\(\int (A+B x+C x^2) (2-4 x+3 x^3)^p \, dx\) [21]

Optimal result

Integrand size = 23, antiderivative size = 936 \[ \int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx =\text {Too large to display} \] Output:

C*(3*x^3-4*x+2)^(p+1)/(9*p+9)+1/27*(9*A-3*(4+(9-17^(1/2))^(2/3))*B/(9-17^( 
1/2))^(1/3)+4*C)*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)*(3*x^3-4* 
x+2)^p*AppellF1(p+1,-p,-p,2+p,2*I*(9-17^(1/2))^(1/3)*((4+(9-17^(1/2))^(2/3 
))/(9-17^(1/2))^(1/3)+3*x)/(12*I-4*3^(1/2)+(3*I+3^(1/2))*(9-17^(1/2))^(2/3 
)),2*I*(9-17^(1/2))^(1/3)*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)/ 
(12*I+4*3^(1/2)+(3*I-3^(1/2))*(9-17^(1/2))^(2/3)))/(p+1)/((1-2*I*(9-17^(1/ 
2))^(1/3)*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)/(12*I+4*3^(1/2)+ 
(3*I-3^(1/2))*(9-17^(1/2))^(2/3)))^p)/((1-2*I*(9-17^(1/2))^(1/3)*((4+(9-17 
^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)/(12*I-4*3^(1/2)+(3*I+3^(1/2))*(9-17 
^(1/2))^(2/3)))^p)+1/9*B*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)^2 
*(3*x^3-4*x+2)^p*AppellF1(2+p,-p,-p,3+p,2*I*(9-17^(1/2))^(1/3)*((4+(9-17^( 
1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)/(12*I-4*3^(1/2)+(3*I+3^(1/2))*(9-17^( 
1/2))^(2/3)),2*I*(9-17^(1/2))^(1/3)*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^( 
1/3)+3*x)/(12*I+4*3^(1/2)+(3*I-3^(1/2))*(9-17^(1/2))^(2/3)))/(2+p)/((1-2*I 
*(9-17^(1/2))^(1/3)*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)/(12*I+ 
4*3^(1/2)+(3*I-3^(1/2))*(9-17^(1/2))^(2/3)))^p)/((1-2*I*(9-17^(1/2))^(1/3) 
*((4+(9-17^(1/2))^(2/3))/(9-17^(1/2))^(1/3)+3*x)/(12*I-4*3^(1/2)+(3*I+3^(1 
/2))*(9-17^(1/2))^(2/3)))^p)
 

Mathematica [F]

\[ \int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx=\int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx \] Input:

Integrate[(A + B*x + C*x^2)*(2 - 4*x + 3*x^3)^p,x]
 

Output:

Integrate[(A + B*x + C*x^2)*(2 - 4*x + 3*x^3)^p, x]
 

Rubi [A] (warning: unable to verify)

Time = 2.26 (sec) , antiderivative size = 1175, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2526, 2486, 1269, 1179, 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.

\(\displaystyle \int \left (3 x^3-4 x+2\right )^p \left (A+B x+C x^2\right ) \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle \frac {1}{9} \int (9 A+4 C+9 B x) \left (3 x^3-4 x+2\right )^pdx+\frac {C \left (3 x^3-4 x+2\right )^{p+1}}{9 (p+1)}\)

\(\Big \downarrow \) 2486

\(\displaystyle \frac {1}{9} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^{-p} \left (3 x^3-4 x+2\right )^p \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^{-p} \int \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^p (9 A+4 C+9 B x) \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^pdx+\frac {C \left (3 x^3-4 x+2\right )^{p+1}}{9 (p+1)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {1}{9} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^{-p} \left (3 x^3-4 x+2\right )^p \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^{-p} \left (\left (9 A-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B}{\sqrt [3]{9-\sqrt {17}}}+4 C\right ) \int \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^p \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^pdx+3 B \int \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^{p+1} \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^pdx\right )+\frac {C \left (3 x^3-4 x+2\right )^{p+1}}{9 (p+1)}\)

