Integrand size = 17, antiderivative size = 39 \[ \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx=-\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} (d-2 e) (1+x)^{12}+\frac {1}{13} e (1+x)^{13} \] Output:
-1/11*(d-e)*(1+x)^11+1/12*(d-2*e)*(1+x)^12+1/13*e*(1+x)^13
Leaf count is larger than twice the leaf count of optimal. \(147\) vs. \(2(39)=78\).
Time = 0.02 (sec) , antiderivative size = 147, normalized size of antiderivative = 3.77 \[ \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^2}{2}+\frac {1}{3} (10 d+e) x^3+\frac {5}{4} (9 d+2 e) x^4+3 (8 d+3 e) x^5+5 (7 d+4 e) x^6+6 (6 d+5 e) x^7+\frac {21}{4} (5 d+6 e) x^8+\frac {10}{3} (4 d+7 e) x^9+\frac {3}{2} (3 d+8 e) x^{10}+\frac {5}{11} (2 d+9 e) x^{11}+\frac {1}{12} (d+10 e) x^{12}+\frac {e x^{13}}{13} \] Input:
Integrate[x*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
(d*x^2)/2 + ((10*d + e)*x^3)/3 + (5*(9*d + 2*e)*x^4)/4 + 3*(8*d + 3*e)*x^5 + 5*(7*d + 4*e)*x^6 + 6*(6*d + 5*e)*x^7 + (21*(5*d + 6*e)*x^8)/4 + (10*(4 *d + 7*e)*x^9)/3 + (3*(3*d + 8*e)*x^10)/2 + (5*(2*d + 9*e)*x^11)/11 + ((d + 10*e)*x^12)/12 + (e*x^13)/13
Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1184, 85, 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.
\(\displaystyle \int x \left (x^2+2 x+1\right )^5 (d+e x) \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x (x+1)^{10} (d+e x)dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left ((x+1)^{11} (d-2 e)+(x+1)^{10} (e-d)+e (x+1)^{12}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{12} (x+1)^{12} (d-2 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{13} e (x+1)^{13}\) |
Input:
Int[x*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
-1/11*((d - e)*(1 + x)^11) + ((d - 2*e)*(1 + x)^12)/12 + (e*(1 + x)^13)/13
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(33)=66\).
Time = 0.88 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.18
method | result | size |
norman | \(\frac {x^{13} e}{13}+\left (\frac {d}{12}+\frac {5 e}{6}\right ) x^{12}+\left (\frac {10 d}{11}+\frac {45 e}{11}\right ) x^{11}+\left (\frac {9 d}{2}+12 e \right ) x^{10}+\left (\frac {40 d}{3}+\frac {70 e}{3}\right ) x^{9}+\left (\frac {105 d}{4}+\frac {63 e}{2}\right ) x^{8}+\left (36 d +30 e \right ) x^{7}+\left (35 d +20 e \right ) x^{6}+\left (24 d +9 e \right ) x^{5}+\left (\frac {45 d}{4}+\frac {5 e}{2}\right ) x^{4}+\left (\frac {10 d}{3}+\frac {e}{3}\right ) x^{3}+\frac {d \,x^{2}}{2}\) | \(124\) |
