Integrand size = 25, antiderivative size = 96 \[ \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^4 \, dx=432 x+1080 x^2+2856 x^3+5144 x^4+\frac {43083 x^5}{5}+\frac {64529 x^6}{6}+\frac {91349 x^7}{7}+\frac {94881 x^8}{8}+\frac {103583 x^9}{9}+\frac {75311 x^{10}}{10}+\frac {68583 x^{11}}{11}+\frac {30395 x^{12}}{12}+\frac {27050 x^{13}}{13}+\frac {2250 x^{14}}{7}+\frac {1000 x^{15}}{3} \]
432*x+1080*x^2+2856*x^3+5144*x^4+43083/5*x^5+64529/6*x^6+91349/7*x^7+94881 /8*x^8+103583/9*x^9+75311/10*x^10+68583/11*x^11+30395/12*x^12+27050/13*x^1 3+2250/7*x^14+1000/3*x^15
Time = 0.00 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00 \[ \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^4 \, dx=432 x+1080 x^2+2856 x^3+5144 x^4+\frac {43083 x^5}{5}+\frac {64529 x^6}{6}+\frac {91349 x^7}{7}+\frac {94881 x^8}{8}+\frac {103583 x^9}{9}+\frac {75311 x^{10}}{10}+\frac {68583 x^{11}}{11}+\frac {30395 x^{12}}{12}+\frac {27050 x^{13}}{13}+\frac {2250 x^{14}}{7}+\frac {1000 x^{15}}{3} \]
432*x + 1080*x^2 + 2856*x^3 + 5144*x^4 + (43083*x^5)/5 + (64529*x^6)/6 + ( 91349*x^7)/7 + (94881*x^8)/8 + (103583*x^9)/9 + (75311*x^10)/10 + (68583*x ^11)/11 + (30395*x^12)/12 + (27050*x^13)/13 + (2250*x^14)/7 + (1000*x^15)/ 3
Time = 0.30 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2188, 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.
\(\displaystyle \int \left (2 x^2-x+3\right )^3 \left (5 x^2+3 x+2\right )^4 \, dx\) |
\(\Big \downarrow \) 2188 |
\(\displaystyle \int \left (5000 x^{14}+4500 x^{13}+27050 x^{12}+30395 x^{11}+68583 x^{10}+75311 x^9+103583 x^8+94881 x^7+91349 x^6+64529 x^5+43083 x^4+20576 x^3+8568 x^2+2160 x+432\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1000 x^{15}}{3}+\frac {2250 x^{14}}{7}+\frac {27050 x^{13}}{13}+\frac {30395 x^{12}}{12}+\frac {68583 x^{11}}{11}+\frac {75311 x^{10}}{10}+\frac {103583 x^9}{9}+\frac {94881 x^8}{8}+\frac {91349 x^7}{7}+\frac {64529 x^6}{6}+\frac {43083 x^5}{5}+5144 x^4+2856 x^3+1080 x^2+432 x\) |
432*x + 1080*x^2 + 2856*x^3 + 5144*x^4 + (43083*x^5)/5 + (64529*x^6)/6 + ( 91349*x^7)/7 + (94881*x^8)/8 + (103583*x^9)/9 + (75311*x^10)/10 + (68583*x ^11)/11 + (30395*x^12)/12 + (27050*x^13)/13 + (2250*x^14)/7 + (1000*x^15)/ 3
3.1.30.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Time = 0.62 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(432 x +1080 x^{2}+2856 x^{3}+5144 x^{4}+\frac {43083}{5} x^{5}+\frac {64529}{6} x^{6}+\frac {91349}{7} x^{7}+\frac {94881}{8} x^{8}+\frac {103583}{9} x^{9}+\frac {75311}{10} x^{10}+\frac {68583}{11} x^{11}+\frac {30395}{12} x^{12}+\frac {27050}{13} x^{13}+\frac {2250}{7} x^{14}+\frac {1000}{3} x^{15}\) | \(75\) |
default | \(432 x +1080 x^{2}+2856 x^{3}+5144 x^{4}+\frac {43083}{5} x^{5}+\frac {64529}{6} x^{6}+\frac {91349}{7} x^{7}+\frac {94881}{8} x^{8}+\frac {103583}{9} x^{9}+\frac {75311}{10} x^{10}+\frac {68583}{11} x^{11}+\frac {30395}{12} x^{12}+\frac {27050}{13} x^{13}+\frac {2250}{7} x^{14}+\frac {1000}{3} x^{15}\) | \(75\) |
norman | \(432 x +1080 x^{2}+2856 x^{3}+5144 x^{4}+\frac {43083}{5} x^{5}+\frac {64529}{6} x^{6}+\frac {91349}{7} x^{7}+\frac {94881}{8} x^{8}+\frac {103583}{9} x^{9}+\frac {75311}{10} x^{10}+\frac {68583}{11} x^{11}+\frac {30395}{12} x^{12}+\frac {27050}{13} x^{13}+\frac {2250}{7} x^{14}+\frac {1000}{3} x^{15}\) | \(75\) |
risch | \(432 x +1080 x^{2}+2856 x^{3}+5144 x^{4}+\frac {43083}{5} x^{5}+\frac {64529}{6} x^{6}+\frac {91349}{7} x^{7}+\frac {94881}{8} x^{8}+\frac {103583}{9} x^{9}+\frac {75311}{10} x^{10}+\frac {68583}{11} x^{11}+\frac {30395}{12} x^{12}+\frac {27050}{13} x^{13}+\frac {2250}{7} x^{14}+\frac {1000}{3} x^{15}\) | \(75\) |
