Integrand size = 18, antiderivative size = 69 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {\left (c d^2-b d e+a e^2\right ) (d+e x)^4}{4 e^3}-\frac {(2 c d-b e) (d+e x)^5}{5 e^3}+\frac {c (d+e x)^6}{6 e^3} \] Output:
1/4*(a*e^2-b*d*e+c*d^2)*(e*x+d)^4/e^3-1/5*(-b*e+2*c*d)*(e*x+d)^5/e^3+1/6*c *(e*x+d)^6/e^3
Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.51 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=a d^3 x+\frac {1}{2} d^2 (b d+3 a e) x^2+\frac {1}{3} d \left (c d^2+3 b d e+3 a e^2\right ) x^3+\frac {1}{4} e \left (3 c d^2+3 b d e+a e^2\right ) x^4+\frac {1}{5} e^2 (3 c d+b e) x^5+\frac {1}{6} c e^3 x^6 \] Input:
Integrate[(d + e*x)^3*(a + b*x + c*x^2),x]
Output:
a*d^3*x + (d^2*(b*d + 3*a*e)*x^2)/2 + (d*(c*d^2 + 3*b*d*e + 3*a*e^2)*x^3)/ 3 + (e*(3*c*d^2 + 3*b*d*e + a*e^2)*x^4)/4 + (e^2*(3*c*d + b*e)*x^5)/5 + (c *e^3*x^6)/6
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1140, 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.
\(\displaystyle \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right )}{e^2}+\frac {(d+e x)^4 (b e-2 c d)}{e^2}+\frac {c (d+e x)^5}{e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right )}{4 e^3}-\frac {(d+e x)^5 (2 c d-b e)}{5 e^3}+\frac {c (d+e x)^6}{6 e^3}\) |
Input:
Int[(d + e*x)^3*(a + b*x + c*x^2),x]
Output:
((c*d^2 - b*d*e + a*e^2)*(d + e*x)^4)/(4*e^3) - ((2*c*d - b*e)*(d + e*x)^5 )/(5*e^3) + (c*(d + e*x)^6)/(6*e^3)
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.47 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.46
method | result | size |
norman | \(\frac {c \,e^{3} x^{6}}{6}+\left (\frac {1}{5} b \,e^{3}+\frac {3}{5} c d \,e^{2}\right ) x^{5}+\left (\frac {1}{4} a \,e^{3}+\frac {3}{4} b d \,e^{2}+\frac {3}{4} c \,d^{2} e \right ) x^{4}+\left (a d \,e^{2}+b \,d^{2} e +\frac {1}{3} c \,d^{3}\right ) x^{3}+\left (\frac {3}{2} a \,d^{2} e +\frac {1}{2} b \,d^{3}\right ) x^{2}+a x \,d^{3}\) | \(101\) |
default | \(\frac {c \,e^{3} x^{6}}{6}+\frac {\left (b \,e^{3}+3 c d \,e^{2}\right ) x^{5}}{5}+\frac {\left (a \,e^{3}+3 b d \,e^{2}+3 c \,d^{2} e \right ) x^{4}}{4}+\frac {\left (3 a d \,e^{2}+3 b \,d^{2} e +c \,d^{3}\right ) x^{3}}{3}+\frac {\left (3 a \,d^{2} e +b \,d^{3}\right ) x^{2}}{2}+a x \,d^{3}\) | \(103\) |
gosper | \(\frac {1}{6} c \,e^{3} x^{6}+\frac {1}{5} x^{5} b \,e^{3}+\frac {3}{5} d \,e^{2} c \,x^{5}+\frac {1}{4} x^{4} a \,e^{3}+\frac {3}{4} x^{4} b d \,e^{2}+\frac {3}{4} x^{4} c \,d^{2} e +x^{3} a d \,e^{2}+x^{3} b \,d^{2} e +\frac {1}{3} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e a \,x^{2}+\frac {1}{2} b \,d^{3} x^{2}+a x \,d^{3}\) | \(111\) |
