Integrand size = 27, antiderivative size = 84 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=-\frac {b^3 \left (b^2-3 a c\right )^3 x}{c^3}+\frac {3 b^2 \left (b^2-3 a c\right )^2 (b+c x)^4}{4 c^4}-\frac {3 b \left (b^2-3 a c\right ) (b+c x)^7}{7 c^4}+\frac {(b+c x)^{10}}{10 c^4} \]
-b^3*(-3*a*c+b^2)^3*x/c^3+3/4*b^2*(-3*a*c+b^2)^2*(c*x+b)^4/c^4-3/7*b*(-3*a *c+b^2)*(c*x+b)^7/c^4+1/10*(c*x+b)^10/c^4
Time = 0.02 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.89 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=27 a^3 b^3 x+\frac {81}{2} a^2 b^4 x^2+27 a b^3 \left (b^2+a c\right ) x^3+\frac {27}{4} b^2 \left (b^4+6 a b^2 c+a^2 c^2\right ) x^4+\frac {27}{5} b^3 c \left (3 b^2+5 a c\right ) x^5+9 b^2 c^2 \left (2 b^2+a c\right ) x^6+\frac {9}{7} b c^3 \left (9 b^2+a c\right ) x^7+\frac {9}{2} b^2 c^4 x^8+b c^5 x^9+\frac {c^6 x^{10}}{10} \]
27*a^3*b^3*x + (81*a^2*b^4*x^2)/2 + 27*a*b^3*(b^2 + a*c)*x^3 + (27*b^2*(b^ 4 + 6*a*b^2*c + a^2*c^2)*x^4)/4 + (27*b^3*c*(3*b^2 + 5*a*c)*x^5)/5 + 9*b^2 *c^2*(2*b^2 + a*c)*x^6 + (9*b*c^3*(9*b^2 + a*c)*x^7)/7 + (9*b^2*c^4*x^8)/2 + b*c^5*x^9 + (c^6*x^10)/10
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2458, 747, 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.
\(\displaystyle \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx\) |
\(\Big \downarrow \) 2458 |
\(\displaystyle \int \left (b \left (3 a-\frac {b^2}{c}\right )+c^2 \left (\frac {b}{c}+x\right )^3\right )^3d\left (\frac {b}{c}+x\right )\) |
\(\Big \downarrow \) 747 |
\(\displaystyle \int \left (\frac {\left (3 a b c-b^3\right )^3}{c^3}+3 \left (b^3-3 a b c\right )^2 \left (\frac {b}{c}+x\right )^3-3 b c^3 \left (b^2-3 a c\right ) \left (\frac {b}{c}+x\right )^6+c^6 \left (\frac {b}{c}+x\right )^9\right )d\left (\frac {b}{c}+x\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3}{7} b c^3 \left (b^2-3 a c\right ) \left (\frac {b}{c}+x\right )^7+\frac {3}{4} b^2 \left (b^2-3 a c\right )^2 \left (\frac {b}{c}+x\right )^4-\frac {b^3 \left (b^2-3 a c\right )^3 \left (\frac {b}{c}+x\right )}{c^3}+\frac {1}{10} c^6 \left (\frac {b}{c}+x\right )^{10}\) |
-((b^3*(b^2 - 3*a*c)^3*(b/c + x))/c^3) + (3*b^2*(b^2 - 3*a*c)^2*(b/c + x)^ 4)/4 - (3*b*c^3*(b^2 - 3*a*c)*(b/c + x)^7)/7 + (c^6*(b/c + x)^10)/10
3.1.9.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b* x^n)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(78)=156\).