default | \(\frac {x^{13} e}{13}+\frac {\left (d +10 e \right ) x^{12}}{12}+\frac {\left (10 d +45 e \right ) x^{11}}{11}+\frac {\left (45 d +120 e \right ) x^{10}}{10}+\frac {\left (120 d +210 e \right ) x^{9}}{9}+\frac {\left (210 d +252 e \right ) x^{8}}{8}+\frac {\left (252 d +210 e \right ) x^{7}}{7}+\frac {\left (210 d +120 e \right ) x^{6}}{6}+\frac {\left (120 d +45 e \right ) x^{5}}{5}+\frac {\left (45 d +10 e \right ) x^{4}}{4}+\frac {\left (10 d +e \right ) x^{3}}{3}+\frac {d \,x^{2}}{2}\) | \(130\) |
gosper | \(\frac {x^{2} \left (132 e \,x^{11}+143 d \,x^{10}+1430 e \,x^{10}+1560 d \,x^{9}+7020 e \,x^{9}+7722 d \,x^{8}+20592 e \,x^{8}+22880 d \,x^{7}+40040 e \,x^{7}+45045 d \,x^{6}+54054 e \,x^{6}+61776 d \,x^{5}+51480 x^{5} e +60060 d \,x^{4}+34320 x^{4} e +41184 d \,x^{3}+15444 x^{3} e +19305 d \,x^{2}+4290 e \,x^{2}+5720 d x +572 e x +858 d \right )}{1716}\) | \(132\) |
risch | \(\frac {1}{13} x^{13} e +\frac {1}{12} d \,x^{12}+\frac {5}{6} x^{12} e +\frac {10}{11} x^{11} d +\frac {45}{11} e \,x^{11}+\frac {9}{2} d \,x^{10}+12 e \,x^{10}+\frac {40}{3} d \,x^{9}+\frac {70}{3} e \,x^{9}+\frac {105}{4} d \,x^{8}+\frac {63}{2} e \,x^{8}+36 d \,x^{7}+30 e \,x^{7}+35 d \,x^{6}+20 e \,x^{6}+24 d \,x^{5}+9 x^{5} e +\frac {45}{4} d \,x^{4}+\frac {5}{2} x^{4} e +\frac {10}{3} d \,x^{3}+\frac {1}{3} x^{3} e +\frac {1}{2} d \,x^{2}\) | \(134\) |
parallelrisch | \(\frac {1}{13} x^{13} e +\frac {1}{12} d \,x^{12}+\frac {5}{6} x^{12} e +\frac {10}{11} x^{11} d +\frac {45}{11} e \,x^{11}+\frac {9}{2} d \,x^{10}+12 e \,x^{10}+\frac {40}{3} d \,x^{9}+\frac {70}{3} e \,x^{9}+\frac {105}{4} d \,x^{8}+\frac {63}{2} e \,x^{8}+36 d \,x^{7}+30 e \,x^{7}+35 d \,x^{6}+20 e \,x^{6}+24 d \,x^{5}+9 x^{5} e +\frac {45}{4} d \,x^{4}+\frac {5}{2} x^{4} e +\frac {10}{3} d \,x^{3}+\frac {1}{3} x^{3} e +\frac {1}{2} d \,x^{2}\) | \(134\) |
orering | \(\frac {x^{2} \left (132 e \,x^{11}+143 d \,x^{10}+1430 e \,x^{10}+1560 d \,x^{9}+7020 e \,x^{9}+7722 d \,x^{8}+20592 e \,x^{8}+22880 d \,x^{7}+40040 e \,x^{7}+45045 d \,x^{6}+54054 e \,x^{6}+61776 d \,x^{5}+51480 x^{5} e +60060 d \,x^{4}+34320 x^{4} e +41184 d \,x^{3}+15444 x^{3} e +19305 d \,x^{2}+4290 e \,x^{2}+5720 d x +572 e x +858 d \right ) \left (x^{2}+2 x +1\right )^{5}}{1716 \left (x +1\right )^{10}}\) | \(147\) |
Input:
int(x*(e*x+d)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/13*x^13*e+(1/12*d+5/6*e)*x^12+(10/11*d+45/11*e)*x^11+(9/2*d+12*e)*x^10+( 40/3*d+70/3*e)*x^9+(105/4*d+63/2*e)*x^8+(36*d+30*e)*x^7+(35*d+20*e)*x^6+(2 4*d+9*e)*x^5+(45/4*d+5/2*e)*x^4+(10/3*d+1/3*e)*x^3+1/2*d*x^2
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (33) = 66\).
