Integrand size = 19, antiderivative size = 69 \[ \int x^3 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=-\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} (3 d-4 e) (1+x)^{12}-\frac {3}{13} (d-2 e) (1+x)^{13}+\frac {1}{14} (d-4 e) (1+x)^{14}+\frac {1}{15} e (1+x)^{15} \] Output:
-1/11*(d-e)*(1+x)^11+1/12*(3*d-4*e)*(1+x)^12-3/13*(d-2*e)*(1+x)^13+1/14*(d -4*e)*(1+x)^14+1/15*e*(1+x)^15
Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(69)=138\).
Time = 0.02 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.22 \[ \int x^3 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^4}{4}+\frac {1}{5} (10 d+e) x^5+\frac {5}{6} (9 d+2 e) x^6+\frac {15}{7} (8 d+3 e) x^7+\frac {15}{4} (7 d+4 e) x^8+\frac {14}{3} (6 d+5 e) x^9+\frac {21}{5} (5 d+6 e) x^{10}+\frac {30}{11} (4 d+7 e) x^{11}+\frac {5}{4} (3 d+8 e) x^{12}+\frac {5}{13} (2 d+9 e) x^{13}+\frac {1}{14} (d+10 e) x^{14}+\frac {e x^{15}}{15} \] Input:
Integrate[x^3*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
(d*x^4)/4 + ((10*d + e)*x^5)/5 + (5*(9*d + 2*e)*x^6)/6 + (15*(8*d + 3*e)*x ^7)/7 + (15*(7*d + 4*e)*x^8)/4 + (14*(6*d + 5*e)*x^9)/3 + (21*(5*d + 6*e)* x^10)/5 + (30*(4*d + 7*e)*x^11)/11 + (5*(3*d + 8*e)*x^12)/4 + (5*(2*d + 9* e)*x^13)/13 + ((d + 10*e)*x^14)/14 + (e*x^15)/15
Time = 0.40 (sec) , antiderivative size = 69, 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.
\(\displaystyle \int x^3 \left (x^2+2 x+1\right )^5 (d+e x) \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int x^3 (x+1)^{10} (d+e x)dx\) |
\(\Big \downarrow \) 85 |
\(\displaystyle \int \left ((x+1)^{13} (d-4 e)-3 (x+1)^{12} (d-2 e)+(x+1)^{11} (3 d-4 e)+(x+1)^{10} (e-d)+e (x+1)^{14}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{14} (x+1)^{14} (d-4 e)-\frac {3}{13} (x+1)^{13} (d-2 e)+\frac {1}{12} (x+1)^{12} (3 d-4 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{15} e (x+1)^{15}\) |
Input:
Int[x^3*(d + e*x)*(1 + 2*x + x^2)^5,x]
Output:
-1/11*((d - e)*(1 + x)^11) + ((3*d - 4*e)*(1 + x)^12)/12 - (3*(d - 2*e)*(1 + x)^13)/13 + ((d - 4*e)*(1 + x)^14)/14 + (e*(1 + x)^15)/15
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. \(2(59)=118\).
Time = 0.87 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.80
method | result | size |
norman | \(\frac {d \,x^{4}}{4}+\left (2 d +\frac {e}{5}\right ) x^{5}+\left (\frac {15 d}{2}+\frac {5 e}{3}\right ) x^{6}+\left (\frac {120 d}{7}+\frac {45 e}{7}\right ) x^{7}+\left (\frac {105 d}{4}+15 e \right ) x^{8}+\left (28 d +\frac {70 e}{3}\right ) x^{9}+\left (21 d +\frac {126 e}{5}\right ) x^{10}+\left (\frac {120 d}{11}+\frac {210 e}{11}\right ) x^{11}+\left (\frac {15 d}{4}+10 e \right ) x^{12}+\left (\frac {10 d}{13}+\frac {45 e}{13}\right ) x^{13}+\left (\frac {d}{14}+\frac {5 e}{7}\right ) x^{14}+\frac {x^{15} e}{15}\) | \(124\) |
