\(\int x^8 (a+b x^3+c x^6)^p \, dx\) [242]

Optimal result

Integrand size = 18, antiderivative size = 224 \[ \int x^8 \left (a+b x^3+c x^6\right )^p \, dx=-\frac {b (2+p) \left (a+b x^3+c x^6\right )^{1+p}}{6 c^2 (1+p) (3+2 p)}+\frac {x^3 \left (a+b x^3+c x^6\right )^{1+p}}{3 c (3+2 p)}+\frac {2^p \left (2 a c-b^2 (2+p)\right ) \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 c^2 \sqrt {b^2-4 a c} (1+p) (3+2 p)} \] Output:

-1/6*b*(2+p)*(c*x^6+b*x^3+a)^(p+1)/c^2/(p+1)/(3+2*p)+1/3*x^3*(c*x^6+b*x^3+ 
a)^(p+1)/c/(3+2*p)+1/3*2^p*(2*a*c-b^2*(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))/c^2/(-4*a*c+b^ 
2)^(1/2)/(p+1)/(3+2*p)
 

Mathematica [C] (verified)

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

Time = 0.52 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.72 \[ \int x^8 \left (a+b x^3+c x^6\right )^p \, dx=\frac {1}{9} x^9 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (3,-p,-p,4,-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right ) \] Input:

Integrate[x^8*(a + b*x^3 + c*x^6)^p,x]
 

Output:

(x^9*(a + b*x^3 + c*x^6)^p*AppellF1[3, -p, -p, 4, (-2*c*x^3)/(b + Sqrt[b^2 
 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(9*((b - Sqrt[b^2 - 4*a*c 
] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3) 
/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1693, 1166, 25, 1160, 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.

\(\displaystyle \int x^8 \left (a+b x^3+c x^6\right )^p \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {1}{3} \int x^6 \left (c x^6+b x^3+a\right )^pdx^3\)

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1096

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

Input:

Int[x^8*(a + b*x^3 + c*x^6)^p,x]
 

Output:

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

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 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 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 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]

\[\int x^{8} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}d x\]

Input:

int(x^8*(c*x^6+b*x^3+a)^p,x)
 

Output:

int(x^8*(c*x^6+b*x^3+a)^p,x)
 

Fricas [F]

\[ \int x^8 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{8} \,d x } \] Input:

integrate(x^8*(c*x^6+b*x^3+a)^p,x, algorithm="fricas")
 

Output:

integral((c*x^6 + b*x^3 + a)^p*x^8, x)
 

Sympy [F(-1)]

Timed out. \[ \int x^8 \left (a+b x^3+c x^6\right )^p \, dx=\text {Timed out} \] Input:

integrate(x**8*(c*x**6+b*x**3+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^8 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{8} \,d x } \] Input:

integrate(x^8*(c*x^6+b*x^3+a)^p,x, algorithm="maxima")
 

Output:

integrate((c*x^6 + b*x^3 + a)^p*x^8, x)
 

Giac [F]

\[ \int x^8 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{8} \,d x } \] Input:

integrate(x^8*(c*x^6+b*x^3+a)^p,x, algorithm="giac")
 

Output:

integrate((c*x^6 + b*x^3 + a)^p*x^8, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^8 \left (a+b x^3+c x^6\right )^p \, dx=\int x^8\,{\left (c\,x^6+b\,x^3+a\right )}^p \,d x \] Input:

int(x^8*(a + b*x^3 + c*x^6)^p,x)
 

Output:

int(x^8*(a + b*x^3 + c*x^6)^p, x)
 

Reduce [F]

\[ \int x^8 \left (a+b x^3+c x^6\right )^p \, dx =\text {Too large to display} \] Input:

int(x^8*(c*x^6+b*x^3+a)^p,x)
 

Output:

( - 4*(a + b*x**3 + c*x**6)**p*a**2*c*p - 4*(a + b*x**3 + c*x**6)**p*a**2* 
c + (a + b*x**3 + c*x**6)**p*a*b**2*p + 2*(a + b*x**3 + c*x**6)**p*a*b**2 
+ 4*(a + b*x**3 + c*x**6)**p*a*b*c*p**2*x**3 + 4*(a + b*x**3 + c*x**6)**p* 
a*b*c*p*x**3 - (a + b*x**3 + c*x**6)**p*b**3*p**2*x**3 - 2*(a + b*x**3 + c 
*x**6)**p*b**3*p*x**3 + 2*(a + b*x**3 + c*x**6)**p*b**2*c*p**2*x**6 + (a + 
 b*x**3 + c*x**6)**p*b**2*c*p*x**6 + 4*(a + b*x**3 + c*x**6)**p*b*c**2*p** 
2*x**9 + 6*(a + b*x**3 + c*x**6)**p*b*c**2*p*x**9 + 2*(a + b*x**3 + c*x**6 
)**p*b*c**2*x**9 + 96*int(((a + b*x**3 + c*x**6)**p*x**5)/(4*a*p**2 + 8*a* 
p + 3*a + 4*b*p**2*x**3 + 8*b*p*x**3 + 3*b*x**3 + 4*c*p**2*x**6 + 8*c*p*x* 
*6 + 3*c*x**6),x)*a**2*c**2*p**4 + 288*int(((a + b*x**3 + c*x**6)**p*x**5) 
/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**3 + 8*b*p*x**3 + 3*b*x**3 + 4*c*p** 
2*x**6 + 8*c*p*x**6 + 3*c*x**6),x)*a**2*c**2*p**3 + 264*int(((a + b*x**3 + 
 c*x**6)**p*x**5)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**3 + 8*b*p*x**3 + 3 
*b*x**3 + 4*c*p**2*x**6 + 8*c*p*x**6 + 3*c*x**6),x)*a**2*c**2*p**2 + 72*in 
t(((a + b*x**3 + c*x**6)**p*x**5)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x**3 
+ 8*b*p*x**3 + 3*b*x**3 + 4*c*p**2*x**6 + 8*c*p*x**6 + 3*c*x**6),x)*a**2*c 
**2*p - 48*int(((a + b*x**3 + c*x**6)**p*x**5)/(4*a*p**2 + 8*a*p + 3*a + 4 
*b*p**2*x**3 + 8*b*p*x**3 + 3*b*x**3 + 4*c*p**2*x**6 + 8*c*p*x**6 + 3*c*x* 
*6),x)*a*b**2*c*p**5 - 264*int(((a + b*x**3 + c*x**6)**p*x**5)/(4*a*p**2 + 
 8*a*p + 3*a + 4*b*p**2*x**3 + 8*b*p*x**3 + 3*b*x**3 + 4*c*p**2*x**6 + ...