Integrand size = 32, antiderivative size = 135 \[ \int \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \, dx=-\frac {2^{-1-p} 3^{-1+2 p} \sqrt {1-6 b} \left (1-\frac {1-6 x}{\sqrt {1-6 b}}\right )^{-2 p} \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right ) \left (-2 (1-6 b)^{3/2}+3 (1-6 b) (1-6 x)-(1-6 x)^3\right )^p \operatorname {Hypergeometric2F1}\left (-2 p,1+p,2+p,\frac {1}{3} \left (2+\frac {1-6 x}{\sqrt {1-6 b}}\right )\right )}{1+p} \] Output:
-2^(-1-p)*3^(-1+2*p)*(1-6*b)^(1/2)*(2+(1-6*x)/(1-6*b)^(1/2))*(-2*(1-6*b)^( 3/2)+3*(1-6*b)*(1-6*x)-(1-6*x)^3)^p*hypergeom([-2*p, p+1],[2+p],2/3+1/3*(1 -6*x)/(1-6*b)^(1/2))/(p+1)/((1-(1-6*x)/(1-6*b)^(1/2))^(2*p))
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.29 (sec) , antiderivative size = 727, normalized size of antiderivative = 5.39 \[ \int \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \, dx=\frac {\left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \operatorname {AppellF1}\left (1+p,-p,-p,2+p,\frac {-x+\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,1\right ]}{\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,1\right ]-\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,2\right ]},\frac {-x+\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,1\right ]}{\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,1\right ]-\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,3\right ]}\right ) \left (x-\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,1\right ]\right ) \left (\frac {x-\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,2\right ]}{\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,1\right ]-\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,2\right ]}\right )^{-p} \left (\frac {x-\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,3\right ]}{\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,1\right ]-\text {Root}\left [1-\sqrt {1-6 b}-9 b+6 \sqrt {1-6 b} b+54 b \text {$\#$1}-54 \text {$\#$1}^2+108 \text {$\#$1}^3\&,3\right ]}\right )^{-p}}{1+p} \] Input:
Integrate[(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)^p,x]
Output:
((1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)^p*AppellF1[1 + p, -p, -p, 2 + p, (-x + Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 5 4*b*#1 - 54*#1^2 + 108*#1^3 & , 1])/(Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt [1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 1] - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 2]), (-x + R oot[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108* #1^3 & , 1])/(Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 1] - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b ]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3])]*(x - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 1]))/((1 + p)*( (x - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 2])/(Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54* b*#1 - 54*#1^2 + 108*#1^3 & , 1] - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 2]))^p*((x - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3])/(R oot[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108* #1^3 & , 1] - Root[1 - Sqrt[1 - 6*b] - 9*b + 6*Sqrt[1 - 6*b]*b + 54*b*#1 - 54*#1^2 + 108*#1^3 & , 3]))^p)
Time = 0.39 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2480, 80, 79}
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.
| \(\displaystyle \int \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^p \, dx\) |
| \(\Big \downarrow \) 2480 |
| \(\displaystyle \left (5832 (1-6 b) x-972 \left (1-\sqrt {1-6 b}\right ) (1-6 b)\right )^{-2 p} \left (314928 (1-6 b) x-52488 \left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{-p} \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^p \int \left (5832 (1-6 b) x-972 \left (1-\sqrt {1-6 b}\right ) (1-6 b)\right )^{2 p} \left (314928 (1-6 b) x-52488 \left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^pdx\) |
| \(\Big \downarrow \) 80 |
| \(\displaystyle 3^{2 p} \left (-\frac {\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x}{(1-6 b)^{3/2}}\right )^{-2 p} \left (314928 (1-6 b) x-52488 \left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^{-p} \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^p \int \left (\frac {1}{3} \left (1-\frac {1}{\sqrt {1-6 b}}\right )+\frac {2 x}{\sqrt {1-6 b}}\right )^{2 p} \left (314928 (1-6 b) x-52488 \left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right )^pdx\) |
| \(\Big \downarrow \) 79 |
