\(\int \frac {x^5 (a+b x^3+c x^6)^p}{(d+e x^3)^2} \, dx\) [101]

Optimal result

Integrand size = 27, antiderivative size = 450 \[ \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx=\frac {d \left (a+b x^3+c x^6\right )^{1+p}}{3 \left (c d^2-b d e+a e^2\right ) \left (d+e x^3\right )}+\frac {2^{-1+2 p} \left (c d^2 (1+2 p)+e (a e-b d (1+p))\right ) \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c \left (d+e x^3\right )},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 \left (d+e x^3\right )}\right )}{3 e^2 \left (c d^2-b d e+a e^2\right ) p}+\frac {2^{1+p} c d (1+2 p) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 \sqrt {b^2-4 a c} e \left (c d^2-b d e+a e^2\right ) (1+p)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.56 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.73 \[ \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx=\frac {2^{-1+2 p} \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (-2 d p \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c \left (d+e x^3\right )},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x^3}\right )+(-1+2 p) \left (d+e x^3\right ) \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c \left (d+e x^3\right )},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x^3}\right )\right )}{3 e^2 p (-1+2 p) \left (d+e x^3\right )} \] Input:

Integrate[(x^5*(a + b*x^3 + c*x^6)^p)/(d + e*x^3)^2,x]
 

Output:

(2^(-1 + 2*p)*(a + b*x^3 + c*x^6)^p*(-2*d*p*AppellF1[1 - 2*p, -p, -p, 2 - 
2*p, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x^3)), (2*c*d - b*e + 
 Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x^3)] + (-1 + 2*p)*(d + e*x^3)*Appell 
F1[-2*p, -p, -p, 1 - 2*p, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e* 
x^3)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x^3)]))/(3*e^2*p 
*(-1 + 2*p)*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3))/(c*(d + e*x^3)))^p*((e* 
(b + Sqrt[b^2 - 4*a*c] + 2*c*x^3))/(c*(d + e*x^3)))^p*(d + e*x^3))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 429, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1802, 1237, 25, 1269, 1096, 1178, 150}

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.

\(\displaystyle \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx\)

\(\Big \downarrow \) 1802

\(\displaystyle \frac {1}{3} \int \frac {x^3 \left (c x^6+b x^3+a\right )^p}{\left (e x^3+d\right )^2}dx^3\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {1}{3} \left (\frac {d \left (a+b x^3+c x^6\right )^{p+1}}{\left (d+e x^3\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {\left (-c d (2 p+1) x^3+a e-b d (p+1)\right ) \left (c x^6+b x^3+a\right )^p}{e x^3+d}dx^3}{a e^2-b d e+c d^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {\left (-c d (2 p+1) x^3+a e-b d (p+1)\right ) \left (c x^6+b x^3+a\right )^p}{e x^3+d}dx^3}{a e^2-b d e+c d^2}+\frac {d \left (a+b x^3+c x^6\right )^{p+1}}{\left (d+e x^3\right ) \left (a e^2-b d e+c d^2\right )}\right )\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{3} \left (\frac {\frac {\left (e (a e-b d (p+1))+c d^2 (2 p+1)\right ) \int \frac {\left (c x^6+b x^3+a\right )^p}{e x^3+d}dx^3}{e}-\frac {c d (2 p+1) \int \left (c x^6+b x^3+a\right )^pdx^3}{e}}{a e^2-b d e+c d^2}+\frac {d \left (a+b x^3+c x^6\right )^{p+1}}{\left (d+e x^3\right ) \left (a e^2-b d e+c d^2\right )}\right )\)

\(\Big \downarrow \) 1096

\(\displaystyle \frac {1}{3} \left (\frac {\frac {\left (e (a e-b d (p+1))+c d^2 (2 p+1)\right ) \int \frac {\left (c x^6+b x^3+a\right )^p}{e x^3+d}dx^3}{e}+\frac {c d 2^{p+1} (2 p+1) \left (a+b x^3+c x^6\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{e (p+1) \sqrt {b^2-4 a c}}}{a e^2-b d e+c d^2}+\frac {d \left (a+b x^3+c x^6\right )^{p+1}}{\left (d+e x^3\right ) \left (a e^2-b d e+c d^2\right )}\right )\)

