Integrand size = 36, antiderivative size = 121 \[ \int (d+e x) \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=18 d x+\frac {3}{2} (11 d+6 e) x^2+\frac {1}{3} (107 d+33 e) x^3+\frac {1}{4} (65 d+107 e) x^4+\frac {1}{5} (148 d+65 e) x^5-\frac {37}{6} (d-4 e) x^6+\frac {37}{7} (3 d-e) x^7-\frac {3}{8} (15 d-37 e) x^8+\frac {5}{9} (20 d-9 e) x^9+10 e x^{10} \]
18*d*x+3/2*(11*d+6*e)*x^2+1/3*(107*d+33*e)*x^3+1/4*(65*d+107*e)*x^4+1/5*(1 48*d+65*e)*x^5-37/6*(d-4*e)*x^6+37/7*(3*d-e)*x^7-3/8*(15*d-37*e)*x^8+5/9*( 20*d-9*e)*x^9+10*e*x^10
Time = 0.01 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=18 d x+\frac {3}{2} (11 d+6 e) x^2+\frac {1}{3} (107 d+33 e) x^3+\frac {1}{4} (65 d+107 e) x^4+\frac {1}{5} (148 d+65 e) x^5-\frac {37}{6} (d-4 e) x^6+\frac {37}{7} (3 d-e) x^7-\frac {3}{8} (15 d-37 e) x^8+\frac {5}{9} (20 d-9 e) x^9+10 e x^{10} \]
18*d*x + (3*(11*d + 6*e)*x^2)/2 + ((107*d + 33*e)*x^3)/3 + ((65*d + 107*e) *x^4)/4 + ((148*d + 65*e)*x^5)/5 - (37*(d - 4*e)*x^6)/6 + (37*(3*d - e)*x^ 7)/7 - (3*(15*d - 37*e)*x^8)/8 + (5*(20*d - 9*e)*x^9)/9 + 10*e*x^10
Time = 0.40 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2159, 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.
\(\displaystyle \int \left (5 x^2+2 x+3\right )^2 \left (4 x^4-5 x^3+3 x^2+x+2\right ) (d+e x) \, dx\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \int \left (5 x^8 (20 d-9 e)-3 x^7 (15 d-37 e)+37 x^6 (3 d-e)-37 x^5 (d-4 e)+x^4 (148 d+65 e)+x^3 (65 d+107 e)+x^2 (107 d+33 e)+3 x (11 d+6 e)+18 d+100 e x^9\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5}{9} x^9 (20 d-9 e)-\frac {3}{8} x^8 (15 d-37 e)+\frac {37}{7} x^7 (3 d-e)-\frac {37}{6} x^6 (d-4 e)+\frac {1}{5} x^5 (148 d+65 e)+\frac {1}{4} x^4 (65 d+107 e)+\frac {1}{3} x^3 (107 d+33 e)+\frac {3}{2} x^2 (11 d+6 e)+18 d x+10 e x^{10}\) |
18*d*x + (3*(11*d + 6*e)*x^2)/2 + ((107*d + 33*e)*x^3)/3 + ((65*d + 107*e) *x^4)/4 + ((148*d + 65*e)*x^5)/5 - (37*(d - 4*e)*x^6)/6 + (37*(3*d - e)*x^ 7)/7 - (3*(15*d - 37*e)*x^8)/8 + (5*(20*d - 9*e)*x^9)/9 + 10*e*x^10
3.3.98.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.53 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83
method | result | size |
norman | \(10 e \,x^{10}+\left (\frac {100 d}{9}-5 e \right ) x^{9}+\left (-\frac {45 d}{8}+\frac {111 e}{8}\right ) x^{8}+\left (\frac {111 d}{7}-\frac {37 e}{7}\right ) x^{7}+\left (-\frac {37 d}{6}+\frac {74 e}{3}\right ) x^{6}+\left (\frac {148 d}{5}+13 e \right ) x^{5}+\left (\frac {65 d}{4}+\frac {107 e}{4}\right ) x^{4}+\left (\frac {107 d}{3}+11 e \right ) x^{3}+\left (\frac {33 d}{2}+9 e \right ) x^{2}+18 d x\) | \(100\) |