Time = 0.04 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.89
method | result | size |
norman | \(\frac {c^{6} x^{10}}{10}+b \,c^{5} x^{9}+\frac {9 b^{2} c^{4} x^{8}}{2}+\left (\frac {9}{7} a b \,c^{4}+\frac {81}{7} b^{3} c^{3}\right ) x^{7}+\left (9 a \,b^{2} c^{3}+18 b^{4} c^{2}\right ) x^{6}+\left (27 a \,b^{3} c^{2}+\frac {81}{5} b^{5} c \right ) x^{5}+\left (\frac {27}{4} a^{2} b^{2} c^{2}+\frac {81}{2} a \,b^{4} c +\frac {27}{4} b^{6}\right ) x^{4}+\left (27 a^{2} b^{3} c +27 b^{5} a \right ) x^{3}+\frac {81 a^{2} b^{4} x^{2}}{2}+27 a^{3} b^{3} x\) | \(159\) |
gosper | \(\frac {1}{10} c^{6} x^{10}+b \,c^{5} x^{9}+\frac {9}{2} b^{2} c^{4} x^{8}+\frac {9}{7} x^{7} a b \,c^{4}+\frac {81}{7} x^{7} b^{3} c^{3}+9 a \,b^{2} c^{3} x^{6}+18 b^{4} c^{2} x^{6}+27 x^{5} a \,b^{3} c^{2}+\frac {81}{5} x^{5} b^{5} c +\frac {27}{4} x^{4} a^{2} b^{2} c^{2}+\frac {81}{2} x^{4} a \,b^{4} c +\frac {27}{4} x^{4} b^{6}+27 a^{2} b^{3} c \,x^{3}+27 a \,b^{5} x^{3}+\frac {81}{2} a^{2} b^{4} x^{2}+27 a^{3} b^{3} x\) | \(167\) |
risch | \(\frac {1}{10} c^{6} x^{10}+b \,c^{5} x^{9}+\frac {9}{2} b^{2} c^{4} x^{8}+\frac {9}{7} x^{7} a b \,c^{4}+\frac {81}{7} x^{7} b^{3} c^{3}+9 a \,b^{2} c^{3} x^{6}+18 b^{4} c^{2} x^{6}+27 x^{5} a \,b^{3} c^{2}+\frac {81}{5} x^{5} b^{5} c +\frac {27}{4} x^{4} a^{2} b^{2} c^{2}+\frac {81}{2} x^{4} a \,b^{4} c +\frac {27}{4} x^{4} b^{6}+27 a^{2} b^{3} c \,x^{3}+27 a \,b^{5} x^{3}+\frac {81}{2} a^{2} b^{4} x^{2}+27 a^{3} b^{3} x\) | \(167\) |
parallelrisch | \(\frac {1}{10} c^{6} x^{10}+b \,c^{5} x^{9}+\frac {9}{2} b^{2} c^{4} x^{8}+\frac {9}{7} x^{7} a b \,c^{4}+\frac {81}{7} x^{7} b^{3} c^{3}+9 a \,b^{2} c^{3} x^{6}+18 b^{4} c^{2} x^{6}+27 x^{5} a \,b^{3} c^{2}+\frac {81}{5} x^{5} b^{5} c +\frac {27}{4} x^{4} a^{2} b^{2} c^{2}+\frac {81}{2} x^{4} a \,b^{4} c +\frac {27}{4} x^{4} b^{6}+27 a^{2} b^{3} c \,x^{3}+27 a \,b^{5} x^{3}+\frac {81}{2} a^{2} b^{4} x^{2}+27 a^{3} b^{3} x\) | \(167\) |
default | \(\frac {c^{6} x^{10}}{10}+b \,c^{5} x^{9}+\frac {9 b^{2} c^{4} x^{8}}{2}+\frac {\left (3 a b \,c^{4}+63 b^{3} c^{3}+c^{2} \left (6 a b \,c^{2}+18 b^{3} c \right )\right ) x^{7}}{7}+\frac {\left (18 a \,b^{2} c^{3}+45 b^{4} c^{2}+3 b c \left (6 a b \,c^{2}+18 b^{3} c \right )+c^{2} \left (18 a \,b^{2} c +9 b^{4}\right )\right ) x^{6}}{6}+\frac {\left (63 a \,b^{3} c^{2}+3 b^{2} \left (6 a b \,c^{2}+18 b^{3} c \right )+3 b c \left (18 a \,b^{2} c +9 b^{4}\right )\right ) x^{5}}{5}+\frac {\left (3 a b \left (6 a b \,c^{2}+18 b^{3} c \right )+3 b^{2} \left (18 a \,b^{2} c +9 b^{4}\right )+54 a \,b^{4} c +9 a^{2} b^{2} c^{2}\right ) x^{4}}{4}+\frac {\left (3 a b \left (18 a \,b^{2} c +9 b^{4}\right )+54 b^{5} a +27 a^{2} b^{3} c \right ) x^{3}}{3}+\frac {81 a^{2} b^{4} x^{2}}{2}+27 a^{3} b^{3} x\) | \(295\) |