default | \(\frac {x^{15} e}{15}+\frac {\left (d +10 e \right ) x^{14}}{14}+\frac {\left (10 d +45 e \right ) x^{13}}{13}+\frac {\left (45 d +120 e \right ) x^{12}}{12}+\frac {\left (120 d +210 e \right ) x^{11}}{11}+\frac {\left (210 d +252 e \right ) x^{10}}{10}+\frac {\left (252 d +210 e \right ) x^{9}}{9}+\frac {\left (210 d +120 e \right ) x^{8}}{8}+\frac {\left (120 d +45 e \right ) x^{7}}{7}+\frac {\left (45 d +10 e \right ) x^{6}}{6}+\frac {\left (10 d +e \right ) x^{5}}{5}+\frac {d \,x^{4}}{4}\) | \(130\) |
gosper | \(\frac {x^{4} \left (4004 e \,x^{11}+4290 d \,x^{10}+42900 e \,x^{10}+46200 d \,x^{9}+207900 e \,x^{9}+225225 d \,x^{8}+600600 e \,x^{8}+655200 d \,x^{7}+1146600 e \,x^{7}+1261260 d \,x^{6}+1513512 e \,x^{6}+1681680 d \,x^{5}+1401400 x^{5} e +1576575 d \,x^{4}+900900 x^{4} e +1029600 d \,x^{3}+386100 x^{3} e +450450 d \,x^{2}+100100 e \,x^{2}+120120 d x +12012 e x +15015 d \right )}{60060}\) | \(132\) |
risch | \(\frac {1}{15} x^{15} e +\frac {1}{14} x^{14} d +\frac {5}{7} x^{14} e +\frac {10}{13} d \,x^{13}+\frac {45}{13} x^{13} e +\frac {15}{4} d \,x^{12}+10 x^{12} e +\frac {120}{11} x^{11} d +\frac {210}{11} e \,x^{11}+21 d \,x^{10}+\frac {126}{5} e \,x^{10}+28 d \,x^{9}+\frac {70}{3} e \,x^{9}+\frac {105}{4} d \,x^{8}+15 e \,x^{8}+\frac {120}{7} d \,x^{7}+\frac {45}{7} e \,x^{7}+\frac {15}{2} d \,x^{6}+\frac {5}{3} e \,x^{6}+2 d \,x^{5}+\frac {1}{5} x^{5} e +\frac {1}{4} d \,x^{4}\) | \(134\) |
parallelrisch | \(\frac {1}{15} x^{15} e +\frac {1}{14} x^{14} d +\frac {5}{7} x^{14} e +\frac {10}{13} d \,x^{13}+\frac {45}{13} x^{13} e +\frac {15}{4} d \,x^{12}+10 x^{12} e +\frac {120}{11} x^{11} d +\frac {210}{11} e \,x^{11}+21 d \,x^{10}+\frac {126}{5} e \,x^{10}+28 d \,x^{9}+\frac {70}{3} e \,x^{9}+\frac {105}{4} d \,x^{8}+15 e \,x^{8}+\frac {120}{7} d \,x^{7}+\frac {45}{7} e \,x^{7}+\frac {15}{2} d \,x^{6}+\frac {5}{3} e \,x^{6}+2 d \,x^{5}+\frac {1}{5} x^{5} e +\frac {1}{4} d \,x^{4}\) | \(134\) |
orering | \(\frac {x^{4} \left (4004 e \,x^{11}+4290 d \,x^{10}+42900 e \,x^{10}+46200 d \,x^{9}+207900 e \,x^{9}+225225 d \,x^{8}+600600 e \,x^{8}+655200 d \,x^{7}+1146600 e \,x^{7}+1261260 d \,x^{6}+1513512 e \,x^{6}+1681680 d \,x^{5}+1401400 x^{5} e +1576575 d \,x^{4}+900900 x^{4} e +1029600 d \,x^{3}+386100 x^{3} e +450450 d \,x^{2}+100100 e \,x^{2}+120120 d x +12012 e x +15015 d \right ) \left (x^{2}+2 x +1\right )^{5}}{60060 \left (x +1\right )^{10}}\) | \(147\) |
Input:
int(x^3*(e*x+d)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)
Output:
1/4*d*x^4+(2*d+1/5*e)*x^5+(15/2*d+5/3*e)*x^6+(120/7*d+45/7*e)*x^7+(105/4*d +15*e)*x^8+(28*d+70/3*e)*x^9+(21*d+126/5*e)*x^10+(120/11*d+210/11*e)*x^11+ (15/4*d+10*e)*x^12+(10/13*d+45/13*e)*x^13+(1/14*d+5/7*e)*x^14+1/15*x^15*e
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (59) = 118\).
