Integrand size = 24, antiderivative size = 124 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^3}{3 e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^4}{4 e^4}-\frac {3 c (2 c d-b e) (d+e x)^5}{5 e^4}+\frac {c^2 (d+e x)^6}{3 e^4} \]
-1/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^3/e^4+1/4*(6*c^2*d^2+b^2*e^2 -2*c*e*(-a*e+3*b*d))*(e*x+d)^4/e^4-3/5*c*(-b*e+2*c*d)*(e*x+d)^5/e^4+1/3*c^ 2*(e*x+d)^6/e^4
Time = 0.04 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx=a b d^2 x+\frac {1}{2} d \left (b^2 d+2 a c d+2 a b e\right ) x^2+\frac {1}{3} \left (3 b c d^2+2 b^2 d e+4 a c d e+a b e^2\right ) x^3+\frac {1}{4} \left (2 c^2 d^2+6 b c d e+b^2 e^2+2 a c e^2\right ) x^4+\frac {1}{5} c e (4 c d+3 b e) x^5+\frac {1}{3} c^2 e^2 x^6 \]
a*b*d^2*x + (d*(b^2*d + 2*a*c*d + 2*a*b*e)*x^2)/2 + ((3*b*c*d^2 + 2*b^2*d* e + 4*a*c*d*e + a*b*e^2)*x^3)/3 + ((2*c^2*d^2 + 6*b*c*d*e + b^2*e^2 + 2*a* c*e^2)*x^4)/4 + (c*e*(4*c*d + 3*b*e)*x^5)/5 + (c^2*e^2*x^6)/3
Time = 0.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.
\(\displaystyle \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {(d+e x)^3 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^3}+\frac {(d+e x)^2 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {3 c (d+e x)^4 (2 c d-b e)}{e^3}+\frac {2 c^2 (d+e x)^5}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^4 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{4 e^4}-\frac {(d+e x)^3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4}-\frac {3 c (d+e x)^5 (2 c d-b e)}{5 e^4}+\frac {c^2 (d+e x)^6}{3 e^4}\) |
-1/3*((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3)/e^4 + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^4)/(4*e^4) - (3*c*(2*c*d - b*e )*(d + e*x)^5)/(5*e^4) + (c^2*(d + e*x)^6)/(3*e^4)
3.15.96.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.53 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03
method | result | size |
norman | \(\frac {c^{2} e^{2} x^{6}}{3}+\left (\frac {3}{5} c \,e^{2} b +\frac {4}{5} c^{2} d e \right ) x^{5}+\left (\frac {1}{2} a c \,e^{2}+\frac {1}{4} b^{2} e^{2}+\frac {3}{2} b c d e +\frac {1}{2} c^{2} d^{2}\right ) x^{4}+\left (\frac {1}{3} a b \,e^{2}+\frac {4}{3} a c d e +\frac {2}{3} b^{2} d e +b c \,d^{2}\right ) x^{3}+\left (a b d e +c \,d^{2} a +\frac {1}{2} b^{2} d^{2}\right ) x^{2}+b \,d^{2} a x\) | \(128\) |
gosper | \(\frac {1}{3} c^{2} e^{2} x^{6}+\frac {3}{5} x^{5} c \,e^{2} b +\frac {4}{5} x^{5} c^{2} d e +\frac {1}{2} x^{4} a c \,e^{2}+\frac {1}{4} x^{4} b^{2} e^{2}+\frac {3}{2} x^{4} b c d e +\frac {1}{2} x^{4} c^{2} d^{2}+\frac {1}{3} x^{3} a b \,e^{2}+\frac {4}{3} x^{3} a c d e +\frac {2}{3} x^{3} b^{2} d e +x^{3} b c \,d^{2}+x^{2} a b d e +x^{2} c \,d^{2} a +\frac {1}{2} x^{2} b^{2} d^{2}+b \,d^{2} a x\) | \(147\) |