\(\Big \downarrow \) 1179

\(\displaystyle \frac {1}{9} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^{-p} \left (3 x^3-4 x+2\right )^p \left (\frac {1}{3} \left (9 A-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B}{\sqrt [3]{9-\sqrt {17}}}+4 C\right ) \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^p \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^{-p} \int \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^p \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^p \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^pd\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^{-p}+B \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^p \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^{-p} \int \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^{p+1} \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^p \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^pd\left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right ) \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^{-p}\right ) \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^{-p}+\frac {C \left (3 x^3-4 x+2\right )^{p+1}}{9 (p+1)}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {1}{9} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^{-p} \left (3 x^3-4 x+2\right )^p \left (\frac {\left (9 A-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) B}{\sqrt [3]{9-\sqrt {17}}}+4 C\right ) \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^{p+1} \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^p \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^{-p} \operatorname {AppellF1}\left (p+1,-p,-p,p+2,\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}},\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right ) \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^{-p}}{3 (p+1)}+\frac {B \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )^{p+2} \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^p \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^{-p} \operatorname {AppellF1}\left (p+2,-p,-p,p+3,\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i+\sqrt {3}\right )+\left (3 i-\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}},\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right ) \left (1-\frac {2 i \sqrt [3]{9-\sqrt {17}} \left (3 x+\frac {4+\left (9-\sqrt {17}\right )^{2/3}}{\sqrt [3]{9-\sqrt {17}}}\right )}{4 \left (3 i-\sqrt {3}\right )+\left (3 i+\sqrt {3}\right ) \left (9-\sqrt {17}\right )^{2/3}}\right )^{-p}}{p+2}\right ) \left (9 x^2-\frac {3 \left (4+\left (9-\sqrt {17}\right )^{2/3}\right ) x}{\sqrt [3]{9-\sqrt {17}}}+\left (9-\sqrt {17}\right )^{2/3}+\frac {16}{\left (9-\sqrt {17}\right )^{2/3}}-4\right )^{-p}+\frac {C \left (3 x^3-4 x+2\right )^{p+1}}{9 (p+1)}\)

Input:

Int[(A + B*x + C*x^2)*(2 - 4*x + 3*x^3)^p,x]
 

Output:

(C*(2 - 4*x + 3*x^3)^(1 + p))/(9*(1 + p)) + ((2 - 4*x + 3*x^3)^p*(((9*A - 
(3*(4 + (9 - Sqrt[17])^(2/3))*B)/(9 - Sqrt[17])^(1/3) + 4*C)*((4 + (9 - Sq 
rt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x)^(1 + p)*(-4 + 16/(9 - Sqrt[17]) 
^(2/3) + (9 - Sqrt[17])^(2/3) - (3*(4 + (9 - Sqrt[17])^(2/3))*x)/(9 - Sqrt 
[17])^(1/3) + 9*x^2)^p*AppellF1[1 + p, -p, -p, 2 + p, ((2*I)*(9 - Sqrt[17] 
)^(1/3)*((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x))/(4*(3*I + 
 Sqrt[3]) + (3*I - Sqrt[3])*(9 - Sqrt[17])^(2/3)), ((2*I)*(9 - Sqrt[17])^( 
1/3)*((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x))/(4*(3*I - Sq 
rt[3]) + (3*I + Sqrt[3])*(9 - Sqrt[17])^(2/3))])/(3*(1 + p)*(1 - ((2*I)*(9 
 - Sqrt[17])^(1/3)*((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x) 
)/(4*(3*I + Sqrt[3]) + (3*I - Sqrt[3])*(9 - Sqrt[17])^(2/3)))^p*(1 - ((2*I 
)*(9 - Sqrt[17])^(1/3)*((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 
3*x))/(4*(3*I - Sqrt[3]) + (3*I + Sqrt[3])*(9 - Sqrt[17])^(2/3)))^p) + (B* 
((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x)^(2 + p)*(-4 + 16/( 
9 - Sqrt[17])^(2/3) + (9 - Sqrt[17])^(2/3) - (3*(4 + (9 - Sqrt[17])^(2/3)) 
*x)/(9 - Sqrt[17])^(1/3) + 9*x^2)^p*AppellF1[2 + p, -p, -p, 3 + p, ((2*I)* 
(9 - Sqrt[17])^(1/3)*((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3* 
x))/(4*(3*I + Sqrt[3]) + (3*I - Sqrt[3])*(9 - Sqrt[17])^(2/3)), ((2*I)*(9 
- Sqrt[17])^(1/3)*((4 + (9 - Sqrt[17])^(2/3))/(9 - Sqrt[17])^(1/3) + 3*x)) 
/(4*(3*I - Sqrt[3]) + (3*I + Sqrt[3])*(9 - Sqrt[17])^(2/3))])/((2 + p)*...
 

Defintions of rubi rules used

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 1179

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[(a + b*x + c*x^2)^p/(e*(1 - ( 
d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) 
^p)   Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d 
- e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m 
, p}, x]
 

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 2486

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> With[{r = Rt[-9*a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]} 
, Simp[(a + b*x + d*x^3)^p/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x 
]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/ 
3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p)   Int[(e + f*x)^m*Sim 
p[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2 
*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/ 
3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && NeQ[4* 
b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [F]

\[\int \left (C \,x^{2}+B x +A \right ) \left (3 x^{3}-4 x +2\right )^{p}d x\]

Input:

int((C*x^2+B*x+A)*(3*x^3-4*x+2)^p,x)
 

Output:

int((C*x^2+B*x+A)*(3*x^3-4*x+2)^p,x)
 