1/10*c^6*x^10+b*c^5*x^9+9/2*b^2*c^4*x^8+(9/7*a*b*c^4+81/7*b^3*c^3)*x^7+(9* a*b^2*c^3+18*b^4*c^2)*x^6+(27*a*b^3*c^2+81/5*b^5*c)*x^5+(27/4*a^2*b^2*c^2+ 81/2*a*b^4*c+27/4*b^6)*x^4+(27*a^2*b^3*c+27*a*b^5)*x^3+81/2*a^2*b^4*x^2+27 *a^3*b^3*x
Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.86 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=\frac {1}{10} \, c^{6} x^{10} + b c^{5} x^{9} + \frac {9}{2} \, b^{2} c^{4} x^{8} + \frac {81}{2} \, a^{2} b^{4} x^{2} + \frac {9}{7} \, {\left (9 \, b^{3} c^{3} + a b c^{4}\right )} x^{7} + 27 \, a^{3} b^{3} x + 9 \, {\left (2 \, b^{4} c^{2} + a b^{2} c^{3}\right )} x^{6} + \frac {27}{5} \, {\left (3 \, b^{5} c + 5 \, a b^{3} c^{2}\right )} x^{5} + \frac {27}{4} \, {\left (b^{6} + 6 \, a b^{4} c + a^{2} b^{2} c^{2}\right )} x^{4} + 27 \, {\left (a b^{5} + a^{2} b^{3} c\right )} x^{3} \]
1/10*c^6*x^10 + b*c^5*x^9 + 9/2*b^2*c^4*x^8 + 81/2*a^2*b^4*x^2 + 9/7*(9*b^ 3*c^3 + a*b*c^4)*x^7 + 27*a^3*b^3*x + 9*(2*b^4*c^2 + a*b^2*c^3)*x^6 + 27/5 *(3*b^5*c + 5*a*b^3*c^2)*x^5 + 27/4*(b^6 + 6*a*b^4*c + a^2*b^2*c^2)*x^4 + 27*(a*b^5 + a^2*b^3*c)*x^3
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (80) = 160\).
Time = 0.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.08 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=27 a^{3} b^{3} x + \frac {81 a^{2} b^{4} x^{2}}{2} + \frac {9 b^{2} c^{4} x^{8}}{2} + b c^{5} x^{9} + \frac {c^{6} x^{10}}{10} + x^{7} \cdot \left (\frac {9 a b c^{4}}{7} + \frac {81 b^{3} c^{3}}{7}\right ) + x^{6} \cdot \left (9 a b^{2} c^{3} + 18 b^{4} c^{2}\right ) + x^{5} \cdot \left (27 a b^{3} c^{2} + \frac {81 b^{5} c}{5}\right ) + x^{4} \cdot \left (\frac {27 a^{2} b^{2} c^{2}}{4} + \frac {81 a b^{4} c}{2} + \frac {27 b^{6}}{4}\right ) + x^{3} \cdot \left (27 a^{2} b^{3} c + 27 a b^{5}\right ) \]
27*a**3*b**3*x + 81*a**2*b**4*x**2/2 + 9*b**2*c**4*x**8/2 + b*c**5*x**9 + c**6*x**10/10 + x**7*(9*a*b*c**4/7 + 81*b**3*c**3/7) + x**6*(9*a*b**2*c**3 + 18*b**4*c**2) + x**5*(27*a*b**3*c**2 + 81*b**5*c/5) + x**4*(27*a**2*b** 2*c**2/4 + 81*a*b**4*c/2 + 27*b**6/4) + x**3*(27*a**2*b**3*c + 27*a*b**5)
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (78) = 156\).