\(\Big \downarrow \) 1178

\(\displaystyle \frac {1}{3} \left (\frac {\frac {c d 2^{p+1} (2 p+1) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{e (p+1) \sqrt {b^2-4 a c}}-\frac {4^p \left (\frac {1}{d+e x^3}\right )^{2 p} \left (a+b x^3+c x^6\right )^p \left (e (a e-b d (p+1))+c d^2 (2 p+1)\right ) \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \int \left (\frac {1}{e x^3+d}\right )^{-2 p-1} \left (1-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{2 \left (e x^3+d\right )}\right )^p \left (1-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 \left (e x^3+d\right )}\right )^pd\frac {1}{e x^3+d}}{e^2}}{a e^2-b d e+c d^2}+\frac {d \left (a+b x^3+c x^6\right )^{p+1}}{\left (d+e x^3\right ) \left (a e^2-b d e+c d^2\right )}\right )\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {1}{3} \left (\frac {\frac {2^{2 p-1} \left (a+b x^3+c x^6\right )^p \left (e (a e-b d (p+1))+c d^2 (2 p+1)\right ) \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x^3\right )}{c \left (d+e x^3\right )}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c \left (e x^3+d\right )},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 \left (e x^3+d\right )}\right )}{e^2 p}+\frac {c d 2^{p+1} (2 p+1) \left (a+b x^3+c x^6\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{e (p+1) \sqrt {b^2-4 a c}}}{a e^2-b d e+c d^2}+\frac {d \left (a+b x^3+c x^6\right )^{p+1}}{\left (d+e x^3\right ) \left (a e^2-b d e+c d^2\right )}\right )\)

Input:

Int[(x^5*(a + b*x^3 + c*x^6)^p)/(d + e*x^3)^2,x]
 

Output:

((d*(a + b*x^3 + c*x^6)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(d + e*x^3)) + ( 
(2^(-1 + 2*p)*(c*d^2*(1 + 2*p) + e*(a*e - b*d*(1 + p)))*(a + b*x^3 + c*x^6 
)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2 
*c*(d + e*x^3)), (2*d - ((b + Sqrt[b^2 - 4*a*c])*e)/c)/(2*(d + e*x^3))])/( 
e^2*p*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3))/(c*(d + e*x^3)))^p*((e*(b + S 
qrt[b^2 - 4*a*c] + 2*c*x^3))/(c*(d + e*x^3)))^p) + (2^(1 + p)*c*d*(1 + 2*p 
)*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b 
*x^3 + c*x^6)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 
4*a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*e*(1 + p)))/( 
c*d^2 - b*d*e + a*e^2))/3
 

Defintions of rubi rules used

rule 25

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

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

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 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

rule 1237

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)/((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*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

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

rule 1802

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

Maple [F]

\[\int \frac {x^{5} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}}{\left (e \,x^{3}+d \right )^{2}}d x\]

Input:

int(x^5*(c*x^6+b*x^3+a)^p/(e*x^3+d)^2,x)
 

Output:

int(x^5*(c*x^6+b*x^3+a)^p/(e*x^3+d)^2,x)
 

Fricas [F]

\[ \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5}}{{\left (e x^{3} + d\right )}^{2}} \,d x } \] Input:

integrate(x^5*(c*x^6+b*x^3+a)^p/(e*x^3+d)^2,x, algorithm="fricas")
 

Output:

integral((c*x^6 + b*x^3 + a)^p*x^5/(e^2*x^6 + 2*d*e*x^3 + d^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**5*(c*x**6+b*x**3+a)**p/(e*x**3+d)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5}}{{\left (e x^{3} + d\right )}^{2}} \,d x } \] Input:

integrate(x^5*(c*x^6+b*x^3+a)^p/(e*x^3+d)^2,x, algorithm="maxima")
 

Output:

integrate((c*x^6 + b*x^3 + a)^p*x^5/(e*x^3 + d)^2, x)
 

Giac [F]

\[ \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{5}}{{\left (e x^{3} + d\right )}^{2}} \,d x } \] Input:

integrate(x^5*(c*x^6+b*x^3+a)^p/(e*x^3+d)^2,x, algorithm="giac")
 

