Integrand size = 19, antiderivative size = 55 \[ \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{11} (d-e) (1+x)^{11}-\frac {1}{12} (2 d-3 e) (1+x)^{12}+\frac {1}{13} (d-3 e) (1+x)^{13}+\frac {1}{14} e (1+x)^{14} \] Output:
1/11*(d-e)*(1+x)^11-1/12*(2*d-3*e)*(1+x)^12+1/13*(d-3*e)*(1+x)^13+1/14*e*( 1+x)^14
Leaf count is larger than twice the leaf count of optimal. \(148\) vs. \(2(55)=110\).
Time = 0.02 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.69 \[ \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^3}{3}+\frac {1}{4} (10 d+e) x^4+(9 d+2 e) x^5+\frac {5}{2} (8 d+3 e) x^6+\frac {30}{7} (7 d+4 e) x^7+\frac {21}{4} (6 d+5 e) x^8+\frac {14}{3} (5 d+6 e) x^9+3 (4 d+7 e) x^{10}+\frac {15}{11} (3 d+8 e) x^{11}+\frac {5}{12} (2 d+9 e) x^{12}+\frac {1}{13} (d+10 e) x^{13}+\frac {e x^{14}}{14} \] Input:
Integrate[x^2*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
(d*x^3)/3 + ((10*d + e)*x^4)/4 + (9*d + 2*e)*x^5 + (5*(8*d + 3*e)*x^6)/2 + (30*(7*d + 4*e)*x^7)/7 + (21*(6*d + 5*e)*x^8)/4 + (14*(5*d + 6*e)*x^9)/3 + 3*(4*d + 7*e)*x^10 + (15*(3*d + 8*e)*x^11)/11 + (5*(2*d + 9*e)*x^12)/12 + ((d + 10*e)*x^13)/13 + (e*x^14)/14
Time = 0.37 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, 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.
\(\displaystyle \int x^2 \left (x^2+2 x+1\right )^5 (d+e x) \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x^2 (x+1)^{10} (d+e x)dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left ((x+1)^{12} (d-3 e)+(x+1)^{11} (3 e-2 d)+(x+1)^{10} (d-e)+e (x+1)^{13}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{13} (x+1)^{13} (d-3 e)-\frac {1}{12} (x+1)^{12} (2 d-3 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{14} e (x+1)^{14}\) |
Input:
Int[x^2*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
((d - e)*(1 + x)^11)/11 - ((2*d - 3*e)*(1 + x)^12)/12 + ((d - 3*e)*(1 + x) ^13)/13 + (e*(1 + x)^14)/14
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. \(2(47)=94\).
Time = 0.80 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.25
method | result | size |
norman | \(\frac {d \,x^{3}}{3}+\left (\frac {5 d}{2}+\frac {e}{4}\right ) x^{4}+\left (9 d +2 e \right ) x^{5}+\left (20 d +\frac {15 e}{2}\right ) x^{6}+\left (30 d +\frac {120 e}{7}\right ) x^{7}+\left (\frac {63 d}{2}+\frac {105 e}{4}\right ) x^{8}+\left (\frac {70 d}{3}+28 e \right ) x^{9}+\left (12 d +21 e \right ) x^{10}+\left (\frac {45 d}{11}+\frac {120 e}{11}\right ) x^{11}+\left (\frac {5 d}{6}+\frac {15 e}{4}\right ) x^{12}+\left (\frac {d}{13}+\frac {10 e}{13}\right ) x^{13}+\frac {x^{14} e}{14}\) | \(124\) |