Time = 0.21 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.43 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=\frac {1}{10} \, c^{6} x^{10} + b c^{5} x^{9} + \frac {27}{8} \, b^{2} c^{4} x^{8} + \frac {27}{7} \, b^{3} c^{3} x^{7} + \frac {27}{4} \, b^{6} x^{4} + 27 \, a^{3} b^{3} x + \frac {27}{4} \, {\left (c^{2} x^{4} + 4 \, b c x^{3} + 6 \, b^{2} x^{2}\right )} a^{2} b^{2} + \frac {9}{10} \, {\left (5 \, c^{2} x^{6} + 18 \, b c x^{5}\right )} b^{4} + \frac {9}{70} \, {\left (10 \, c^{4} x^{7} + 70 \, b c^{3} x^{6} + 126 \, b^{2} c^{2} x^{5} + 210 \, b^{4} x^{3} + 21 \, {\left (4 \, c^{2} x^{5} + 15 \, b c x^{4}\right )} b^{2}\right )} a b + \frac {9}{56} \, {\left (7 \, c^{4} x^{8} + 48 \, b c^{3} x^{7} + 84 \, b^{2} c^{2} x^{6}\right )} b^{2} \]
1/10*c^6*x^10 + b*c^5*x^9 + 27/8*b^2*c^4*x^8 + 27/7*b^3*c^3*x^7 + 27/4*b^6 *x^4 + 27*a^3*b^3*x + 27/4*(c^2*x^4 + 4*b*c*x^3 + 6*b^2*x^2)*a^2*b^2 + 9/1 0*(5*c^2*x^6 + 18*b*c*x^5)*b^4 + 9/70*(10*c^4*x^7 + 70*b*c^3*x^6 + 126*b^2 *c^2*x^5 + 210*b^4*x^3 + 21*(4*c^2*x^5 + 15*b*c*x^4)*b^2)*a*b + 9/56*(7*c^ 4*x^8 + 48*b*c^3*x^7 + 84*b^2*c^2*x^6)*b^2
Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.98 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=\frac {1}{10} \, c^{6} x^{10} + b c^{5} x^{9} + \frac {9}{2} \, b^{2} c^{4} x^{8} + \frac {81}{7} \, b^{3} c^{3} x^{7} + \frac {9}{7} \, a b c^{4} x^{7} + 18 \, b^{4} c^{2} x^{6} + 9 \, a b^{2} c^{3} x^{6} + \frac {81}{5} \, b^{5} c x^{5} + 27 \, a b^{3} c^{2} x^{5} + \frac {27}{4} \, b^{6} x^{4} + \frac {81}{2} \, a b^{4} c x^{4} + \frac {27}{4} \, a^{2} b^{2} c^{2} x^{4} + 27 \, a b^{5} x^{3} + 27 \, a^{2} b^{3} c x^{3} + \frac {81}{2} \, a^{2} b^{4} x^{2} + 27 \, a^{3} b^{3} x \]
1/10*c^6*x^10 + b*c^5*x^9 + 9/2*b^2*c^4*x^8 + 81/7*b^3*c^3*x^7 + 9/7*a*b*c ^4*x^7 + 18*b^4*c^2*x^6 + 9*a*b^2*c^3*x^6 + 81/5*b^5*c*x^5 + 27*a*b^3*c^2* x^5 + 27/4*b^6*x^4 + 81/2*a*b^4*c*x^4 + 27/4*a^2*b^2*c^2*x^4 + 27*a*b^5*x^ 3 + 27*a^2*b^3*c*x^3 + 81/2*a^2*b^4*x^2 + 27*a^3*b^3*x
Time = 9.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.77 \[ \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx=x^4\,\left (\frac {27\,a^2\,b^2\,c^2}{4}+\frac {81\,a\,b^4\,c}{2}+\frac {27\,b^6}{4}\right )+\frac {c^6\,x^{10}}{10}+27\,a^3\,b^3\,x+b\,c^5\,x^9+\frac {81\,a^2\,b^4\,x^2}{2}+\frac {9\,b^2\,c^4\,x^8}{2}+9\,b^2\,c^2\,x^6\,\left (2\,b^2+a\,c\right )+27\,a\,b^3\,x^3\,\left (b^2+a\,c\right )+\frac {27\,b^3\,c\,x^5\,\left (3\,b^2+5\,a\,c\right )}{5}+\frac {9\,b\,c^3\,x^7\,\left (9\,b^2+a\,c\right )}{7} \]