Integrand size = 24, antiderivative size = 124 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=-\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^4}{4 e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^5}{5 e^4}-\frac {c (2 c d-b e) (d+e x)^6}{2 e^4}+\frac {2 c^2 (d+e x)^7}{7 e^4} \]
-1/4*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^4/e^4+1/5*(6*c^2*d^2+b^2*e^2 -2*c*e*(-a*e+3*b*d))*(e*x+d)^5/e^4-1/2*c*(-b*e+2*c*d)*(e*x+d)^6/e^4+2/7*c^ 2*(e*x+d)^7/e^4
Time = 0.06 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.41 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=a b d^3 x+\frac {1}{2} d^2 \left (b^2 d+2 a c d+3 a b e\right ) x^2+d \left (b c d^2+b^2 d e+2 a c d e+a b e^2\right ) x^3+\frac {1}{4} \left (2 c^2 d^3+b e^2 (3 b d+a e)+3 c d e (3 b d+2 a e)\right ) x^4+\frac {1}{5} e \left (6 c^2 d^2+b^2 e^2+c e (9 b d+2 a e)\right ) x^5+\frac {1}{2} c e^2 (2 c d+b e) x^6+\frac {2}{7} c^2 e^3 x^7 \]
a*b*d^3*x + (d^2*(b^2*d + 2*a*c*d + 3*a*b*e)*x^2)/2 + d*(b*c*d^2 + b^2*d*e + 2*a*c*d*e + a*b*e^2)*x^3 + ((2*c^2*d^3 + b*e^2*(3*b*d + a*e) + 3*c*d*e* (3*b*d + 2*a*e))*x^4)/4 + (e*(6*c^2*d^2 + b^2*e^2 + c*e*(9*b*d + 2*a*e))*x ^5)/5 + (c*e^2*(2*c*d + b*e)*x^6)/2 + (2*c^2*e^3*x^7)/7
Time = 0.36 (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.
| \(\displaystyle \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^4 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^3}+\frac {(d+e x)^3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )}{e^3}-\frac {3 c (d+e x)^5 (2 c d-b e)}{e^3}+\frac {2 c^2 (d+e x)^6}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^5 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^4}-\frac {(d+e x)^4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4}-\frac {c (d+e x)^6 (2 c d-b e)}{2 e^4}+\frac {2 c^2 (d+e x)^7}{7 e^4}\) |
-1/4*((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^4)/e^4 + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^5)/(5*e^4) - (c*(2*c*d - b*e)* (d + e*x)^6)/(2*e^4) + (2*c^2*(d + e*x)^7)/(7*e^4)
3.15.95.3.1 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 = 0.33 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.48
| method | result | size |
| norman | \(\frac {2 c^{2} e^{3} x^{7}}{7}+\left (\frac {1}{2} c \,e^{3} b +c^{2} d \,e^{2}\right ) x^{6}+\left (\frac {2}{5} c \,e^{3} a +\frac {1}{5} b^{2} e^{3}+\frac {9}{5} b c d \,e^{2}+\frac {6}{5} c^{2} d^{2} e \right ) x^{5}+\left (\frac {1}{4} a b \,e^{3}+\frac {3}{2} a c d \,e^{2}+\frac {3}{4} b^{2} d \,e^{2}+\frac {9}{4} b c \,d^{2} e +\frac {1}{2} c^{2} d^{3}\right ) x^{4}+\left (a b d \,e^{2}+2 a c \,d^{2} e +b^{2} d^{2} e +b \,d^{3} c \right ) x^{3}+\left (\frac {3}{2} a b \,d^{2} e +a c \,d^{3}+\frac {1}{2} b^{2} d^{3}\right ) x^{2}+b \,d^{3} a x\) | \(183\) |