Output:

integrate((c*x^6 + b*x^3 + a)^p*x^5/(e*x^3 + d)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx=\int \frac {x^5\,{\left (c\,x^6+b\,x^3+a\right )}^p}{{\left (e\,x^3+d\right )}^2} \,d x \] Input:

int((x^5*(a + b*x^3 + c*x^6)^p)/(d + e*x^3)^2,x)
 

Output:

int((x^5*(a + b*x^3 + c*x^6)^p)/(d + e*x^3)^2, x)
 

Reduce [F]

\[ \int \frac {x^5 \left (a+b x^3+c x^6\right )^p}{\left (d+e x^3\right )^2} \, dx=\text {too large to display} \] Input:

int(x^5*(c*x^6+b*x^3+a)^p/(e*x^3+d)^2,x)
 

Output:

(2*(a + b*x**3 + c*x**6)**p*a*e*p - (a + b*x**3 + c*x**6)**p*b*d*p - (a + 
b*x**3 + c*x**6)**p*b*d + (a + b*x**3 + c*x**6)**p*b*e*p*x**3 - (a + b*x** 
3 + c*x**6)**p*b*e*x**3 + 2*(a + b*x**3 + c*x**6)**p*c*d*p*x**3 - 12*int(( 
(a + b*x**3 + c*x**6)**p*x**8)/(a*b*d**2*e*p - a*b*d**2*e + 2*a*b*d*e**2*p 
*x**3 - 2*a*b*d*e**2*x**3 + a*b*e**3*p*x**6 - a*b*e**3*x**6 + 2*a*c*d**3*p 
 + 4*a*c*d**2*e*p*x**3 + 2*a*c*d*e**2*p*x**6 + b**2*d**2*e*p*x**3 - b**2*d 
**2*e*x**3 + 2*b**2*d*e**2*p*x**6 - 2*b**2*d*e**2*x**6 + b**2*e**3*p*x**9 
- b**2*e**3*x**9 + 2*b*c*d**3*p*x**3 + 5*b*c*d**2*e*p*x**6 - b*c*d**2*e*x* 
*6 + 4*b*c*d*e**2*p*x**9 - 2*b*c*d*e**2*x**9 + b*c*e**3*p*x**12 - b*c*e**3 
*x**12 + 2*c**2*d**3*p*x**6 + 4*c**2*d**2*e*p*x**9 + 2*c**2*d*e**2*p*x**12 
),x)*a*b*c*d*e**3*p**3 + 18*int(((a + b*x**3 + c*x**6)**p*x**8)/(a*b*d**2* 
e*p - a*b*d**2*e + 2*a*b*d*e**2*p*x**3 - 2*a*b*d*e**2*x**3 + a*b*e**3*p*x* 
*6 - a*b*e**3*x**6 + 2*a*c*d**3*p + 4*a*c*d**2*e*p*x**3 + 2*a*c*d*e**2*p*x 
**6 + b**2*d**2*e*p*x**3 - b**2*d**2*e*x**3 + 2*b**2*d*e**2*p*x**6 - 2*b** 
2*d*e**2*x**6 + b**2*e**3*p*x**9 - b**2*e**3*x**9 + 2*b*c*d**3*p*x**3 + 5* 
b*c*d**2*e*p*x**6 - b*c*d**2*e*x**6 + 4*b*c*d*e**2*p*x**9 - 2*b*c*d*e**2*x 
**9 + b*c*e**3*p*x**12 - b*c*e**3*x**12 + 2*c**2*d**3*p*x**6 + 4*c**2*d**2 
*e*p*x**9 + 2*c**2*d*e**2*p*x**12),x)*a*b*c*d*e**3*p**2 - 6*int(((a + b*x* 
*3 + c*x**6)**p*x**8)/(a*b*d**2*e*p - a*b*d**2*e + 2*a*b*d*e**2*p*x**3 - 2 
*a*b*d*e**2*x**3 + a*b*e**3*p*x**6 - a*b*e**3*x**6 + 2*a*c*d**3*p + 4*a...