Fricas [F]

\[ \int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 4 \, x + 2\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(3*x^3-4*x+2)^p,x, algorithm="fricas")
 

Output:

integral((C*x^2 + B*x + A)*(3*x^3 - 4*x + 2)^p, x)
 

Sympy [F]

\[ \int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx=\int \left (A + B x + C x^{2}\right ) \left (3 x^{3} - 4 x + 2\right )^{p}\, dx \] Input:

integrate((C*x**2+B*x+A)*(3*x**3-4*x+2)**p,x)
 

Output:

Integral((A + B*x + C*x**2)*(3*x**3 - 4*x + 2)**p, x)
 

Maxima [F]

\[ \int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 4 \, x + 2\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(3*x^3-4*x+2)^p,x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(3*x^3 - 4*x + 2)^p, x)
 

Giac [F]

\[ \int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx=\int { {\left (C x^{2} + B x + A\right )} {\left (3 \, x^{3} - 4 \, x + 2\right )}^{p} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(3*x^3-4*x+2)^p,x, algorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*(3*x^3 - 4*x + 2)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx=\int \left (C\,x^2+B\,x+A\right )\,{\left (3\,x^3-4\,x+2\right )}^p \,d x \] Input:

int((A + B*x + C*x^2)*(3*x^3 - 4*x + 2)^p,x)
 

Output:

int((A + B*x + C*x^2)*(3*x^3 - 4*x + 2)^p, x)
 

Reduce [F]

\[ \int \left (A+B x+C x^2\right ) \left (2-4 x+3 x^3\right )^p \, dx=\text {too large to display} \] Input:

int((C*x^2+B*x+A)*(3*x^3-4*x+2)^p,x)
 

Output:

(27*(3*x**3 - 4*x + 2)**p*a*p**2*x + 45*(3*x**3 - 4*x + 2)**p*a*p*x + 18*( 
3*x**3 - 4*x + 2)**p*a*x + 27*(3*x**3 - 4*x + 2)**p*b*p**2*x**2 - 24*(3*x* 
*3 - 4*x + 2)**p*b*p**2 + 36*(3*x**3 - 4*x + 2)**p*b*p*x**2 - 32*(3*x**3 - 
 4*x + 2)**p*b*p + 9*(3*x**3 - 4*x + 2)**p*b*x**2 - 8*(3*x**3 - 4*x + 2)** 
p*b + 27*(3*x**3 - 4*x + 2)**p*c*p**2*x**3 - 24*(3*x**3 - 4*x + 2)**p*c*p* 
*2*x + 18*(3*x**3 - 4*x + 2)**p*c*p**2 + 27*(3*x**3 - 4*x + 2)**p*c*p*x**3 
 - 16*(3*x**3 - 4*x + 2)**p*c*p*x + 18*(3*x**3 - 4*x + 2)**p*c*p + 6*(3*x* 
*3 - 4*x + 2)**p*c*x**3 + 4*(3*x**3 - 4*x + 2)**p*c + 1458*int((3*x**3 - 4 
*x + 2)**p/(27*p**2*x**3 - 36*p**2*x + 18*p**2 + 27*p*x**3 - 36*p*x + 18*p 
 + 6*x**3 - 8*x + 4),x)*a*p**5 + 3888*int((3*x**3 - 4*x + 2)**p/(27*p**2*x 
**3 - 36*p**2*x + 18*p**2 + 27*p*x**3 - 36*p*x + 18*p + 6*x**3 - 8*x + 4), 
x)*a*p**4 + 3726*int((3*x**3 - 4*x + 2)**p/(27*p**2*x**3 - 36*p**2*x + 18* 
p**2 + 27*p*x**3 - 36*p*x + 18*p + 6*x**3 - 8*x + 4),x)*a*p**3 + 1512*int( 
(3*x**3 - 4*x + 2)**p/(27*p**2*x**3 - 36*p**2*x + 18*p**2 + 27*p*x**3 - 36 
*p*x + 18*p + 6*x**3 - 8*x + 4),x)*a*p**2 + 216*int((3*x**3 - 4*x + 2)**p/ 
(27*p**2*x**3 - 36*p**2*x + 18*p**2 + 27*p*x**3 - 36*p*x + 18*p + 6*x**3 - 
 8*x + 4),x)*a*p - 864*int((3*x**3 - 4*x + 2)**p/(27*p**2*x**3 - 36*p**2*x 
 + 18*p**2 + 27*p*x**3 - 36*p*x + 18*p + 6*x**3 - 8*x + 4),x)*b*p**5 - 201 
6*int((3*x**3 - 4*x + 2)**p/(27*p**2*x**3 - 36*p**2*x + 18*p**2 + 27*p*x** 
3 - 36*p*x + 18*p + 6*x**3 - 8*x + 4),x)*b*p**4 - 1632*int((3*x**3 - 4*...