| gosper | \(\frac {2}{7} c^{2} e^{3} x^{7}+\frac {1}{2} x^{6} c \,e^{3} b +x^{6} c^{2} d \,e^{2}+\frac {2}{5} x^{5} c \,e^{3} a +\frac {1}{5} x^{5} b^{2} e^{3}+\frac {9}{5} x^{5} b c d \,e^{2}+\frac {6}{5} x^{5} c^{2} d^{2} e +\frac {1}{4} x^{4} a b \,e^{3}+\frac {3}{2} x^{4} a c d \,e^{2}+\frac {3}{4} x^{4} b^{2} d \,e^{2}+\frac {9}{4} x^{4} b c \,d^{2} e +\frac {1}{2} x^{4} c^{2} d^{3}+a b d \,e^{2} x^{3}+2 a c \,d^{2} e \,x^{3}+b^{2} d^{2} e \,x^{3}+b c \,d^{3} x^{3}+\frac {3}{2} x^{2} a b \,d^{2} e +x^{2} a c \,d^{3}+\frac {1}{2} x^{2} b^{2} d^{3}+b \,d^{3} a x\) | \(212\) |
| risch | \(\frac {2}{7} c^{2} e^{3} x^{7}+\frac {1}{2} x^{6} c \,e^{3} b +x^{6} c^{2} d \,e^{2}+\frac {2}{5} x^{5} c \,e^{3} a +\frac {1}{5} x^{5} b^{2} e^{3}+\frac {9}{5} x^{5} b c d \,e^{2}+\frac {6}{5} x^{5} c^{2} d^{2} e +\frac {1}{4} x^{4} a b \,e^{3}+\frac {3}{2} x^{4} a c d \,e^{2}+\frac {3}{4} x^{4} b^{2} d \,e^{2}+\frac {9}{4} x^{4} b c \,d^{2} e +\frac {1}{2} x^{4} c^{2} d^{3}+a b d \,e^{2} x^{3}+2 a c \,d^{2} e \,x^{3}+b^{2} d^{2} e \,x^{3}+b c \,d^{3} x^{3}+\frac {3}{2} x^{2} a b \,d^{2} e +x^{2} a c \,d^{3}+\frac {1}{2} x^{2} b^{2} d^{3}+b \,d^{3} a x\) | \(212\) |
| parallelrisch | \(\frac {2}{7} c^{2} e^{3} x^{7}+\frac {1}{2} x^{6} c \,e^{3} b +x^{6} c^{2} d \,e^{2}+\frac {2}{5} x^{5} c \,e^{3} a +\frac {1}{5} x^{5} b^{2} e^{3}+\frac {9}{5} x^{5} b c d \,e^{2}+\frac {6}{5} x^{5} c^{2} d^{2} e +\frac {1}{4} x^{4} a b \,e^{3}+\frac {3}{2} x^{4} a c d \,e^{2}+\frac {3}{4} x^{4} b^{2} d \,e^{2}+\frac {9}{4} x^{4} b c \,d^{2} e +\frac {1}{2} x^{4} c^{2} d^{3}+a b d \,e^{2} x^{3}+2 a c \,d^{2} e \,x^{3}+b^{2} d^{2} e \,x^{3}+b c \,d^{3} x^{3}+\frac {3}{2} x^{2} a b \,d^{2} e +x^{2} a c \,d^{3}+\frac {1}{2} x^{2} b^{2} d^{3}+b \,d^{3} a x\) | \(212\) |
| default | \(\frac {2 c^{2} e^{3} x^{7}}{7}+\frac {\left (\left (b \,e^{3}+6 c d \,e^{2}\right ) c +2 c \,e^{3} b \right ) x^{6}}{6}+\frac {\left (\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) c +\left (b \,e^{3}+6 c d \,e^{2}\right ) b +2 c \,e^{3} a \right ) x^{5}}{5}+\frac {\left (\left (3 b \,d^{2} e +2 c \,d^{3}\right ) c +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) b +\left (b \,e^{3}+6 c d \,e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (b \,d^{3} c +\left (3 b \,d^{2} e +2 c \,d^{3}\right ) b +\left (3 b d \,e^{2}+6 c \,d^{2} e \right ) a \right ) x^{3}}{3}+\frac {\left (b^{2} d^{3}+\left (3 b \,d^{2} e +2 c \,d^{3}\right ) a \right ) x^{2}}{2}+b \,d^{3} a x\) | \(221\) |