risch | \(\frac {1}{3} c^{2} e^{2} x^{6}+\frac {3}{5} x^{5} c \,e^{2} b +\frac {4}{5} x^{5} c^{2} d e +\frac {1}{2} x^{4} a c \,e^{2}+\frac {1}{4} x^{4} b^{2} e^{2}+\frac {3}{2} x^{4} b c d e +\frac {1}{2} x^{4} c^{2} d^{2}+\frac {1}{3} x^{3} a b \,e^{2}+\frac {4}{3} x^{3} a c d e +\frac {2}{3} x^{3} b^{2} d e +x^{3} b c \,d^{2}+x^{2} a b d e +x^{2} c \,d^{2} a +\frac {1}{2} x^{2} b^{2} d^{2}+b \,d^{2} a x\) | \(147\) |
parallelrisch | \(\frac {1}{3} c^{2} e^{2} x^{6}+\frac {3}{5} x^{5} c \,e^{2} b +\frac {4}{5} x^{5} c^{2} d e +\frac {1}{2} x^{4} a c \,e^{2}+\frac {1}{4} x^{4} b^{2} e^{2}+\frac {3}{2} x^{4} b c d e +\frac {1}{2} x^{4} c^{2} d^{2}+\frac {1}{3} x^{3} a b \,e^{2}+\frac {4}{3} x^{3} a c d e +\frac {2}{3} x^{3} b^{2} d e +x^{3} b c \,d^{2}+x^{2} a b d e +x^{2} c \,d^{2} a +\frac {1}{2} x^{2} b^{2} d^{2}+b \,d^{2} a x\) | \(147\) |
default | \(\frac {c^{2} e^{2} x^{6}}{3}+\frac {\left (\left (b \,e^{2}+4 c d e \right ) c +2 c \,e^{2} b \right ) x^{5}}{5}+\frac {\left (\left (2 b d e +2 c \,d^{2}\right ) c +\left (b \,e^{2}+4 c d e \right ) b +2 a c \,e^{2}\right ) x^{4}}{4}+\frac {\left (b c \,d^{2}+\left (2 b d e +2 c \,d^{2}\right ) b +\left (b \,e^{2}+4 c d e \right ) a \right ) x^{3}}{3}+\frac {\left (b^{2} d^{2}+\left (2 b d e +2 c \,d^{2}\right ) a \right ) x^{2}}{2}+b \,d^{2} a x\) | \(152\) |
1/3*c^2*e^2*x^6+(3/5*c*e^2*b+4/5*c^2*d*e)*x^5+(1/2*a*c*e^2+1/4*b^2*e^2+3/2 *b*c*d*e+1/2*c^2*d^2)*x^4+(1/3*a*b*e^2+4/3*a*c*d*e+2/3*b^2*d*e+b*c*d^2)*x^ 3+(a*b*d*e+c*d^2*a+1/2*b^2*d^2)*x^2+b*d^2*a*x
Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx=\frac {1}{3} \, c^{2} e^{2} x^{6} + \frac {1}{5} \, {\left (4 \, c^{2} d e + 3 \, b c e^{2}\right )} x^{5} + a b d^{2} x + \frac {1}{4} \, {\left (2 \, c^{2} d^{2} + 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, b c d^{2} + a b e^{2} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d e + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{2} \]
1/3*c^2*e^2*x^6 + 1/5*(4*c^2*d*e + 3*b*c*e^2)*x^5 + a*b*d^2*x + 1/4*(2*c^2 *d^2 + 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*x^4 + 1/3*(3*b*c*d^2 + a*b*e^2 + 2*( b^2 + 2*a*c)*d*e)*x^3 + 1/2*(2*a*b*d*e + (b^2 + 2*a*c)*d^2)*x^2
Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.18 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx=a b d^{2} x + \frac {c^{2} e^{2} x^{6}}{3} + x^{5} \cdot \left (\frac {3 b c e^{2}}{5} + \frac {4 c^{2} d e}{5}\right ) + x^{4} \left (\frac {a c e^{2}}{2} + \frac {b^{2} e^{2}}{4} + \frac {3 b c d e}{2} + \frac {c^{2} d^{2}}{2}\right ) + x^{3} \left (\frac {a b e^{2}}{3} + \frac {4 a c d e}{3} + \frac {2 b^{2} d e}{3} + b c d^{2}\right ) + x^{2} \left (a b d e + a c d^{2} + \frac {b^{2} d^{2}}{2}\right ) \]
a*b*d**2*x + c**2*e**2*x**6/3 + x**5*(3*b*c*e**2/5 + 4*c**2*d*e/5) + x**4* (a*c*e**2/2 + b**2*e**2/4 + 3*b*c*d*e/2 + c**2*d**2/2) + x**3*(a*b*e**2/3 + 4*a*c*d*e/3 + 2*b**2*d*e/3 + b*c*d**2) + x**2*(a*b*d*e + a*c*d**2 + b**2 *d**2/2)
Time = 0.20 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx=\frac {1}{3} \, c^{2} e^{2} x^{6} + \frac {1}{5} \, {\left (4 \, c^{2} d e + 3 \, b c e^{2}\right )} x^{5} + a b d^{2} x + \frac {1}{4} \, {\left (2 \, c^{2} d^{2} + 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, b c d^{2} + a b e^{2} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d e + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} x^{2} \]
1/3*c^2*e^2*x^6 + 1/5*(4*c^2*d*e + 3*b*c*e^2)*x^5 + a*b*d^2*x + 1/4*(2*c^2 *d^2 + 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*x^4 + 1/3*(3*b*c*d^2 + a*b*e^2 + 2*( b^2 + 2*a*c)*d*e)*x^3 + 1/2*(2*a*b*d*e + (b^2 + 2*a*c)*d^2)*x^2
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.18 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx=\frac {1}{3} \, c^{2} e^{2} x^{6} + \frac {4}{5} \, c^{2} d e x^{5} + \frac {3}{5} \, b c e^{2} x^{5} + \frac {1}{2} \, c^{2} d^{2} x^{4} + \frac {3}{2} \, b c d e x^{4} + \frac {1}{4} \, b^{2} e^{2} x^{4} + \frac {1}{2} \, a c e^{2} x^{4} + b c d^{2} x^{3} + \frac {2}{3} \, b^{2} d e x^{3} + \frac {4}{3} \, a c d e x^{3} + \frac {1}{3} \, a b e^{2} x^{3} + \frac {1}{2} \, b^{2} d^{2} x^{2} + a c d^{2} x^{2} + a b d e x^{2} + a b d^{2} x \]
1/3*c^2*e^2*x^6 + 4/5*c^2*d*e*x^5 + 3/5*b*c*e^2*x^5 + 1/2*c^2*d^2*x^4 + 3/ 2*b*c*d*e*x^4 + 1/4*b^2*e^2*x^4 + 1/2*a*c*e^2*x^4 + b*c*d^2*x^3 + 2/3*b^2* d*e*x^3 + 4/3*a*c*d*e*x^3 + 1/3*a*b*e^2*x^3 + 1/2*b^2*d^2*x^2 + a*c*d^2*x^ 2 + a*b*d*e*x^2 + a*b*d^2*x
Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00 \[ \int (b+2 c x) (d+e x)^2 \left (a+b x+c x^2\right ) \, dx=x^4\,\left (\frac {b^2\,e^2}{4}+\frac {3\,b\,c\,d\,e}{2}+\frac {c^2\,d^2}{2}+\frac {a\,c\,e^2}{2}\right )+x^3\,\left (\frac {2\,b^2\,d\,e}{3}+c\,b\,d^2+\frac {a\,b\,e^2}{3}+\frac {4\,a\,c\,d\,e}{3}\right )+x^2\,\left (\frac {b^2\,d^2}{2}+a\,e\,b\,d+a\,c\,d^2\right )+\frac {c^2\,e^2\,x^6}{3}+a\,b\,d^2\,x+\frac {c\,e\,x^5\,\left (3\,b\,e+4\,c\,d\right )}{5} \]