\(\int (A+B x+C x^2) (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3)^p \, dx\) [80]

Optimal result

Integrand size = 41, antiderivative size = 177 \[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx=\frac {C \left (-64+(b+c x)^3\right )^{1+p}}{3 c^3 (1+p)}-\frac {64^p \left (b B c-A c^2-b^2 C\right ) (b+c x) \left (64-(b+c x)^3\right )^{-p} \left (-64+(b+c x)^3\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-p,\frac {4}{3},\frac {1}{64} (b+c x)^3\right )}{c^3}+\frac {2^{-1+6 p} (B c-2 b C) (b+c x)^2 \left (64-(b+c x)^3\right )^{-p} \left (-64+(b+c x)^3\right )^p \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},\frac {1}{64} (b+c x)^3\right )}{c^3} \] Output:

1/3*C*(-64+(c*x+b)^3)^(p+1)/c^3/(p+1)-64^p*(-A*c^2+B*b*c-C*b^2)*(c*x+b)*(- 
64+(c*x+b)^3)^p*hypergeom([1/3, -p],[4/3],1/64*(c*x+b)^3)/c^3/((64-(c*x+b) 
^3)^p)+2^(-1+6*p)*(B*c-2*C*b)*(c*x+b)^2*(-64+(c*x+b)^3)^p*hypergeom([2/3, 
-p],[5/3],1/64*(c*x+b)^3)/c^3/((64-(c*x+b)^3)^p)
 

Mathematica [F]

\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx \] Input:

Integrate[(A + B*x + C*x^2)*(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3 
)^p,x]
 

Output:

Integrate[(A + B*x + C*x^2)*(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3 
)^p, x]
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2459, 2425, 793, 2432, 2009}

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.

\(\displaystyle \int \left (A+B x+C x^2\right ) \left (b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64\right )^p \, dx\)

\(\Big \downarrow \) 2459

\(\displaystyle \int \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^p \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )+C \left (\frac {b}{c}+x\right )^2\right )d\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 2425

\(\displaystyle \int \left (A-\frac {b (B c-b C)}{c^2}+\left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^pd\left (\frac {b}{c}+x\right )+C \int \left (\frac {b}{c}+x\right )^2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^pd\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 793

\(\displaystyle \int \left (A-\frac {b (B c-b C)}{c^2}+\left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^pd\left (\frac {b}{c}+x\right )+\frac {C \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{p+1}}{3 c^3 (p+1)}\)

\(\Big \downarrow \) 2432

\(\displaystyle \int \left (A \left (\frac {b (b C-B c)}{A c^2}+1\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^p+\frac {(B c-2 b C) \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^p}{c}\right )d\left (\frac {b}{c}+x\right )+\frac {C \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{p+1}}{3 c^3 (p+1)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^p \left (1-\frac {1}{64} c^3 \left (\frac {b}{c}+x\right )^3\right )^{-p} \left (-A c^2+b^2 (-C)+b B c\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-p,\frac {4}{3},\frac {1}{64} c^3 \left (\frac {b}{c}+x\right )^3\right )}{c^2}+\frac {\left (\frac {b}{c}+x\right )^2 (B c-2 b C) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^p \left (1-\frac {1}{64} c^3 \left (\frac {b}{c}+x\right )^3\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-p,\frac {5}{3},\frac {1}{64} c^3 \left (\frac {b}{c}+x\right )^3\right )}{2 c}+\frac {C \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{p+1}}{3 c^3 (p+1)}\)

Input:

Int[(A + B*x + C*x^2)*(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^p,x]
 

Output:

(C*(-64 + c^3*(b/c + x)^3)^(1 + p))/(3*c^3*(1 + p)) - ((b*B*c - A*c^2 - b^ 
2*C)*(b/c + x)*(-64 + c^3*(b/c + x)^3)^p*Hypergeometric2F1[1/3, -p, 4/3, ( 
c^3*(b/c + x)^3)/64])/(c^2*(1 - (c^3*(b/c + x)^3)/64)^p) + ((B*c - 2*b*C)* 
(b/c + x)^2*(-64 + c^3*(b/c + x)^3)^p*Hypergeometric2F1[2/3, -p, 5/3, (c^3 
*(b/c + x)^3)/64])/(2*c*(1 - (c^3*(b/c + x)^3)/64)^p)
 

Defintions of rubi rules used

rule 793

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) 
^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && 
 NeQ[p, -1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2425

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Pq, x, n - 
 1]   Int[x^(n - 1)*(a + b*x^n)^p, x], x] + Int[ExpandToSum[Pq - Coeff[Pq, 
x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[P 
q, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1
 

rule 2432

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[ 
Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n, p}, x] && (PolyQ[Pq, x] || Poly 
Q[Pq, x^n])
 

rule 2459

Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 
]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x 
 -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial 
Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - 
> x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ 
[Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] &&  !(MonomialQ[Qx, x] 
&& IGtQ[p, 0])
 

Maple [F]

\[\int \left (C \,x^{2}+B x +A \right ) \left (c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64\right )^{p}d x\]

Input:

int((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^p,x)
 

Output:

int((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^p,x)
 

Fricas [F]

\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^p,x, algori 
thm="fricas")
 

Output:

integral((C*x^2 + B*x + A)*(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^ 
p, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx=\text {Timed out} \] Input:

integrate((C*x**2+B*x+A)*(c**3*x**3+3*b*c**2*x**2+3*b**2*c*x+b**3-64)**p,x 
)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^p,x, algori 
thm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64) 
^p, x)
 

Giac [F]

\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^p,x, algori 
thm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64) 
^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64\right )}^p \,d x \] Input:

int((A + B*x + C*x^2)*(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^p,x)
 

Output:

int((A + B*x + C*x^2)*(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^p, x)
 

Reduce [F]

\[ \int \left (A+B x+C x^2\right ) \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^p \, dx=\text {too large to display} \] Input:

int((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^p,x)
 

Output:

(9*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**p*a*b*c*p**2 + 15 
*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**p*a*b*c*p + 6*(b**3 
 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**p*a*b*c + 9*(b**3 + 3*b** 
2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**p*a*c**2*p**2*x + 15*(b**3 + 3*b* 
*2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**p*a*c**2*p*x + 6*(b**3 + 3*b**2* 
c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**p*a*c**2*x - 3*(b**3 + 3*b**2*c*x + 
 3*b*c**2*x**2 + c**3*x**3 - 64)**p*b**3*p - (b**3 + 3*b**2*c*x + 3*b*c**2 
*x**2 + c**3*x**3 - 64)**p*b**3 + 9*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c 
**3*x**3 - 64)**p*b**2*c*p**2*x + 3*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c 
**3*x**3 - 64)**p*b**2*c*p*x + 18*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c** 
3*x**3 - 64)**p*b*c**2*p**2*x**2 + 15*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + 
 c**3*x**3 - 64)**p*b*c**2*p*x**2 + 3*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + 
 c**3*x**3 - 64)**p*b*c**2*x**2 + 9*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c 
**3*x**3 - 64)**p*c**3*p**2*x**3 + 9*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + 
c**3*x**3 - 64)**p*c**3*p*x**3 + 2*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c* 
*3*x**3 - 64)**p*c**3*x**3 - 288*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3 
*x**3 - 64)**p*p**2 - 192*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 
 64)**p*p - 32*(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**p - 1 
5552*int((b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**p/(9*b**3*p 
**2 + 9*b**3*p + 2*b**3 + 27*b**2*c*p**2*x + 27*b**2*c*p*x + 6*b**2*c*x...