2/7*c^2*e^3*x^7+(1/2*c*e^3*b+c^2*d*e^2)*x^6+(2/5*c*e^3*a+1/5*b^2*e^3+9/5*b *c*d*e^2+6/5*c^2*d^2*e)*x^5+(1/4*a*b*e^3+3/2*a*c*d*e^2+3/4*b^2*d*e^2+9/4*b *c*d^2*e+1/2*c^2*d^3)*x^4+(a*b*d*e^2+2*a*c*d^2*e+b^2*d^2*e+b*c*d^3)*x^3+(3 /2*a*b*d^2*e+a*c*d^3+1/2*b^2*d^3)*x^2+b*d^3*a*x
Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {2}{7} \, c^{2} e^{3} x^{7} + \frac {1}{2} \, {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{6} + a b d^{3} x + \frac {1}{5} \, {\left (6 \, c^{2} d^{2} e + 9 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, c^{2} d^{3} + 9 \, b c d^{2} e + a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{4} + {\left (b c d^{3} + a b d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a b d^{2} e + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{2} \]
2/7*c^2*e^3*x^7 + 1/2*(2*c^2*d*e^2 + b*c*e^3)*x^6 + a*b*d^3*x + 1/5*(6*c^2 *d^2*e + 9*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x^5 + 1/4*(2*c^2*d^3 + 9*b*c*d^2 *e + a*b*e^3 + 3*(b^2 + 2*a*c)*d*e^2)*x^4 + (b*c*d^3 + a*b*d*e^2 + (b^2 + 2*a*c)*d^2*e)*x^3 + 1/2*(3*a*b*d^2*e + (b^2 + 2*a*c)*d^3)*x^2
Time = 0.03 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.70 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=a b d^{3} x + \frac {2 c^{2} e^{3} x^{7}}{7} + x^{6} \left (\frac {b c e^{3}}{2} + c^{2} d e^{2}\right ) + x^{5} \cdot \left (\frac {2 a c e^{3}}{5} + \frac {b^{2} e^{3}}{5} + \frac {9 b c d e^{2}}{5} + \frac {6 c^{2} d^{2} e}{5}\right ) + x^{4} \left (\frac {a b e^{3}}{4} + \frac {3 a c d e^{2}}{2} + \frac {3 b^{2} d e^{2}}{4} + \frac {9 b c d^{2} e}{4} + \frac {c^{2} d^{3}}{2}\right ) + x^{3} \left (a b d e^{2} + 2 a c d^{2} e + b^{2} d^{2} e + b c d^{3}\right ) + x^{2} \cdot \left (\frac {3 a b d^{2} e}{2} + a c d^{3} + \frac {b^{2} d^{3}}{2}\right ) \]
a*b*d**3*x + 2*c**2*e**3*x**7/7 + x**6*(b*c*e**3/2 + c**2*d*e**2) + x**5*( 2*a*c*e**3/5 + b**2*e**3/5 + 9*b*c*d*e**2/5 + 6*c**2*d**2*e/5) + x**4*(a*b *e**3/4 + 3*a*c*d*e**2/2 + 3*b**2*d*e**2/4 + 9*b*c*d**2*e/4 + c**2*d**3/2) + x**3*(a*b*d*e**2 + 2*a*c*d**2*e + b**2*d**2*e + b*c*d**3) + x**2*(3*a*b *d**2*e/2 + a*c*d**3 + b**2*d**3/2)
Time = 0.18 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {2}{7} \, c^{2} e^{3} x^{7} + \frac {1}{2} \, {\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{6} + a b d^{3} x + \frac {1}{5} \, {\left (6 \, c^{2} d^{2} e + 9 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, c^{2} d^{3} + 9 \, b c d^{2} e + a b e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x^{4} + {\left (b c d^{3} + a b d e^{2} + {\left (b^{2} + 2 \, a c\right )} d^{2} e\right )} x^{3} + \frac {1}{2} \, {\left (3 \, a b d^{2} e + {\left (b^{2} + 2 \, a c\right )} d^{3}\right )} x^{2} \]