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.87 \[ \int x^3 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{15} \, e x^{15} + \frac {1}{14} \, {\left (d + 10 \, e\right )} x^{14} + \frac {5}{13} \, {\left (2 \, d + 9 \, e\right )} x^{13} + \frac {5}{4} \, {\left (3 \, d + 8 \, e\right )} x^{12} + \frac {30}{11} \, {\left (4 \, d + 7 \, e\right )} x^{11} + \frac {21}{5} \, {\left (5 \, d + 6 \, e\right )} x^{10} + \frac {14}{3} \, {\left (6 \, d + 5 \, e\right )} x^{9} + \frac {15}{4} \, {\left (7 \, d + 4 \, e\right )} x^{8} + \frac {15}{7} \, {\left (8 \, d + 3 \, e\right )} x^{7} + \frac {5}{6} \, {\left (9 \, d + 2 \, e\right )} x^{6} + \frac {1}{5} \, {\left (10 \, d + e\right )} x^{5} + \frac {1}{4} \, d x^{4} \] Input:
integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")
Output:
1/15*e*x^15 + 1/14*(d + 10*e)*x^14 + 5/13*(2*d + 9*e)*x^13 + 5/4*(3*d + 8* e)*x^12 + 30/11*(4*d + 7*e)*x^11 + 21/5*(5*d + 6*e)*x^10 + 14/3*(6*d + 5*e )*x^9 + 15/4*(7*d + 4*e)*x^8 + 15/7*(8*d + 3*e)*x^7 + 5/6*(9*d + 2*e)*x^6 + 1/5*(10*d + e)*x^5 + 1/4*d*x^4
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (60) = 120\).
Time = 0.04 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.97 \[ \int x^3 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {d x^{4}}{4} + \frac {e x^{15}}{15} + x^{14} \left (\frac {d}{14} + \frac {5 e}{7}\right ) + x^{13} \cdot \left (\frac {10 d}{13} + \frac {45 e}{13}\right ) + x^{12} \cdot \left (\frac {15 d}{4} + 10 e\right ) + x^{11} \cdot \left (\frac {120 d}{11} + \frac {210 e}{11}\right ) + x^{10} \cdot \left (21 d + \frac {126 e}{5}\right ) + x^{9} \cdot \left (28 d + \frac {70 e}{3}\right ) + x^{8} \cdot \left (\frac {105 d}{4} + 15 e\right ) + x^{7} \cdot \left (\frac {120 d}{7} + \frac {45 e}{7}\right ) + x^{6} \cdot \left (\frac {15 d}{2} + \frac {5 e}{3}\right ) + x^{5} \cdot \left (2 d + \frac {e}{5}\right ) \] Input:
integrate(x**3*(e*x+d)*(x**2+2*x+1)**5,x)
Output:
d*x**4/4 + e*x**15/15 + x**14*(d/14 + 5*e/7) + x**13*(10*d/13 + 45*e/13) + x**12*(15*d/4 + 10*e) + x**11*(120*d/11 + 210*e/11) + x**10*(21*d + 126*e /5) + x**9*(28*d + 70*e/3) + x**8*(105*d/4 + 15*e) + x**7*(120*d/7 + 45*e/ 7) + x**6*(15*d/2 + 5*e/3) + x**5*(2*d + e/5)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (59) = 118\).
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.87 \[ \int x^3 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{15} \, e x^{15} + \frac {1}{14} \, {\left (d + 10 \, e\right )} x^{14} + \frac {5}{13} \, {\left (2 \, d + 9 \, e\right )} x^{13} + \frac {5}{4} \, {\left (3 \, d + 8 \, e\right )} x^{12} + \frac {30}{11} \, {\left (4 \, d + 7 \, e\right )} x^{11} + \frac {21}{5} \, {\left (5 \, d + 6 \, e\right )} x^{10} + \frac {14}{3} \, {\left (6 \, d + 5 \, e\right )} x^{9} + \frac {15}{4} \, {\left (7 \, d + 4 \, e\right )} x^{8} + \frac {15}{7} \, {\left (8 \, d + 3 \, e\right )} x^{7} + \frac {5}{6} \, {\left (9 \, d + 2 \, e\right )} x^{6} + \frac {1}{5} \, {\left (10 \, d + e\right )} x^{5} + \frac {1}{4} \, d x^{4} \] Input:
integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")
Output:
1/15*e*x^15 + 1/14*(d + 10*e)*x^14 + 5/13*(2*d + 9*e)*x^13 + 5/4*(3*d + 8* e)*x^12 + 30/11*(4*d + 7*e)*x^11 + 21/5*(5*d + 6*e)*x^10 + 14/3*(6*d + 5*e )*x^9 + 15/4*(7*d + 4*e)*x^8 + 15/7*(8*d + 3*e)*x^7 + 5/6*(9*d + 2*e)*x^6 + 1/5*(10*d + e)*x^5 + 1/4*d*x^4
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (59) = 118\).
Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.93 \[ \int x^3 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {1}{15} \, e x^{15} + \frac {1}{14} \, d x^{14} + \frac {5}{7} \, e x^{14} + \frac {10}{13} \, d x^{13} + \frac {45}{13} \, e x^{13} + \frac {15}{4} \, d x^{12} + 10 \, e x^{12} + \frac {120}{11} \, d x^{11} + \frac {210}{11} \, e x^{11} + 21 \, d x^{10} + \frac {126}{5} \, e x^{10} + 28 \, d x^{9} + \frac {70}{3} \, e x^{9} + \frac {105}{4} \, d x^{8} + 15 \, e x^{8} + \frac {120}{7} \, d x^{7} + \frac {45}{7} \, e x^{7} + \frac {15}{2} \, d x^{6} + \frac {5}{3} \, e x^{6} + 2 \, d x^{5} + \frac {1}{5} \, e x^{5} + \frac {1}{4} \, d x^{4} \] Input:
integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")
Output:
1/15*e*x^15 + 1/14*d*x^14 + 5/7*e*x^14 + 10/13*d*x^13 + 45/13*e*x^13 + 15/ 4*d*x^12 + 10*e*x^12 + 120/11*d*x^11 + 210/11*e*x^11 + 21*d*x^10 + 126/5*e *x^10 + 28*d*x^9 + 70/3*e*x^9 + 105/4*d*x^8 + 15*e*x^8 + 120/7*d*x^7 + 45/ 7*e*x^7 + 15/2*d*x^6 + 5/3*e*x^6 + 2*d*x^5 + 1/5*e*x^5 + 1/4*d*x^4
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.78 \[ \int x^3 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {e\,x^{15}}{15}+\left (\frac {d}{14}+\frac {5\,e}{7}\right )\,x^{14}+\left (\frac {10\,d}{13}+\frac {45\,e}{13}\right )\,x^{13}+\left (\frac {15\,d}{4}+10\,e\right )\,x^{12}+\left (\frac {120\,d}{11}+\frac {210\,e}{11}\right )\,x^{11}+\left (21\,d+\frac {126\,e}{5}\right )\,x^{10}+\left (28\,d+\frac {70\,e}{3}\right )\,x^9+\left (\frac {105\,d}{4}+15\,e\right )\,x^8+\left (\frac {120\,d}{7}+\frac {45\,e}{7}\right )\,x^7+\left (\frac {15\,d}{2}+\frac {5\,e}{3}\right )\,x^6+\left (2\,d+\frac {e}{5}\right )\,x^5+\frac {d\,x^4}{4} \] Input:
int(x^3*(d + e*x)*(2*x + x^2 + 1)^5,x)
Output:
x^5*(2*d + e/5) + x^6*((15*d)/2 + (5*e)/3) + x^12*((15*d)/4 + 10*e) + x^14 *(d/14 + (5*e)/7) + x^13*((10*d)/13 + (45*e)/13) + x^9*(28*d + (70*e)/3) + x^8*((105*d)/4 + 15*e) + x^10*(21*d + (126*e)/5) + x^7*((120*d)/7 + (45*e )/7) + x^11*((120*d)/11 + (210*e)/11) + (d*x^4)/4 + (e*x^15)/15
Time = 0.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.90 \[ \int x^3 (d+e x) \left (1+2 x+x^2\right )^5 \, dx=\frac {x^{4} \left (4004 e \,x^{11}+4290 d \,x^{10}+42900 e \,x^{10}+46200 d \,x^{9}+207900 e \,x^{9}+225225 d \,x^{8}+600600 e \,x^{8}+655200 d \,x^{7}+1146600 e \,x^{7}+1261260 d \,x^{6}+1513512 e \,x^{6}+1681680 d \,x^{5}+1401400 e \,x^{5}+1576575 d \,x^{4}+900900 e \,x^{4}+1029600 d \,x^{3}+386100 e \,x^{3}+450450 d \,x^{2}+100100 e \,x^{2}+120120 d x +12012 e x +15015 d \right )}{60060} \] Input:
int(x^3*(e*x+d)*(x^2+2*x+1)^5,x)
Output:
(x**4*(4290*d*x**10 + 46200*d*x**9 + 225225*d*x**8 + 655200*d*x**7 + 12612 60*d*x**6 + 1681680*d*x**5 + 1576575*d*x**4 + 1029600*d*x**3 + 450450*d*x* *2 + 120120*d*x + 15015*d + 4004*e*x**11 + 42900*e*x**10 + 207900*e*x**9 + 600600*e*x**8 + 1146600*e*x**7 + 1513512*e*x**6 + 1401400*e*x**5 + 900900 *e*x**4 + 386100*e*x**3 + 100100*e*x**2 + 12012*e*x))/60060