| \(\displaystyle \frac {3^{2 p-9} \left (314928 (1-6 b) x-52488 \left (2 \sqrt {1-6 b}+1\right ) (1-6 b)\right ) \left (-\frac {\left (1-\sqrt {1-6 b}\right ) (1-6 b)-6 (1-6 b) x}{(1-6 b)^{3/2}}\right )^{-2 p} \left (54 b x-(1-6 b)^{3/2}-9 b+108 x^3-54 x^2+1\right )^p \operatorname {Hypergeometric2F1}\left (-2 p,p+1,p+2,\frac {\left (2 \sqrt {1-6 b}+1\right ) (1-6 b)-6 (1-6 b) x}{3 (1-6 b)^{3/2}}\right )}{16 (1-6 b) (p+1)}\) |
Input:
Int[(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)^p,x]
Output:
(3^(-9 + 2*p)*(-52488*(1 + 2*Sqrt[1 - 6*b])*(1 - 6*b) + 314928*(1 - 6*b)*x )*(1 - (1 - 6*b)^(3/2) - 9*b + 54*b*x - 54*x^2 + 108*x^3)^p*Hypergeometric 2F1[-2*p, 1 + p, 2 + p, ((1 + 2*Sqrt[1 - 6*b])*(1 - 6*b) - 6*(1 - 6*b)*x)/ (3*(1 - 6*b)^(3/2))])/(16*(1 - 6*b)*(1 + p)*(-(((1 - Sqrt[1 - 6*b])*(1 - 6 *b) - 6*(1 - 6*b)*x)/(1 - 6*b)^(3/2)))^(2*p))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
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]}, Simp[Px^p/((c^3 - 4*b*c*d + 9* a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p)) Int [(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3* b*d)*x)^(2*p), x], x] /; EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 27* a^2*d^2, 0] && NeQ[c^2 - 3*b*d, 0]] /; FreeQ[p, x] && PolyQ[Px, x, 3] && ! IntegerQ[p]
\[\int \left (1-\left (1-6 b \right )^{\frac {3}{2}}-9 b +54 b x -54 x^{2}+108 x^{3}\right )^{p}d x\]
Input:
int((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^p,x)
Output:
int((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^p,x)
\[ \int \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \, dx=\int { {\left (108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1\right )}^{p} \,d x } \] Input:
integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^p,x, algorithm="fric as")
Output:
integral((108*x^3 + 54*b*x - 54*x^2 + (6*b - 1)*sqrt(-6*b + 1) - 9*b + 1)^ p, x)
Exception generated. \[ \int \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \, dx=\text {Exception raised: AttributeError} \] Input:
integrate((1-(1-6*b)**(3/2)-9*b+54*b*x-54*x**2+108*x**3)**p,x)
Output:
Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr imitive'
\[ \int \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \, dx=\int { {\left (108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1\right )}^{p} \,d x } \] Input:
integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^p,x, algorithm="maxi ma")
Output:
integrate((108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1)^p, x)
\[ \int \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \, dx=\int { {\left (108 \, x^{3} + 54 \, b x - 54 \, x^{2} - {\left (-6 \, b + 1\right )}^{\frac {3}{2}} - 9 \, b + 1\right )}^{p} \,d x } \] Input:
integrate((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^p,x, algorithm="giac ")
Output:
integrate((108*x^3 + 54*b*x - 54*x^2 - (-6*b + 1)^(3/2) - 9*b + 1)^p, x)
Timed out. \[ \int \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \, dx=\int {\left (54\,b\,x-9\,b-{\left (1-6\,b\right )}^{3/2}-54\,x^2+108\,x^3+1\right )}^p \,d x \] Input:
int((54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^p,x)
Output:
int((54*b*x - 9*b - (1 - 6*b)^(3/2) - 54*x^2 + 108*x^3 + 1)^p, x)
\[ \int \left (1-(1-6 b)^{3/2}-9 b+54 b x-54 x^2+108 x^3\right )^p \, dx=\text {too large to display} \] Input:
int((1-(1-6*b)^(3/2)-9*b+54*b*x-54*x^2+108*x^3)^p,x)
Output:
( - 432*sqrt( - 6*b + 1)*int((6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54 *b*x - 9*b + 108*x**3 - 54*x**2 + 1)**p/(24*b**2*p + 8*b**2 + 180*b*p*x**2 - 60*b*p*x - 3*b*p + 60*b*x**2 - 20*b*x - b + 216*p*x**4 - 144*p*x**3 + 6 *p*x**2 + 6*p*x + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b**2*p**2 - 144*sqr t( - 6*b + 1)*int((6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)**p/(24*b**2*p + 8*b**2 + 180*b*p*x**2 - 60*b*p*x - 3*b*p + 60*b*x**2 - 20*b*x - b + 216*p*x**4 - 144*p*x**3 + 6*p*x**2 + 6 *p*x + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b**2*p + 36*sqrt( - 6*b + 1)*i nt((6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54 *x**2 + 1)**p/(24*b**2*p + 8*b**2 + 180*b*p*x**2 - 60*b*p*x - 3*b*p + 60*b *x**2 - 20*b*x - b + 216*p*x**4 - 144*p*x**3 + 6*p*x**2 + 6*p*x + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*b*p**2 + 12*sqrt( - 6*b + 1)*int((6*sqrt( - 6 *b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)**p/( 24*b**2*p + 8*b**2 + 180*b*p*x**2 - 60*b*p*x - 3*b*p + 60*b*x**2 - 20*b*x - b + 216*p*x**4 - 144*p*x**3 + 6*p*x**2 + 6*p*x + 72*x**4 - 48*x**3 + 2*x **2 + 2*x),x)*b*p + 6*sqrt( - 6*b + 1)*int((6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*b*x - 9*b + 108*x**3 - 54*x**2 + 1)**p/(24*b**2*p + 8*b**2 + 180*b*p*x**2 - 60*b*p*x - 3*b*p + 60*b*x**2 - 20*b*x - b + 216*p*x**4 - 144*p*x**3 + 6*p*x**2 + 6*p*x + 72*x**4 - 48*x**3 + 2*x**2 + 2*x),x)*p**2 + 2*sqrt( - 6*b + 1)*int((6*sqrt( - 6*b + 1)*b - sqrt( - 6*b + 1) + 54*...