Integrand size = 14, antiderivative size = 250 \[ \int \left (2-6 x^2+3 x^3\right )^p \, dx=\frac {3^{-1+\frac {p}{2}} \left (2-6 x^2+3 x^3\right )^p \operatorname {AppellF1}\left (1+p,-p,-p,2+p,-\frac {\left (3 x-2 \left (1+2 \cos \left (\frac {1}{6} \left (\pi +2 \arcsin \left (\frac {1}{8}\right )\right )\right )\right )\right ) \sec \left (\frac {1}{3} \arcsin \left (\frac {1}{8}\right )\right )}{4 \sqrt {3}},-\frac {3 x-2 \left (1+2 \cos \left (\frac {1}{6} \left (\pi +2 \arcsin \left (\frac {1}{8}\right )\right )\right )\right )}{2 \left (\sqrt {3} \cos \left (\frac {1}{3} \arcsin \left (\frac {1}{8}\right )\right )-3 \sin \left (\frac {1}{3} \arcsin \left (\frac {1}{8}\right )\right )\right )}\right ) \left (3 x-2 \left (1+2 \cos \left (\frac {1}{6} \left (\pi +2 \arcsin \left (\frac {1}{8}\right )\right )\right )\right )\right ) \left (8 \cos \left (\frac {1}{3} \arcsin \left (\frac {1}{8}\right )\right ) \left (\sqrt {3} \cos \left (\frac {1}{3} \arcsin \left (\frac {1}{8}\right )\right )-3 \sin \left (\frac {1}{3} \arcsin \left (\frac {1}{8}\right )\right )\right )\right )^p \left (3 x-2 \left (1+2 \sin \left (\frac {1}{3} \arcsin \left (\frac {1}{8}\right )\right )\right )\right )^{-p} \left (3 x-2 \left (1-2 \sin \left (\frac {1}{3} \left (\pi +\arcsin \left (\frac {1}{8}\right )\right )\right )\right )\right )^{-p}}{1+p} \] Output:
3^(-1+1/2*p)*(3*x^3-6*x^2+2)^p*AppellF1(p+1,-p,-p,2+p,-1/2*(3*x-2-4*cos(1/ 6*Pi+1/3*arcsin(1/8)))/(3^(1/2)*cos(1/3*arcsin(1/8))-3*sin(1/3*arcsin(1/8) )),-1/12*(3*x-2-4*cos(1/6*Pi+1/3*arcsin(1/8)))*sec(1/3*arcsin(1/8))*3^(1/2 ))*(3*x-2-4*cos(1/6*Pi+1/3*arcsin(1/8)))*(8*cos(1/3*arcsin(1/8))*(3^(1/2)* cos(1/3*arcsin(1/8))-3*sin(1/3*arcsin(1/8))))^p/(p+1)/((3*x-2-4*sin(1/3*ar csin(1/8)))^p)/((3*x-2+4*sin(1/3*Pi+1/3*arcsin(1/8)))^p)
Result contains higher order function than in optimal. Order 9 vs. order 6 in optimal.
Time = 0.08 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.26 \[ \int \left (2-6 x^2+3 x^3\right )^p \, dx=\frac {\operatorname {AppellF1}\left (1+p,-p,-p,2+p,\frac {-x+\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,1\right ]}{\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,2\right ]},\frac {-x+\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,1\right ]}{\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,3\right ]}\right ) \left (x-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,1\right ]\right ) \left (-x+\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,2\right ]\right )^{-p} \left (\left (2-6 x^2+3 x^3\right ) \left (\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,2\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,3\right ]\right )\right )^p \left (-x+\text {Root}\left [3 \text {$\#$1}^3-6 \text {$\#$1}^2+2\&,3\right ]\right )^{-p}}{1+p} \] Input:
Integrate[(2 - 6*x^2 + 3*x^3)^p,x]
Output:
(AppellF1[1 + p, -p, -p, 2 + p, (-x + Root[2 - 6*#1^2 + 3*#1^3 & , 1, 0])/ (Root[2 - 6*#1^2 + 3*#1^3 & , 1, 0] - Root[2 - 6*#1^2 + 3*#1^3 & , 2, 0]), (-x + Root[2 - 6*#1^2 + 3*#1^3 & , 1, 0])/(Root[2 - 6*#1^2 + 3*#1^3 & , 1 , 0] - Root[2 - 6*#1^2 + 3*#1^3 & , 3, 0])]*(x - Root[2 - 6*#1^2 + 3*#1^3 & , 1, 0])*((2 - 6*x^2 + 3*x^3)*(Root[2 - 6*#1^2 + 3*#1^3 & , 1, 0] - Root [2 - 6*#1^2 + 3*#1^3 & , 2, 0])*(Root[2 - 6*#1^2 + 3*#1^3 & , 1, 0] - Root [2 - 6*#1^2 + 3*#1^3 & , 3, 0]))^p)/((1 + p)*(-x + Root[2 - 6*#1^2 + 3*#1^ 3 & , 2, 0])^p*(-x + Root[2 - 6*#1^2 + 3*#1^3 & , 3, 0])^p)
Result contains complex when optimal does not.
