Integrand size = 18, antiderivative size = 96 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 d x+\frac {1}{2} a (2 b d+a e) x^2+\frac {1}{3} \left (b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{4} \left (2 b c d+b^2 e+2 a c e\right ) x^4+\frac {1}{5} c (c d+2 b e) x^5+\frac {1}{6} c^2 e x^6 \] Output:
a^2*d*x+1/2*a*(a*e+2*b*d)*x^2+1/3*(2*a*b*e+2*a*c*d+b^2*d)*x^3+1/4*(2*a*c*e +b^2*e+2*b*c*d)*x^4+1/5*c*(2*b*e+c*d)*x^5+1/6*c^2*e*x^6
Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^2 d x+\frac {1}{2} a (2 b d+a e) x^2+\frac {1}{3} \left (b^2 d+2 a c d+2 a b e\right ) x^3+\frac {1}{4} \left (2 b c d+b^2 e+2 a c e\right ) x^4+\frac {1}{5} c (c d+2 b e) x^5+\frac {1}{6} c^2 e x^6 \] Input:
Integrate[(d + e*x)*(a + b*x + c*x^2)^2,x]
Output:
a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*c*d + b^2*e + 2*a*c*e)*x^4)/4 + (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6 )/6
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
\(\displaystyle \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (a^2 d+x^3 \left (2 a c e+b^2 e+2 b c d\right )+x^2 \left (2 a b e+2 a c d+b^2 d\right )+a x (a e+2 b d)+c x^4 (2 b e+c d)+c^2 e x^5\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 d x+\frac {1}{4} x^4 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{3} x^3 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{2} a x^2 (a e+2 b d)+\frac {1}{5} c x^5 (2 b e+c d)+\frac {1}{6} c^2 e x^6\) |
Input:
Int[(d + e*x)*(a + b*x + c*x^2)^2,x]
Output:
a^2*d*x + (a*(2*b*d + a*e)*x^2)/2 + ((b^2*d + 2*a*c*d + 2*a*b*e)*x^3)/3 + ((2*b*c*d + b^2*e + 2*a*c*e)*x^4)/4 + (c*(c*d + 2*b*e)*x^5)/5 + (c^2*e*x^6 )/6
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 = 0.79 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94
method | result | size |
norman | \(\frac {c^{2} e \,x^{6}}{6}+\left (\frac {2}{5} b c e +\frac {1}{5} c^{2} d \right ) x^{5}+\left (\frac {1}{2} a c e +\frac {1}{4} e \,b^{2}+\frac {1}{2} d b c \right ) x^{4}+\left (\frac {2}{3} a b e +\frac {2}{3} a c d +\frac {1}{3} b^{2} d \right ) x^{3}+\left (\frac {1}{2} a^{2} e +d a b \right ) x^{2}+a^{2} d x\) | \(90\) |
default | \(\frac {c^{2} e \,x^{6}}{6}+\frac {\left (2 b c e +c^{2} d \right ) x^{5}}{5}+\frac {\left (2 d b c +e \left (2 a c +b^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (2 a c +b^{2}\right ) d +2 a b e \right ) x^{3}}{3}+\frac {\left (a^{2} e +2 d a b \right ) x^{2}}{2}+a^{2} d x\) | \(91\) |
gosper | \(\frac {1}{6} c^{2} e \,x^{6}+\frac {2}{5} x^{5} b c e +\frac {1}{5} c^{2} d \,x^{5}+\frac {1}{2} a c e \,x^{4}+\frac {1}{4} b^{2} e \,x^{4}+\frac {1}{2} d b c \,x^{4}+\frac {2}{3} a b e \,x^{3}+\frac {2}{3} a c d \,x^{3}+\frac {1}{3} b^{2} d \,x^{3}+\frac {1}{2} a^{2} e \,x^{2}+a b d \,x^{2}+a^{2} d x\) | \(100\) |
risch | \(\frac {1}{6} c^{2} e \,x^{6}+\frac {2}{5} x^{5} b c e +\frac {1}{5} c^{2} d \,x^{5}+\frac {1}{2} a c e \,x^{4}+\frac {1}{4} b^{2} e \,x^{4}+\frac {1}{2} d b c \,x^{4}+\frac {2}{3} a b e \,x^{3}+\frac {2}{3} a c d \,x^{3}+\frac {1}{3} b^{2} d \,x^{3}+\frac {1}{2} a^{2} e \,x^{2}+a b d \,x^{2}+a^{2} d x\) | \(100\) |
parallelrisch | \(\frac {1}{6} c^{2} e \,x^{6}+\frac {2}{5} x^{5} b c e +\frac {1}{5} c^{2} d \,x^{5}+\frac {1}{2} a c e \,x^{4}+\frac {1}{4} b^{2} e \,x^{4}+\frac {1}{2} d b c \,x^{4}+\frac {2}{3} a b e \,x^{3}+\frac {2}{3} a c d \,x^{3}+\frac {1}{3} b^{2} d \,x^{3}+\frac {1}{2} a^{2} e \,x^{2}+a b d \,x^{2}+a^{2} d x\) | \(100\) |