risch | \(\frac {1}{6} c \,e^{3} x^{6}+\frac {1}{5} x^{5} b \,e^{3}+\frac {3}{5} d \,e^{2} c \,x^{5}+\frac {1}{4} x^{4} a \,e^{3}+\frac {3}{4} x^{4} b d \,e^{2}+\frac {3}{4} x^{4} c \,d^{2} e +x^{3} a d \,e^{2}+x^{3} b \,d^{2} e +\frac {1}{3} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e a \,x^{2}+\frac {1}{2} b \,d^{3} x^{2}+a x \,d^{3}\) | \(111\) |
parallelrisch | \(\frac {1}{6} c \,e^{3} x^{6}+\frac {1}{5} x^{5} b \,e^{3}+\frac {3}{5} d \,e^{2} c \,x^{5}+\frac {1}{4} x^{4} a \,e^{3}+\frac {3}{4} x^{4} b d \,e^{2}+\frac {3}{4} x^{4} c \,d^{2} e +x^{3} a d \,e^{2}+x^{3} b \,d^{2} e +\frac {1}{3} c \,d^{3} x^{3}+\frac {3}{2} d^{2} e a \,x^{2}+\frac {1}{2} b \,d^{3} x^{2}+a x \,d^{3}\) | \(111\) |
orering | \(\frac {x \left (10 e^{3} c \,x^{5}+12 b \,e^{3} x^{4}+36 c d \,e^{2} x^{4}+15 x^{3} a \,e^{3}+45 x^{3} b d \,e^{2}+45 c \,d^{2} e \,x^{3}+60 a d \,e^{2} x^{2}+60 b \,d^{2} e \,x^{2}+20 d^{3} c \,x^{2}+90 x a \,d^{2} e +30 b \,d^{3} x +60 a \,d^{3}\right )}{60}\) | \(112\) |
Input:
int((e*x+d)^3*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
1/6*c*e^3*x^6+(1/5*b*e^3+3/5*c*d*e^2)*x^5+(1/4*a*e^3+3/4*b*d*e^2+3/4*c*d^2 *e)*x^4+(a*d*e^2+b*d^2*e+1/3*c*d^3)*x^3+(3/2*a*d^2*e+1/2*b*d^3)*x^2+a*x*d^ 3
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.48 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{6} \, c e^{3} x^{6} + \frac {1}{5} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + a d^{3} x + \frac {1}{4} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a),x, algorithm="fricas")
Output:
1/6*c*e^3*x^6 + 1/5*(3*c*d*e^2 + b*e^3)*x^5 + a*d^3*x + 1/4*(3*c*d^2*e + 3 *b*d*e^2 + a*e^3)*x^4 + 1/3*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*x^3 + 1/2*(b*d ^3 + 3*a*d^2*e)*x^2
Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=a d^{3} x + \frac {c e^{3} x^{6}}{6} + x^{5} \left (\frac {b e^{3}}{5} + \frac {3 c d e^{2}}{5}\right ) + x^{4} \left (\frac {a e^{3}}{4} + \frac {3 b d e^{2}}{4} + \frac {3 c d^{2} e}{4}\right ) + x^{3} \left (a d e^{2} + b d^{2} e + \frac {c d^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 a d^{2} e}{2} + \frac {b d^{3}}{2}\right ) \] Input:
integrate((e*x+d)**3*(c*x**2+b*x+a),x)
Output:
a*d**3*x + c*e**3*x**6/6 + x**5*(b*e**3/5 + 3*c*d*e**2/5) + x**4*(a*e**3/4 + 3*b*d*e**2/4 + 3*c*d**2*e/4) + x**3*(a*d*e**2 + b*d**2*e + c*d**3/3) + x**2*(3*a*d**2*e/2 + b*d**3/2)