Time = 3.71 (sec) , antiderivative size = 677, normalized size of antiderivative = 2.71, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2481, 2475, 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.
| \(\displaystyle \int \left (3 x^3-6 x^2+2\right )^p \, dx\) |
| \(\Big \downarrow \) 2481 |
| \(\displaystyle \int \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^pd\left (x-\frac {2}{3}\right )\) |
| \(\Big \downarrow \) 2475 |
| \(\displaystyle \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{-p} \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^p \int \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^p \left (9 \left (x-\frac {2}{3}\right )^2-\frac {3 \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right ) \left (x-\frac {2}{3}\right )}{\sqrt [3]{1-3 i \sqrt {7}}}+\left (1-3 i \sqrt {7}\right )^{2/3}+\frac {16}{\left (1-3 i \sqrt {7}\right )^{2/3}}-4\right )^pd\left (x-\frac {2}{3}\right )\) |
| \(\Big \downarrow \) 1179 |
| \(\displaystyle \frac {1}{3} \left (1+\frac {2 \left (3 \sqrt {7}+i\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^{-p} \left (1-\frac {2 \left (3 \sqrt {7}+i\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^p \int \left (\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}+1\right )^p \left (1-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^p \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )^pd\left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {\left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right ) \left (1+\frac {2 \left (3 \sqrt {7}+i\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^{-p} \left (1-\frac {2 \left (3 \sqrt {7}+i\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )^{-p} \left (3 \left (x-\frac {2}{3}\right )^3-4 \left (x-\frac {2}{3}\right )+\frac {2}{9}\right )^p \operatorname {AppellF1}\left (p+1,-p,-p,p+2,-\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )-3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )},\frac {2 \left (i+3 \sqrt {7}\right ) \left (3 \left (x-\frac {2}{3}\right )+\frac {4+\left (1-3 i \sqrt {7}\right )^{2/3}}{\sqrt [3]{1-3 i \sqrt {7}}}\right )}{\sqrt {-3 \left (1-3 i \sqrt {7}\right )^{2/3}} \left (i+3 \sqrt {7}-4 i \sqrt [3]{1-3 i \sqrt {7}}\right )+3 i \left (1-3 i \sqrt {7}\right )^{2/3} \left (4+\left (1-3 i \sqrt {7}\right )^{2/3}\right )}\right )}{3 (p+1)}\) |
Input:
Int[(2 - 6*x^2 + 3*x^3)^p,x]
Output:
(((4 + (1 - (3*I)*Sqrt[7])^(2/3))/(1 - (3*I)*Sqrt[7])^(1/3) + 3*(-2/3 + x) )*(2/9 - 4*(-2/3 + x) + 3*(-2/3 + x)^3)^p*AppellF1[1 + p, -p, -p, 2 + p, ( -2*(I + 3*Sqrt[7])*((4 + (1 - (3*I)*Sqrt[7])^(2/3))/(1 - (3*I)*Sqrt[7])^(1 /3) + 3*(-2/3 + x)))/(Sqrt[-3*(1 - (3*I)*Sqrt[7])^(2/3)]*(I + 3*Sqrt[7] - (4*I)*(1 - (3*I)*Sqrt[7])^(1/3)) - (3*I)*(1 - (3*I)*Sqrt[7])^(2/3)*(4 + (1 - (3*I)*Sqrt[7])^(2/3))), (2*(I + 3*Sqrt[7])*((4 + (1 - (3*I)*Sqrt[7])^(2 /3))/(1 - (3*I)*Sqrt[7])^(1/3) + 3*(-2/3 + x)))/(Sqrt[-3*(1 - (3*I)*Sqrt[7 ])^(2/3)]*(I + 3*Sqrt[7] - (4*I)*(1 - (3*I)*Sqrt[7])^(1/3)) + (3*I)*(1 - ( 3*I)*Sqrt[7])^(2/3)*(4 + (1 - (3*I)*Sqrt[7])^(2/3)))])/(3*(1 + p)*(1 + (2* (I + 3*Sqrt[7])*((4 + (1 - (3*I)*Sqrt[7])^(2/3))/(1 - (3*I)*Sqrt[7])^(1/3) + 3*(-2/3 + x)))/(Sqrt[-3*(1 - (3*I)*Sqrt[7])^(2/3)]*(I + 3*Sqrt[7] - (4* I)*(1 - (3*I)*Sqrt[7])^(1/3)) - (3*I)*(1 - (3*I)*Sqrt[7])^(2/3)*(4 + (1 - (3*I)*Sqrt[7])^(2/3))))^p*(1 - (2*(I + 3*Sqrt[7])*((4 + (1 - (3*I)*Sqrt[7] )^(2/3))/(1 - (3*I)*Sqrt[7])^(1/3) + 3*(-2/3 + x)))/(Sqrt[-3*(1 - (3*I)*Sq rt[7])^(2/3)]*(I + 3*Sqrt[7] - (4*I)*(1 - (3*I)*Sqrt[7])^(1/3)) + (3*I)*(1 - (3*I)*Sqrt[7])^(2/3)*(4 + (1 - (3*I)*Sqrt[7])^(2/3))))^p)