gosper | \(10 e \,x^{10}+\frac {100}{9} d \,x^{9}-5 e \,x^{9}-\frac {45}{8} d \,x^{8}+\frac {111}{8} e \,x^{8}+\frac {111}{7} d \,x^{7}-\frac {37}{7} e \,x^{7}-\frac {37}{6} d \,x^{6}+\frac {74}{3} e \,x^{6}+\frac {148}{5} d \,x^{5}+13 e \,x^{5}+\frac {65}{4} d \,x^{4}+\frac {107}{4} e \,x^{4}+\frac {107}{3} d \,x^{3}+11 e \,x^{3}+\frac {33}{2} d \,x^{2}+9 e \,x^{2}+18 d x\) | \(108\) |
default | \(10 e \,x^{10}+\frac {\left (100 d -45 e \right ) x^{9}}{9}+\frac {\left (-45 d +111 e \right ) x^{8}}{8}+\frac {\left (111 d -37 e \right ) x^{7}}{7}+\frac {\left (-37 d +148 e \right ) x^{6}}{6}+\frac {\left (148 d +65 e \right ) x^{5}}{5}+\frac {\left (65 d +107 e \right ) x^{4}}{4}+\frac {\left (107 d +33 e \right ) x^{3}}{3}+\frac {\left (33 d +18 e \right ) x^{2}}{2}+18 d x\) | \(108\) |
risch | \(10 e \,x^{10}+\frac {100}{9} d \,x^{9}-5 e \,x^{9}-\frac {45}{8} d \,x^{8}+\frac {111}{8} e \,x^{8}+\frac {111}{7} d \,x^{7}-\frac {37}{7} e \,x^{7}-\frac {37}{6} d \,x^{6}+\frac {74}{3} e \,x^{6}+\frac {148}{5} d \,x^{5}+13 e \,x^{5}+\frac {65}{4} d \,x^{4}+\frac {107}{4} e \,x^{4}+\frac {107}{3} d \,x^{3}+11 e \,x^{3}+\frac {33}{2} d \,x^{2}+9 e \,x^{2}+18 d x\) | \(108\) |
parallelrisch | \(10 e \,x^{10}+\frac {100}{9} d \,x^{9}-5 e \,x^{9}-\frac {45}{8} d \,x^{8}+\frac {111}{8} e \,x^{8}+\frac {111}{7} d \,x^{7}-\frac {37}{7} e \,x^{7}-\frac {37}{6} d \,x^{6}+\frac {74}{3} e \,x^{6}+\frac {148}{5} d \,x^{5}+13 e \,x^{5}+\frac {65}{4} d \,x^{4}+\frac {107}{4} e \,x^{4}+\frac {107}{3} d \,x^{3}+11 e \,x^{3}+\frac {33}{2} d \,x^{2}+9 e \,x^{2}+18 d x\) | \(108\) |
10*e*x^10+(100/9*d-5*e)*x^9+(-45/8*d+111/8*e)*x^8+(111/7*d-37/7*e)*x^7+(-3 7/6*d+74/3*e)*x^6+(148/5*d+13*e)*x^5+(65/4*d+107/4*e)*x^4+(107/3*d+11*e)*x ^3+(33/2*d+9*e)*x^2+18*d*x
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int (d+e x) \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=10 \, e x^{10} + \frac {5}{9} \, {\left (20 \, d - 9 \, e\right )} x^{9} - \frac {3}{8} \, {\left (15 \, d - 37 \, e\right )} x^{8} + \frac {37}{7} \, {\left (3 \, d - e\right )} x^{7} - \frac {37}{6} \, {\left (d - 4 \, e\right )} x^{6} + \frac {1}{5} \, {\left (148 \, d + 65 \, e\right )} x^{5} + \frac {1}{4} \, {\left (65 \, d + 107 \, e\right )} x^{4} + \frac {1}{3} \, {\left (107 \, d + 33 \, e\right )} x^{3} + \frac {3}{2} \, {\left (11 \, d + 6 \, e\right )} x^{2} + 18 \, d x \]
10*e*x^10 + 5/9*(20*d - 9*e)*x^9 - 3/8*(15*d - 37*e)*x^8 + 37/7*(3*d - e)* x^7 - 37/6*(d - 4*e)*x^6 + 1/5*(148*d + 65*e)*x^5 + 1/4*(65*d + 107*e)*x^4 + 1/3*(107*d + 33*e)*x^3 + 3/2*(11*d + 6*e)*x^2 + 18*d*x