Time = 0.04 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.48 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{6} \, c e^{3} x^{6} + \frac {1}{5} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{5} + a d^{3} x + \frac {1}{4} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a),x, algorithm="maxima")
Output:
1/6*c*e^3*x^6 + 1/5*(3*c*d*e^2 + b*e^3)*x^5 + a*d^3*x + 1/4*(3*c*d^2*e + 3 *b*d*e^2 + a*e^3)*x^4 + 1/3*(c*d^3 + 3*b*d^2*e + 3*a*d*e^2)*x^3 + 1/2*(b*d ^3 + 3*a*d^2*e)*x^2
Time = 0.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {1}{6} \, c e^{3} x^{6} + \frac {3}{5} \, c d e^{2} x^{5} + \frac {1}{5} \, b e^{3} x^{5} + \frac {3}{4} \, c d^{2} e x^{4} + \frac {3}{4} \, b d e^{2} x^{4} + \frac {1}{4} \, a e^{3} x^{4} + \frac {1}{3} \, c d^{3} x^{3} + b d^{2} e x^{3} + a d e^{2} x^{3} + \frac {1}{2} \, b d^{3} x^{2} + \frac {3}{2} \, a d^{2} e x^{2} + a d^{3} x \] Input:
integrate((e*x+d)^3*(c*x^2+b*x+a),x, algorithm="giac")
Output:
1/6*c*e^3*x^6 + 3/5*c*d*e^2*x^5 + 1/5*b*e^3*x^5 + 3/4*c*d^2*e*x^4 + 3/4*b* d*e^2*x^4 + 1/4*a*e^3*x^4 + 1/3*c*d^3*x^3 + b*d^2*e*x^3 + a*d*e^2*x^3 + 1/ 2*b*d^3*x^2 + 3/2*a*d^2*e*x^2 + a*d^3*x
Time = 5.42 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.45 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=x^2\,\left (\frac {b\,d^3}{2}+\frac {3\,a\,e\,d^2}{2}\right )+x^5\,\left (\frac {b\,e^3}{5}+\frac {3\,c\,d\,e^2}{5}\right )+x^3\,\left (\frac {c\,d^3}{3}+b\,d^2\,e+a\,d\,e^2\right )+x^4\,\left (\frac {3\,c\,d^2\,e}{4}+\frac {3\,b\,d\,e^2}{4}+\frac {a\,e^3}{4}\right )+\frac {c\,e^3\,x^6}{6}+a\,d^3\,x \] Input:
int((d + e*x)^3*(a + b*x + c*x^2),x)
Output:
x^2*((b*d^3)/2 + (3*a*d^2*e)/2) + x^5*((b*e^3)/5 + (3*c*d*e^2)/5) + x^3*(( c*d^3)/3 + a*d*e^2 + b*d^2*e) + x^4*((a*e^3)/4 + (3*b*d*e^2)/4 + (3*c*d^2* e)/4) + (c*e^3*x^6)/6 + a*d^3*x
Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.61 \[ \int (d+e x)^3 \left (a+b x+c x^2\right ) \, dx=\frac {x \left (10 c \,e^{3} x^{5}+12 b \,e^{3} x^{4}+36 c d \,e^{2} x^{4}+15 a \,e^{3} x^{3}+45 b d \,e^{2} x^{3}+45 c \,d^{2} e \,x^{3}+60 a d \,e^{2} x^{2}+60 b \,d^{2} e \,x^{2}+20 c \,d^{3} x^{2}+90 a \,d^{2} e x +30 b \,d^{3} x +60 a \,d^{3}\right )}{60} \] Input:
int((e*x+d)^3*(c*x^2+b*x+a),x)
Output:
(x*(60*a*d**3 + 90*a*d**2*e*x + 60*a*d*e**2*x**2 + 15*a*e**3*x**3 + 30*b*d **3*x + 60*b*d**2*e*x**2 + 45*b*d*e**2*x**3 + 12*b*e**3*x**4 + 20*c*d**3*x **2 + 45*c*d**2*e*x**3 + 36*c*d*e**2*x**4 + 10*c*e**3*x**5))/60