default | \(\frac {x^{14} e}{14}+\frac {\left (d +10 e \right ) x^{13}}{13}+\frac {\left (10 d +45 e \right ) x^{12}}{12}+\frac {\left (45 d +120 e \right ) x^{11}}{11}+\frac {\left (120 d +210 e \right ) x^{10}}{10}+\frac {\left (210 d +252 e \right ) x^{9}}{9}+\frac {\left (252 d +210 e \right ) x^{8}}{8}+\frac {\left (210 d +120 e \right ) x^{7}}{7}+\frac {\left (120 d +45 e \right ) x^{6}}{6}+\frac {\left (45 d +10 e \right ) x^{5}}{5}+\frac {\left (10 d +e \right ) x^{4}}{4}+\frac {d \,x^{3}}{3}\) | \(130\) |
gosper | \(\frac {x^{3} \left (858 e \,x^{11}+924 d \,x^{10}+9240 e \,x^{10}+10010 d \,x^{9}+45045 e \,x^{9}+49140 d \,x^{8}+131040 e \,x^{8}+144144 d \,x^{7}+252252 e \,x^{7}+280280 d \,x^{6}+336336 e \,x^{6}+378378 d \,x^{5}+315315 x^{5} e +360360 d \,x^{4}+205920 x^{4} e +240240 d \,x^{3}+90090 x^{3} e +108108 d \,x^{2}+24024 e \,x^{2}+30030 d x +3003 e x +4004 d \right )}{12012}\) | \(132\) |
risch | \(\frac {1}{14} x^{14} e +\frac {1}{13} d \,x^{13}+\frac {10}{13} x^{13} e +\frac {5}{6} d \,x^{12}+\frac {15}{4} x^{12} e +\frac {45}{11} x^{11} d +\frac {120}{11} e \,x^{11}+12 d \,x^{10}+21 e \,x^{10}+\frac {70}{3} d \,x^{9}+28 e \,x^{9}+\frac {63}{2} d \,x^{8}+\frac {105}{4} e \,x^{8}+30 d \,x^{7}+\frac {120}{7} e \,x^{7}+20 d \,x^{6}+\frac {15}{2} e \,x^{6}+9 d \,x^{5}+2 x^{5} e +\frac {5}{2} d \,x^{4}+\frac {1}{4} x^{4} e +\frac {1}{3} d \,x^{3}\) | \(134\) |
parallelrisch | \(\frac {1}{14} x^{14} e +\frac {1}{13} d \,x^{13}+\frac {10}{13} x^{13} e +\frac {5}{6} d \,x^{12}+\frac {15}{4} x^{12} e +\frac {45}{11} x^{11} d +\frac {120}{11} e \,x^{11}+12 d \,x^{10}+21 e \,x^{10}+\frac {70}{3} d \,x^{9}+28 e \,x^{9}+\frac {63}{2} d \,x^{8}+\frac {105}{4} e \,x^{8}+30 d \,x^{7}+\frac {120}{7} e \,x^{7}+20 d \,x^{6}+\frac {15}{2} e \,x^{6}+9 d \,x^{5}+2 x^{5} e +\frac {5}{2} d \,x^{4}+\frac {1}{4} x^{4} e +\frac {1}{3} d \,x^{3}\) | \(134\) |
orering | \(\frac {x^{3} \left (858 e \,x^{11}+924 d \,x^{10}+9240 e \,x^{10}+10010 d \,x^{9}+45045 e \,x^{9}+49140 d \,x^{8}+131040 e \,x^{8}+144144 d \,x^{7}+252252 e \,x^{7}+280280 d \,x^{6}+336336 e \,x^{6}+378378 d \,x^{5}+315315 x^{5} e +360360 d \,x^{4}+205920 x^{4} e +240240 d \,x^{3}+90090 x^{3} e +108108 d \,x^{2}+24024 e \,x^{2}+30030 d x +3003 e x +4004 d \right ) \left (x^{2}+2 x +1\right )^{5}}{12012 \left (x +1\right )^{10}}\) | \(147\) |
Input:
int(x^2*(e*x+d)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/3*d*x^3+(5/2*d+1/4*e)*x^4+(9*d+2*e)*x^5+(20*d+15/2*e)*x^6+(30*d+120/7*e) *x^7+(63/2*d+105/4*e)*x^8+(70/3*d+28*e)*x^9+(12*d+21*e)*x^10+(45/11*d+120/ 11*e)*x^11+(5/6*d+15/4*e)*x^12+(1/13*d+10/13*e)*x^13+1/14*x^14*e
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (47) = 94\).
Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.33 \[ \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{14} \, e x^{14} + \frac {1}{13} \, {\left (d + 10 \, e\right )} x^{13} + \frac {5}{12} \, {\left (2 \, d + 9 \, e\right )} x^{12} + \frac {15}{11} \, {\left (3 \, d + 8 \, e\right )} x^{11} + 3 \, {\left (4 \, d + 7 \, e\right )} x^{10} + \frac {14}{3} \, {\left (5 \, d + 6 \, e\right )} x^{9} + \frac {21}{4} \, {\left (6 \, d + 5 \, e\right )} x^{8} + \frac {30}{7} \, {\left (7 \, d + 4 \, e\right )} x^{7} + \frac {5}{2} \, {\left (8 \, d + 3 \, e\right )} x^{6} + {\left (9 \, d + 2 \, e\right )} x^{5} + \frac {1}{4} \, {\left (10 \, d + e\right )} x^{4} + \frac {1}{3} \, d x^{3} \] Input:
integrate(x^2*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/14*e*x^14 + 1/13*(d + 10*e)*x^13 + 5/12*(2*d + 9*e)*x^12 + 15/11*(3*d + 8*e)*x^11 + 3*(4*d + 7*e)*x^10 + 14/3*(5*d + 6*e)*x^9 + 21/4*(6*d + 5*e)*x ^8 + 30/7*(7*d + 4*e)*x^7 + 5/2*(8*d + 3*e)*x^6 + (9*d + 2*e)*x^5 + 1/4*(1 0*d + e)*x^4 + 1/3*d*x^3
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (44) = 88\).
Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.42 \[ \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^{3}}{3} + \frac {e x^{14}}{14} + x^{13} \left (\frac {d}{13} + \frac {10 e}{13}\right ) + x^{12} \cdot \left (\frac {5 d}{6} + \frac {15 e}{4}\right ) + x^{11} \cdot \left (\frac {45 d}{11} + \frac {120 e}{11}\right ) + x^{10} \cdot \left (12 d + 21 e\right ) + x^{9} \cdot \left (\frac {70 d}{3} + 28 e\right ) + x^{8} \cdot \left (\frac {63 d}{2} + \frac {105 e}{4}\right ) + x^{7} \cdot \left (30 d + \frac {120 e}{7}\right ) + x^{6} \cdot \left (20 d + \frac {15 e}{2}\right ) + x^{5} \cdot \left (9 d + 2 e\right ) + x^{4} \cdot \left (\frac {5 d}{2} + \frac {e}{4}\right ) \] Input:
integrate(x**2*(e*x+d)*(x**2+2*x+1)**5,x)
Output:
d*x**3/3 + e*x**14/14 + x**13*(d/13 + 10*e/13) + x**12*(5*d/6 + 15*e/4) + x**11*(45*d/11 + 120*e/11) + x**10*(12*d + 21*e) + x**9*(70*d/3 + 28*e) + x**8*(63*d/2 + 105*e/4) + x**7*(30*d + 120*e/7) + x**6*(20*d + 15*e/2) + x **5*(9*d + 2*e) + x**4*(5*d/2 + e/4)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (47) = 94\).
Time = 0.03 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.33 \[ \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{14} \, e x^{14} + \frac {1}{13} \, {\left (d + 10 \, e\right )} x^{13} + \frac {5}{12} \, {\left (2 \, d + 9 \, e\right )} x^{12} + \frac {15}{11} \, {\left (3 \, d + 8 \, e\right )} x^{11} + 3 \, {\left (4 \, d + 7 \, e\right )} x^{10} + \frac {14}{3} \, {\left (5 \, d + 6 \, e\right )} x^{9} + \frac {21}{4} \, {\left (6 \, d + 5 \, e\right )} x^{8} + \frac {30}{7} \, {\left (7 \, d + 4 \, e\right )} x^{7} + \frac {5}{2} \, {\left (8 \, d + 3 \, e\right )} x^{6} + {\left (9 \, d + 2 \, e\right )} x^{5} + \frac {1}{4} \, {\left (10 \, d + e\right )} x^{4} + \frac {1}{3} \, d x^{3} \] Input:
integrate(x^2*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/14*e*x^14 + 1/13*(d + 10*e)*x^13 + 5/12*(2*d + 9*e)*x^12 + 15/11*(3*d + 8*e)*x^11 + 3*(4*d + 7*e)*x^10 + 14/3*(5*d + 6*e)*x^9 + 21/4*(6*d + 5*e)*x ^8 + 30/7*(7*d + 4*e)*x^7 + 5/2*(8*d + 3*e)*x^6 + (9*d + 2*e)*x^5 + 1/4*(1 0*d + e)*x^4 + 1/3*d*x^3
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (47) = 94\).
Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.42 \[ \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{14} \, e x^{14} + \frac {1}{13} \, d x^{13} + \frac {10}{13} \, e x^{13} + \frac {5}{6} \, d x^{12} + \frac {15}{4} \, e x^{12} + \frac {45}{11} \, d x^{11} + \frac {120}{11} \, e x^{11} + 12 \, d x^{10} + 21 \, e x^{10} + \frac {70}{3} \, d x^{9} + 28 \, e x^{9} + \frac {63}{2} \, d x^{8} + \frac {105}{4} \, e x^{8} + 30 \, d x^{7} + \frac {120}{7} \, e x^{7} + 20 \, d x^{6} + \frac {15}{2} \, e x^{6} + 9 \, d x^{5} + 2 \, e x^{5} + \frac {5}{2} \, d x^{4} + \frac {1}{4} \, e x^{4} + \frac {1}{3} \, d x^{3} \] Input:
integrate(x^2*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/14*e*x^14 + 1/13*d*x^13 + 10/13*e*x^13 + 5/6*d*x^12 + 15/4*e*x^12 + 45/1 1*d*x^11 + 120/11*e*x^11 + 12*d*x^10 + 21*e*x^10 + 70/3*d*x^9 + 28*e*x^9 + 63/2*d*x^8 + 105/4*e*x^8 + 30*d*x^7 + 120/7*e*x^7 + 20*d*x^6 + 15/2*e*x^6 + 9*d*x^5 + 2*e*x^5 + 5/2*d*x^4 + 1/4*e*x^4 + 1/3*d*x^3
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.24 \[ \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {e\,x^{14}}{14}+\left (\frac {d}{13}+\frac {10\,e}{13}\right )\,x^{13}+\left (\frac {5\,d}{6}+\frac {15\,e}{4}\right )\,x^{12}+\left (\frac {45\,d}{11}+\frac {120\,e}{11}\right )\,x^{11}+\left (12\,d+21\,e\right )\,x^{10}+\left (\frac {70\,d}{3}+28\,e\right )\,x^9+\left (\frac {63\,d}{2}+\frac {105\,e}{4}\right )\,x^8+\left (30\,d+\frac {120\,e}{7}\right )\,x^7+\left (20\,d+\frac {15\,e}{2}\right )\,x^6+\left (9\,d+2\,e\right )\,x^5+\left (\frac {5\,d}{2}+\frac {e}{4}\right )\,x^4+\frac {d\,x^3}{3} \] Input:
int(x^2*(d + e*x)*(2*x + x^2 + 1)^5,x)
Output:
x^4*((5*d)/2 + e/4) + x^5*(9*d + 2*e) + x^12*((5*d)/6 + (15*e)/4) + x^6*(2 0*d + (15*e)/2) + x^10*(12*d + 21*e) + x^13*(d/13 + (10*e)/13) + x^9*((70* d)/3 + 28*e) + x^7*(30*d + (120*e)/7) + x^8*((63*d)/2 + (105*e)/4) + x^11* ((45*d)/11 + (120*e)/11) + (d*x^3)/3 + (e*x^14)/14
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.38 \[ \int x^2 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{3} \left (858 e \,x^{11}+924 d \,x^{10}+9240 e \,x^{10}+10010 d \,x^{9}+45045 e \,x^{9}+49140 d \,x^{8}+131040 e \,x^{8}+144144 d \,x^{7}+252252 e \,x^{7}+280280 d \,x^{6}+336336 e \,x^{6}+378378 d \,x^{5}+315315 e \,x^{5}+360360 d \,x^{4}+205920 e \,x^{4}+240240 d \,x^{3}+90090 e \,x^{3}+108108 d \,x^{2}+24024 e \,x^{2}+30030 d x +3003 e x +4004 d \right )}{12012} \] Input:
int(x^2*(e*x+d)*(x^2+2*x+1)^5,x)
Output:
(x**3*(924*d*x**10 + 10010*d*x**9 + 49140*d*x**8 + 144144*d*x**7 + 280280* d*x**6 + 378378*d*x**5 + 360360*d*x**4 + 240240*d*x**3 + 108108*d*x**2 + 3 0030*d*x + 4004*d + 858*e*x**11 + 9240*e*x**10 + 45045*e*x**9 + 131040*e*x **8 + 252252*e*x**7 + 336336*e*x**6 + 315315*e*x**5 + 205920*e*x**4 + 9009 0*e*x**3 + 24024*e*x**2 + 3003*e*x))/12012