orering | \(\frac {x \left (10 e \,c^{2} x^{5}+24 b c e \,x^{4}+12 c^{2} d \,x^{4}+30 a c e \,x^{3}+15 b^{2} e \,x^{3}+30 b c d \,x^{3}+40 a \,x^{2} e b +40 a d \,x^{2} c +20 b^{2} d \,x^{2}+30 a^{2} e x +60 a b d x +60 a^{2} d \right )}{60}\) | \(100\) |
Input:
int((e*x+d)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/6*c^2*e*x^6+(2/5*b*c*e+1/5*c^2*d)*x^5+(1/2*a*c*e+1/4*e*b^2+1/2*d*b*c)*x^ 4+(2/3*a*b*e+2/3*a*c*d+1/3*b^2*d)*x^3+(1/2*a^2*e+d*a*b)*x^2+a^2*d*x
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{5} \, {\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
1/6*c^2*e*x^6 + 1/5*(c^2*d + 2*b*c*e)*x^5 + 1/4*(2*b*c*d + (b^2 + 2*a*c)*e )*x^4 + a^2*d*x + 1/3*(2*a*b*e + (b^2 + 2*a*c)*d)*x^3 + 1/2*(2*a*b*d + a^2 *e)*x^2
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=a^{2} d x + \frac {c^{2} e x^{6}}{6} + x^{5} \cdot \left (\frac {2 b c e}{5} + \frac {c^{2} d}{5}\right ) + x^{4} \left (\frac {a c e}{2} + \frac {b^{2} e}{4} + \frac {b c d}{2}\right ) + x^{3} \cdot \left (\frac {2 a b e}{3} + \frac {2 a c d}{3} + \frac {b^{2} d}{3}\right ) + x^{2} \left (\frac {a^{2} e}{2} + a b d\right ) \] Input:
integrate((e*x+d)*(c*x**2+b*x+a)**2,x)
Output:
a**2*d*x + c**2*e*x**6/6 + x**5*(2*b*c*e/5 + c**2*d/5) + x**4*(a*c*e/2 + b **2*e/4 + b*c*d/2) + x**3*(2*a*b*e/3 + 2*a*c*d/3 + b**2*d/3) + x**2*(a**2* e/2 + a*b*d)
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{5} \, {\left (c^{2} d + 2 \, b c e\right )} x^{5} + \frac {1}{4} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{3} + \frac {1}{2} \, {\left (2 \, a b d + a^{2} e\right )} x^{2} \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
1/6*c^2*e*x^6 + 1/5*(c^2*d + 2*b*c*e)*x^5 + 1/4*(2*b*c*d + (b^2 + 2*a*c)*e )*x^4 + a^2*d*x + 1/3*(2*a*b*e + (b^2 + 2*a*c)*d)*x^3 + 1/2*(2*a*b*d + a^2 *e)*x^2
Time = 0.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{6} \, c^{2} e x^{6} + \frac {1}{5} \, c^{2} d x^{5} + \frac {2}{5} \, b c e x^{5} + \frac {1}{2} \, b c d x^{4} + \frac {1}{4} \, b^{2} e x^{4} + \frac {1}{2} \, a c e x^{4} + \frac {1}{3} \, b^{2} d x^{3} + \frac {2}{3} \, a c d x^{3} + \frac {2}{3} \, a b e x^{3} + a b d x^{2} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x \] Input:
integrate((e*x+d)*(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
1/6*c^2*e*x^6 + 1/5*c^2*d*x^5 + 2/5*b*c*e*x^5 + 1/2*b*c*d*x^4 + 1/4*b^2*e* x^4 + 1/2*a*c*e*x^4 + 1/3*b^2*d*x^3 + 2/3*a*c*d*x^3 + 2/3*a*b*e*x^3 + a*b* d*x^2 + 1/2*a^2*e*x^2 + a^2*d*x
Time = 6.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=x^3\,\left (\frac {d\,b^2}{3}+\frac {2\,a\,e\,b}{3}+\frac {2\,a\,c\,d}{3}\right )+x^4\,\left (\frac {e\,b^2}{4}+\frac {c\,d\,b}{2}+\frac {a\,c\,e}{2}\right )+x^2\,\left (\frac {e\,a^2}{2}+b\,d\,a\right )+x^5\,\left (\frac {d\,c^2}{5}+\frac {2\,b\,e\,c}{5}\right )+\frac {c^2\,e\,x^6}{6}+a^2\,d\,x \] Input:
int((d + e*x)*(a + b*x + c*x^2)^2,x)
Output:
x^3*((b^2*d)/3 + (2*a*b*e)/3 + (2*a*c*d)/3) + x^4*((b^2*e)/4 + (a*c*e)/2 + (b*c*d)/2) + x^2*((a^2*e)/2 + a*b*d) + x^5*((c^2*d)/5 + (2*b*c*e)/5) + (c ^2*e*x^6)/6 + a^2*d*x
Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (a+b x+c x^2\right )^2 \, dx=\frac {x \left (10 c^{2} e \,x^{5}+24 b c e \,x^{4}+12 c^{2} d \,x^{4}+30 a c e \,x^{3}+15 b^{2} e \,x^{3}+30 b c d \,x^{3}+40 a b e \,x^{2}+40 a c d \,x^{2}+20 b^{2} d \,x^{2}+30 a^{2} e x +60 a b d x +60 a^{2} d \right )}{60} \] Input:
int((e*x+d)*(c*x^2+b*x+a)^2,x)
Output:
(x*(60*a**2*d + 30*a**2*e*x + 60*a*b*d*x + 40*a*b*e*x**2 + 40*a*c*d*x**2 + 30*a*c*e*x**3 + 20*b**2*d*x**2 + 15*b**2*e*x**3 + 30*b*c*d*x**3 + 24*b*c* e*x**4 + 12*c**2*d*x**4 + 10*c**2*e*x**5))/60