parallelrisch | \(432 x +1080 x^{2}+2856 x^{3}+5144 x^{4}+\frac {43083}{5} x^{5}+\frac {64529}{6} x^{6}+\frac {91349}{7} x^{7}+\frac {94881}{8} x^{8}+\frac {103583}{9} x^{9}+\frac {75311}{10} x^{10}+\frac {68583}{11} x^{11}+\frac {30395}{12} x^{12}+\frac {27050}{13} x^{13}+\frac {2250}{7} x^{14}+\frac {1000}{3} x^{15}\) | \(75\) |
432*x+1080*x^2+2856*x^3+5144*x^4+43083/5*x^5+64529/6*x^6+91349/7*x^7+94881 /8*x^8+103583/9*x^9+75311/10*x^10+68583/11*x^11+30395/12*x^12+27050/13*x^1 3+2250/7*x^14+1000/3*x^15
Time = 0.37 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1000}{3} \, x^{15} + \frac {2250}{7} \, x^{14} + \frac {27050}{13} \, x^{13} + \frac {30395}{12} \, x^{12} + \frac {68583}{11} \, x^{11} + \frac {75311}{10} \, x^{10} + \frac {103583}{9} \, x^{9} + \frac {94881}{8} \, x^{8} + \frac {91349}{7} \, x^{7} + \frac {64529}{6} \, x^{6} + \frac {43083}{5} \, x^{5} + 5144 \, x^{4} + 2856 \, x^{3} + 1080 \, x^{2} + 432 \, x \]
1000/3*x^15 + 2250/7*x^14 + 27050/13*x^13 + 30395/12*x^12 + 68583/11*x^11 + 75311/10*x^10 + 103583/9*x^9 + 94881/8*x^8 + 91349/7*x^7 + 64529/6*x^6 + 43083/5*x^5 + 5144*x^4 + 2856*x^3 + 1080*x^2 + 432*x
Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96 \[ \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1000 x^{15}}{3} + \frac {2250 x^{14}}{7} + \frac {27050 x^{13}}{13} + \frac {30395 x^{12}}{12} + \frac {68583 x^{11}}{11} + \frac {75311 x^{10}}{10} + \frac {103583 x^{9}}{9} + \frac {94881 x^{8}}{8} + \frac {91349 x^{7}}{7} + \frac {64529 x^{6}}{6} + \frac {43083 x^{5}}{5} + 5144 x^{4} + 2856 x^{3} + 1080 x^{2} + 432 x \]
1000*x**15/3 + 2250*x**14/7 + 27050*x**13/13 + 30395*x**12/12 + 68583*x**1 1/11 + 75311*x**10/10 + 103583*x**9/9 + 94881*x**8/8 + 91349*x**7/7 + 6452 9*x**6/6 + 43083*x**5/5 + 5144*x**4 + 2856*x**3 + 1080*x**2 + 432*x
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1000}{3} \, x^{15} + \frac {2250}{7} \, x^{14} + \frac {27050}{13} \, x^{13} + \frac {30395}{12} \, x^{12} + \frac {68583}{11} \, x^{11} + \frac {75311}{10} \, x^{10} + \frac {103583}{9} \, x^{9} + \frac {94881}{8} \, x^{8} + \frac {91349}{7} \, x^{7} + \frac {64529}{6} \, x^{6} + \frac {43083}{5} \, x^{5} + 5144 \, x^{4} + 2856 \, x^{3} + 1080 \, x^{2} + 432 \, x \]
1000/3*x^15 + 2250/7*x^14 + 27050/13*x^13 + 30395/12*x^12 + 68583/11*x^11 + 75311/10*x^10 + 103583/9*x^9 + 94881/8*x^8 + 91349/7*x^7 + 64529/6*x^6 + 43083/5*x^5 + 5144*x^4 + 2856*x^3 + 1080*x^2 + 432*x
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1000}{3} \, x^{15} + \frac {2250}{7} \, x^{14} + \frac {27050}{13} \, x^{13} + \frac {30395}{12} \, x^{12} + \frac {68583}{11} \, x^{11} + \frac {75311}{10} \, x^{10} + \frac {103583}{9} \, x^{9} + \frac {94881}{8} \, x^{8} + \frac {91349}{7} \, x^{7} + \frac {64529}{6} \, x^{6} + \frac {43083}{5} \, x^{5} + 5144 \, x^{4} + 2856 \, x^{3} + 1080 \, x^{2} + 432 \, x \]
1000/3*x^15 + 2250/7*x^14 + 27050/13*x^13 + 30395/12*x^12 + 68583/11*x^11 + 75311/10*x^10 + 103583/9*x^9 + 94881/8*x^8 + 91349/7*x^7 + 64529/6*x^6 + 43083/5*x^5 + 5144*x^4 + 2856*x^3 + 1080*x^2 + 432*x
Time = 0.15 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right )^4 \, dx=\frac {1000\,x^{15}}{3}+\frac {2250\,x^{14}}{7}+\frac {27050\,x^{13}}{13}+\frac {30395\,x^{12}}{12}+\frac {68583\,x^{11}}{11}+\frac {75311\,x^{10}}{10}+\frac {103583\,x^9}{9}+\frac {94881\,x^8}{8}+\frac {91349\,x^7}{7}+\frac {64529\,x^6}{6}+\frac {43083\,x^5}{5}+5144\,x^4+2856\,x^3+1080\,x^2+432\,x \]