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.31 \[ \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{13} \, e x^{13} + \frac {1}{12} \, {\left (d + 10 \, e\right )} x^{12} + \frac {5}{11} \, {\left (2 \, d + 9 \, e\right )} x^{11} + \frac {3}{2} \, {\left (3 \, d + 8 \, e\right )} x^{10} + \frac {10}{3} \, {\left (4 \, d + 7 \, e\right )} x^{9} + \frac {21}{4} \, {\left (5 \, d + 6 \, e\right )} x^{8} + 6 \, {\left (6 \, d + 5 \, e\right )} x^{7} + 5 \, {\left (7 \, d + 4 \, e\right )} x^{6} + 3 \, {\left (8 \, d + 3 \, e\right )} x^{5} + \frac {5}{4} \, {\left (9 \, d + 2 \, e\right )} x^{4} + \frac {1}{3} \, {\left (10 \, d + e\right )} x^{3} + \frac {1}{2} \, d x^{2} \] Input:
integrate(x*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/13*e*x^13 + 1/12*(d + 10*e)*x^12 + 5/11*(2*d + 9*e)*x^11 + 3/2*(3*d + 8* e)*x^10 + 10/3*(4*d + 7*e)*x^9 + 21/4*(5*d + 6*e)*x^8 + 6*(6*d + 5*e)*x^7 + 5*(7*d + 4*e)*x^6 + 3*(8*d + 3*e)*x^5 + 5/4*(9*d + 2*e)*x^4 + 1/3*(10*d + e)*x^3 + 1/2*d*x^2
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (31) = 62\).
Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.41 \[ \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^{2}}{2} + \frac {e x^{13}}{13} + x^{12} \left (\frac {d}{12} + \frac {5 e}{6}\right ) + x^{11} \cdot \left (\frac {10 d}{11} + \frac {45 e}{11}\right ) + x^{10} \cdot \left (\frac {9 d}{2} + 12 e\right ) + x^{9} \cdot \left (\frac {40 d}{3} + \frac {70 e}{3}\right ) + x^{8} \cdot \left (\frac {105 d}{4} + \frac {63 e}{2}\right ) + x^{7} \cdot \left (36 d + 30 e\right ) + x^{6} \cdot \left (35 d + 20 e\right ) + x^{5} \cdot \left (24 d + 9 e\right ) + x^{4} \cdot \left (\frac {45 d}{4} + \frac {5 e}{2}\right ) + x^{3} \cdot \left (\frac {10 d}{3} + \frac {e}{3}\right ) \] Input:
integrate(x*(e*x+d)*(x**2+2*x+1)**5,x)
Output:
d*x**2/2 + e*x**13/13 + x**12*(d/12 + 5*e/6) + x**11*(10*d/11 + 45*e/11) + x**10*(9*d/2 + 12*e) + x**9*(40*d/3 + 70*e/3) + x**8*(105*d/4 + 63*e/2) + x**7*(36*d + 30*e) + x**6*(35*d + 20*e) + x**5*(24*d + 9*e) + x**4*(45*d/ 4 + 5*e/2) + x**3*(10*d/3 + e/3)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (33) = 66\).
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.31 \[ \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{13} \, e x^{13} + \frac {1}{12} \, {\left (d + 10 \, e\right )} x^{12} + \frac {5}{11} \, {\left (2 \, d + 9 \, e\right )} x^{11} + \frac {3}{2} \, {\left (3 \, d + 8 \, e\right )} x^{10} + \frac {10}{3} \, {\left (4 \, d + 7 \, e\right )} x^{9} + \frac {21}{4} \, {\left (5 \, d + 6 \, e\right )} x^{8} + 6 \, {\left (6 \, d + 5 \, e\right )} x^{7} + 5 \, {\left (7 \, d + 4 \, e\right )} x^{6} + 3 \, {\left (8 \, d + 3 \, e\right )} x^{5} + \frac {5}{4} \, {\left (9 \, d + 2 \, e\right )} x^{4} + \frac {1}{3} \, {\left (10 \, d + e\right )} x^{3} + \frac {1}{2} \, d x^{2} \] Input:
integrate(x*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/13*e*x^13 + 1/12*(d + 10*e)*x^12 + 5/11*(2*d + 9*e)*x^11 + 3/2*(3*d + 8* e)*x^10 + 10/3*(4*d + 7*e)*x^9 + 21/4*(5*d + 6*e)*x^8 + 6*(6*d + 5*e)*x^7 + 5*(7*d + 4*e)*x^6 + 3*(8*d + 3*e)*x^5 + 5/4*(9*d + 2*e)*x^4 + 1/3*(10*d + e)*x^3 + 1/2*d*x^2
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (33) = 66\).
Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.41 \[ \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{13} \, e x^{13} + \frac {1}{12} \, d x^{12} + \frac {5}{6} \, e x^{12} + \frac {10}{11} \, d x^{11} + \frac {45}{11} \, e x^{11} + \frac {9}{2} \, d x^{10} + 12 \, e x^{10} + \frac {40}{3} \, d x^{9} + \frac {70}{3} \, e x^{9} + \frac {105}{4} \, d x^{8} + \frac {63}{2} \, e x^{8} + 36 \, d x^{7} + 30 \, e x^{7} + 35 \, d x^{6} + 20 \, e x^{6} + 24 \, d x^{5} + 9 \, e x^{5} + \frac {45}{4} \, d x^{4} + \frac {5}{2} \, e x^{4} + \frac {10}{3} \, d x^{3} + \frac {1}{3} \, e x^{3} + \frac {1}{2} \, d x^{2} \] Input:
integrate(x*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/13*e*x^13 + 1/12*d*x^12 + 5/6*e*x^12 + 10/11*d*x^11 + 45/11*e*x^11 + 9/2 *d*x^10 + 12*e*x^10 + 40/3*d*x^9 + 70/3*e*x^9 + 105/4*d*x^8 + 63/2*e*x^8 + 36*d*x^7 + 30*e*x^7 + 35*d*x^6 + 20*e*x^6 + 24*d*x^5 + 9*e*x^5 + 45/4*d*x ^4 + 5/2*e*x^4 + 10/3*d*x^3 + 1/3*e*x^3 + 1/2*d*x^2
Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.15 \[ \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {e\,x^{13}}{13}+\left (\frac {d}{12}+\frac {5\,e}{6}\right )\,x^{12}+\left (\frac {10\,d}{11}+\frac {45\,e}{11}\right )\,x^{11}+\left (\frac {9\,d}{2}+12\,e\right )\,x^{10}+\left (\frac {40\,d}{3}+\frac {70\,e}{3}\right )\,x^9+\left (\frac {105\,d}{4}+\frac {63\,e}{2}\right )\,x^8+\left (36\,d+30\,e\right )\,x^7+\left (35\,d+20\,e\right )\,x^6+\left (24\,d+9\,e\right )\,x^5+\left (\frac {45\,d}{4}+\frac {5\,e}{2}\right )\,x^4+\left (\frac {10\,d}{3}+\frac {e}{3}\right )\,x^3+\frac {d\,x^2}{2} \] Input:
int(x*(d + e*x)*(2*x + x^2 + 1)^5,x)
Output:
x^3*((10*d)/3 + e/3) + x^10*((9*d)/2 + 12*e) + x^12*(d/12 + (5*e)/6) + x^5 *(24*d + 9*e) + x^4*((45*d)/4 + (5*e)/2) + x^6*(35*d + 20*e) + x^7*(36*d + 30*e) + x^11*((10*d)/11 + (45*e)/11) + x^9*((40*d)/3 + (70*e)/3) + x^8*(( 105*d)/4 + (63*e)/2) + (d*x^2)/2 + (e*x^13)/13
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.36 \[ \int x (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{2} \left (132 e \,x^{11}+143 d \,x^{10}+1430 e \,x^{10}+1560 d \,x^{9}+7020 e \,x^{9}+7722 d \,x^{8}+20592 e \,x^{8}+22880 d \,x^{7}+40040 e \,x^{7}+45045 d \,x^{6}+54054 e \,x^{6}+61776 d \,x^{5}+51480 e \,x^{5}+60060 d \,x^{4}+34320 e \,x^{4}+41184 d \,x^{3}+15444 e \,x^{3}+19305 d \,x^{2}+4290 e \,x^{2}+5720 d x +572 e x +858 d \right )}{1716} \] Input:
int(x*(e*x+d)*(x^2+2*x+1)^5,x)
Output:
(x**2*(143*d*x**10 + 1560*d*x**9 + 7722*d*x**8 + 22880*d*x**7 + 45045*d*x* *6 + 61776*d*x**5 + 60060*d*x**4 + 41184*d*x**3 + 19305*d*x**2 + 5720*d*x + 858*d + 132*e*x**11 + 1430*e*x**10 + 7020*e*x**9 + 20592*e*x**8 + 40040* e*x**7 + 54054*e*x**6 + 51480*e*x**5 + 34320*e*x**4 + 15444*e*x**3 + 4290* e*x**2 + 572*e*x))/1716