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=18 d x + 10 e x^{10} + x^{9} \cdot \left (\frac {100 d}{9} - 5 e\right ) + x^{8} \left (- \frac {45 d}{8} + \frac {111 e}{8}\right ) + x^{7} \cdot \left (\frac {111 d}{7} - \frac {37 e}{7}\right ) + x^{6} \left (- \frac {37 d}{6} + \frac {74 e}{3}\right ) + x^{5} \cdot \left (\frac {148 d}{5} + 13 e\right ) + x^{4} \cdot \left (\frac {65 d}{4} + \frac {107 e}{4}\right ) + x^{3} \cdot \left (\frac {107 d}{3} + 11 e\right ) + x^{2} \cdot \left (\frac {33 d}{2} + 9 e\right ) \]
18*d*x + 10*e*x**10 + x**9*(100*d/9 - 5*e) + x**8*(-45*d/8 + 111*e/8) + x* *7*(111*d/7 - 37*e/7) + x**6*(-37*d/6 + 74*e/3) + x**5*(148*d/5 + 13*e) + x**4*(65*d/4 + 107*e/4) + x**3*(107*d/3 + 11*e) + x**2*(33*d/2 + 9*e)
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.87 \[ \int (d+e x) \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=10 \, e x^{10} + \frac {5}{9} \, {\left (20 \, d - 9 \, e\right )} x^{9} - \frac {3}{8} \, {\left (15 \, d - 37 \, e\right )} x^{8} + \frac {37}{7} \, {\left (3 \, d - e\right )} x^{7} - \frac {37}{6} \, {\left (d - 4 \, e\right )} x^{6} + \frac {1}{5} \, {\left (148 \, d + 65 \, e\right )} x^{5} + \frac {1}{4} \, {\left (65 \, d + 107 \, e\right )} x^{4} + \frac {1}{3} \, {\left (107 \, d + 33 \, e\right )} x^{3} + \frac {3}{2} \, {\left (11 \, d + 6 \, e\right )} x^{2} + 18 \, d x \]
10*e*x^10 + 5/9*(20*d - 9*e)*x^9 - 3/8*(15*d - 37*e)*x^8 + 37/7*(3*d - e)* x^7 - 37/6*(d - 4*e)*x^6 + 1/5*(148*d + 65*e)*x^5 + 1/4*(65*d + 107*e)*x^4 + 1/3*(107*d + 33*e)*x^3 + 3/2*(11*d + 6*e)*x^2 + 18*d*x
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88 \[ \int (d+e x) \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=10 \, e x^{10} + \frac {100}{9} \, d x^{9} - 5 \, e x^{9} - \frac {45}{8} \, d x^{8} + \frac {111}{8} \, e x^{8} + \frac {111}{7} \, d x^{7} - \frac {37}{7} \, e x^{7} - \frac {37}{6} \, d x^{6} + \frac {74}{3} \, e x^{6} + \frac {148}{5} \, d x^{5} + 13 \, e x^{5} + \frac {65}{4} \, d x^{4} + \frac {107}{4} \, e x^{4} + \frac {107}{3} \, d x^{3} + 11 \, e x^{3} + \frac {33}{2} \, d x^{2} + 9 \, e x^{2} + 18 \, d x \]
10*e*x^10 + 100/9*d*x^9 - 5*e*x^9 - 45/8*d*x^8 + 111/8*e*x^8 + 111/7*d*x^7 - 37/7*e*x^7 - 37/6*d*x^6 + 74/3*e*x^6 + 148/5*d*x^5 + 13*e*x^5 + 65/4*d* x^4 + 107/4*e*x^4 + 107/3*d*x^3 + 11*e*x^3 + 33/2*d*x^2 + 9*e*x^2 + 18*d*x
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.83 \[ \int (d+e x) \left (3+2 x+5 x^2\right )^2 \left (2+x+3 x^2-5 x^3+4 x^4\right ) \, dx=10\,e\,x^{10}+\left (\frac {100\,d}{9}-5\,e\right )\,x^9+\left (\frac {111\,e}{8}-\frac {45\,d}{8}\right )\,x^8+\left (\frac {111\,d}{7}-\frac {37\,e}{7}\right )\,x^7+\left (\frac {74\,e}{3}-\frac {37\,d}{6}\right )\,x^6+\left (\frac {148\,d}{5}+13\,e\right )\,x^5+\left (\frac {65\,d}{4}+\frac {107\,e}{4}\right )\,x^4+\left (\frac {107\,d}{3}+11\,e\right )\,x^3+\left (\frac {33\,d}{2}+9\,e\right )\,x^2+18\,d\,x \]