2/7*c^2*e^3*x^7 + 1/2*(2*c^2*d*e^2 + b*c*e^3)*x^6 + a*b*d^3*x + 1/5*(6*c^2 *d^2*e + 9*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x^5 + 1/4*(2*c^2*d^3 + 9*b*c*d^2 *e + a*b*e^3 + 3*(b^2 + 2*a*c)*d*e^2)*x^4 + (b*c*d^3 + a*b*d*e^2 + (b^2 + 2*a*c)*d^2*e)*x^3 + 1/2*(3*a*b*d^2*e + (b^2 + 2*a*c)*d^3)*x^2
Time = 0.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.70 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {2}{7} \, c^{2} e^{3} x^{7} + c^{2} d e^{2} x^{6} + \frac {1}{2} \, b c e^{3} x^{6} + \frac {6}{5} \, c^{2} d^{2} e x^{5} + \frac {9}{5} \, b c d e^{2} x^{5} + \frac {1}{5} \, b^{2} e^{3} x^{5} + \frac {2}{5} \, a c e^{3} x^{5} + \frac {1}{2} \, c^{2} d^{3} x^{4} + \frac {9}{4} \, b c d^{2} e x^{4} + \frac {3}{4} \, b^{2} d e^{2} x^{4} + \frac {3}{2} \, a c d e^{2} x^{4} + \frac {1}{4} \, a b e^{3} x^{4} + b c d^{3} x^{3} + b^{2} d^{2} e x^{3} + 2 \, a c d^{2} e x^{3} + a b d e^{2} x^{3} + \frac {1}{2} \, b^{2} d^{3} x^{2} + a c d^{3} x^{2} + \frac {3}{2} \, a b d^{2} e x^{2} + a b d^{3} x \]
2/7*c^2*e^3*x^7 + c^2*d*e^2*x^6 + 1/2*b*c*e^3*x^6 + 6/5*c^2*d^2*e*x^5 + 9/ 5*b*c*d*e^2*x^5 + 1/5*b^2*e^3*x^5 + 2/5*a*c*e^3*x^5 + 1/2*c^2*d^3*x^4 + 9/ 4*b*c*d^2*e*x^4 + 3/4*b^2*d*e^2*x^4 + 3/2*a*c*d*e^2*x^4 + 1/4*a*b*e^3*x^4 + b*c*d^3*x^3 + b^2*d^2*e*x^3 + 2*a*c*d^2*e*x^3 + a*b*d*e^2*x^3 + 1/2*b^2* d^3*x^2 + a*c*d^3*x^2 + 3/2*a*b*d^2*e*x^2 + a*b*d^3*x
Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.44 \[ \int (b+2 c x) (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=x^4\,\left (\frac {3\,b^2\,d\,e^2}{4}+\frac {9\,b\,c\,d^2\,e}{4}+\frac {a\,b\,e^3}{4}+\frac {c^2\,d^3}{2}+\frac {3\,a\,c\,d\,e^2}{2}\right )+x^3\,\left (b^2\,d^2\,e+c\,b\,d^3+a\,b\,d\,e^2+2\,a\,c\,d^2\,e\right )+x^2\,\left (\frac {b^2\,d^3}{2}+\frac {3\,a\,e\,b\,d^2}{2}+a\,c\,d^3\right )+x^5\,\left (\frac {b^2\,e^3}{5}+\frac {9\,b\,c\,d\,e^2}{5}+\frac {6\,c^2\,d^2\,e}{5}+\frac {2\,a\,c\,e^3}{5}\right )+\frac {2\,c^2\,e^3\,x^7}{7}+\frac {c\,e^2\,x^6\,\left (b\,e+2\,c\,d\right )}{2}+a\,b\,d^3\,x \]
x^4*((c^2*d^3)/2 + (3*b^2*d*e^2)/4 + (a*b*e^3)/4 + (3*a*c*d*e^2)/2 + (9*b* c*d^2*e)/4) + x^3*(b^2*d^2*e + b*c*d^3 + a*b*d*e^2 + 2*a*c*d^2*e) + x^2*(( b^2*d^3)/2 + a*c*d^3 + (3*a*b*d^2*e)/2) + x^5*((b^2*e^3)/5 + (6*c^2*d^2*e) /5 + (2*a*c*e^3)/5 + (9*b*c*d*e^2)/5) + (2*c^2*e^3*x^7)/7 + (c*e^2*x^6*(b* e + 2*c*d))/2 + a*b*d^3*x