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])
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]
Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> 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/1 8^(1/3))*x + d^2*x^2, x]^p) Int[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, x], x]] /; Free Q[{a, b, d, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && !IntegerQ[p]
Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c* d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, c/(3 *d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
\[\int \left (3 x^{3}-6 x^{2}+2\right )^{p}d x\]
Input:
int((3*x^3-6*x^2+2)^p,x)
Output:
int((3*x^3-6*x^2+2)^p,x)
\[ \int \left (2-6 x^2+3 x^3\right )^p \, dx=\int { {\left (3 \, x^{3} - 6 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((3*x^3-6*x^2+2)^p,x, algorithm="fricas")
Output:
integral((3*x^3 - 6*x^2 + 2)^p, x)
\[ \int \left (2-6 x^2+3 x^3\right )^p \, dx=\int \left (3 x^{3} - 6 x^{2} + 2\right )^{p}\, dx \] Input:
integrate((3*x**3-6*x**2+2)**p,x)
Output:
Integral((3*x**3 - 6*x**2 + 2)**p, x)
\[ \int \left (2-6 x^2+3 x^3\right )^p \, dx=\int { {\left (3 \, x^{3} - 6 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((3*x^3-6*x^2+2)^p,x, algorithm="maxima")
Output:
integrate((3*x^3 - 6*x^2 + 2)^p, x)
\[ \int \left (2-6 x^2+3 x^3\right )^p \, dx=\int { {\left (3 \, x^{3} - 6 \, x^{2} + 2\right )}^{p} \,d x } \] Input:
integrate((3*x^3-6*x^2+2)^p,x, algorithm="giac")
Output:
integrate((3*x^3 - 6*x^2 + 2)^p, x)
Timed out. \[ \int \left (2-6 x^2+3 x^3\right )^p \, dx=\int {\left (3\,x^3-6\,x^2+2\right )}^p \,d x \] Input:
int((3*x^3 - 6*x^2 + 2)^p,x)
Output:
int((3*x^3 - 6*x^2 + 2)^p, x)
\[ \int \left (2-6 x^2+3 x^3\right )^p \, dx=\frac {3 \left (3 x^{3}-6 x^{2}+2\right )^{p} x -2 \left (3 x^{3}-6 x^{2}+2\right )^{p}+54 \left (\int \frac {\left (3 x^{3}-6 x^{2}+2\right )^{p}}{9 p \,x^{3}-18 p \,x^{2}+3 x^{3}-6 x^{2}+6 p +2}d x \right ) p^{2}+18 \left (\int \frac {\left (3 x^{3}-6 x^{2}+2\right )^{p}}{9 p \,x^{3}-18 p \,x^{2}+3 x^{3}-6 x^{2}+6 p +2}d x \right ) p -72 \left (\int \frac {\left (3 x^{3}-6 x^{2}+2\right )^{p} x}{9 p \,x^{3}-18 p \,x^{2}+3 x^{3}-6 x^{2}+6 p +2}d x \right ) p^{2}-24 \left (\int \frac {\left (3 x^{3}-6 x^{2}+2\right )^{p} x}{9 p \,x^{3}-18 p \,x^{2}+3 x^{3}-6 x^{2}+6 p +2}d x \right ) p}{9 p +3} \] Input:
int((3*x^3-6*x^2+2)^p,x)
Output:
(3*(3*x**3 - 6*x**2 + 2)**p*x - 2*(3*x**3 - 6*x**2 + 2)**p + 54*int((3*x** 3 - 6*x**2 + 2)**p/(9*p*x**3 - 18*p*x**2 + 6*p + 3*x**3 - 6*x**2 + 2),x)*p **2 + 18*int((3*x**3 - 6*x**2 + 2)**p/(9*p*x**3 - 18*p*x**2 + 6*p + 3*x**3 - 6*x**2 + 2),x)*p - 72*int(((3*x**3 - 6*x**2 + 2)**p*x)/(9*p*x**3 - 18*p *x**2 + 6*p + 3*x**3 - 6*x**2 + 2),x)*p**2 - 24*int(((3*x**3 - 6*x**2 + 2) **p*x)/(9*p*x**3 - 18*p*x**2 + 6*p + 3*x**3 - 6*x**2 + 2),x)*p)/(3*(3*p + 1))