Integrand size = 22, antiderivative size = 100 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} A b^2 d x^3+\frac {1}{4} b (b B d+2 A c d+A b e) x^4+\frac {1}{5} \left (A c^2 d+b^2 B e+2 b c (B d+A e)\right ) x^5+\frac {1}{6} c (B c d+2 b B e+A c e) x^6+\frac {1}{7} B c^2 e x^7 \] Output:
1/3*A*b^2*d*x^3+1/4*b*(A*b*e+2*A*c*d+B*b*d)*x^4+1/5*(A*c^2*d+b^2*B*e+2*b*c *(A*e+B*d))*x^5+1/6*c*(A*c*e+2*B*b*e+B*c*d)*x^6+1/7*B*c^2*e*x^7
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{3} A b^2 d x^3+\frac {1}{4} b (b B d+2 A c d+A b e) x^4+\frac {1}{5} \left (2 b B c d+A c^2 d+b^2 B e+2 A b c e\right ) x^5+\frac {1}{6} c (B c d+2 b B e+A c e) x^6+\frac {1}{7} B c^2 e x^7 \] Input:
Integrate[(A + B*x)*(d + e*x)*(b*x + c*x^2)^2,x]
Output:
(A*b^2*d*x^3)/3 + (b*(b*B*d + 2*A*c*d + A*b*e)*x^4)/4 + ((2*b*B*c*d + A*c^ 2*d + b^2*B*e + 2*A*b*c*e)*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c^2*e*x^7)/7
Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int (A+B x) \left (b x+c x^2\right )^2 (d+e x) \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (x^4 \left (2 b c (A e+B d)+A c^2 d+b^2 B e\right )+A b^2 d x^2+c x^5 (A c e+2 b B e+B c d)+b x^3 (A b e+2 A c d+b B d)+B c^2 e x^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} x^5 \left (2 b c (A e+B d)+A c^2 d+b^2 B e\right )+\frac {1}{3} A b^2 d x^3+\frac {1}{6} c x^6 (A c e+2 b B e+B c d)+\frac {1}{4} b x^4 (A b e+2 A c d+b B d)+\frac {1}{7} B c^2 e x^7\) |
Input:
Int[(A + B*x)*(d + e*x)*(b*x + c*x^2)^2,x]
Output:
(A*b^2*d*x^3)/3 + (b*(b*B*d + 2*A*c*d + A*b*e)*x^4)/4 + ((A*c^2*d + b^2*B* e + 2*b*c*(B*d + A*e))*x^5)/5 + (c*(B*c*d + 2*b*B*e + A*c*e)*x^6)/6 + (B*c ^2*e*x^7)/7
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 = 0.56 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {B \,c^{2} e \,x^{7}}{7}+\frac {\left (\left (A e +B d \right ) c^{2}+2 B e b c \right ) x^{6}}{6}+\frac {\left (A \,c^{2} d +b^{2} B e +2 b c \left (A e +B d \right )\right ) x^{5}}{5}+\frac {\left (2 A b c d +\left (A e +B d \right ) b^{2}\right ) x^{4}}{4}+\frac {A \,b^{2} d \,x^{3}}{3}\) | \(97\) |
norman | \(\frac {B \,c^{2} e \,x^{7}}{7}+\left (\frac {1}{6} A \,c^{2} e +\frac {1}{3} B e b c +\frac {1}{6} B \,c^{2} d \right ) x^{6}+\left (\frac {2}{5} A b c e +\frac {1}{5} A \,c^{2} d +\frac {1}{5} b^{2} B e +\frac {2}{5} B b c d \right ) x^{5}+\left (\frac {1}{4} A \,b^{2} e +\frac {1}{2} A b c d +\frac {1}{4} B \,b^{2} d \right ) x^{4}+\frac {A \,b^{2} d \,x^{3}}{3}\) | \(103\) |
gosper | \(\frac {x^{3} \left (60 B \,c^{2} e \,x^{4}+70 x^{3} A \,c^{2} e +140 x^{3} B e b c +70 x^{3} B \,c^{2} d +168 x^{2} A b c e +84 A \,c^{2} d \,x^{2}+84 x^{2} b^{2} B e +168 x^{2} B b c d +105 x A \,b^{2} e +210 A b c d x +105 x B \,b^{2} d +140 A d \,b^{2}\right )}{420}\) | \(114\) |
risch | \(\frac {1}{7} B \,c^{2} e \,x^{7}+\frac {1}{6} x^{6} A \,c^{2} e +\frac {1}{3} x^{6} B e b c +\frac {1}{6} x^{6} B \,c^{2} d +\frac {2}{5} x^{5} A b c e +\frac {1}{5} x^{5} A \,c^{2} d +\frac {1}{5} x^{5} b^{2} B e +\frac {2}{5} x^{5} B b c d +\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} A b c d +\frac {1}{4} B \,b^{2} d \,x^{4}+\frac {1}{3} A \,b^{2} d \,x^{3}\) | \(118\) |
parallelrisch | \(\frac {1}{7} B \,c^{2} e \,x^{7}+\frac {1}{6} x^{6} A \,c^{2} e +\frac {1}{3} x^{6} B e b c +\frac {1}{6} x^{6} B \,c^{2} d +\frac {2}{5} x^{5} A b c e +\frac {1}{5} x^{5} A \,c^{2} d +\frac {1}{5} x^{5} b^{2} B e +\frac {2}{5} x^{5} B b c d +\frac {1}{4} x^{4} A \,b^{2} e +\frac {1}{2} x^{4} A b c d +\frac {1}{4} B \,b^{2} d \,x^{4}+\frac {1}{3} A \,b^{2} d \,x^{3}\) | \(118\) |
orering | \(\frac {x \left (60 B \,c^{2} e \,x^{4}+70 x^{3} A \,c^{2} e +140 x^{3} B e b c +70 x^{3} B \,c^{2} d +168 x^{2} A b c e +84 A \,c^{2} d \,x^{2}+84 x^{2} b^{2} B e +168 x^{2} B b c d +105 x A \,b^{2} e +210 A b c d x +105 x B \,b^{2} d +140 A d \,b^{2}\right ) \left (c \,x^{2}+b x \right )^{2}}{420 \left (c x +b \right )^{2}}\) | \(130\) |
Input:
int((B*x+A)*(e*x+d)*(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
Output:
1/7*B*c^2*e*x^7+1/6*((A*e+B*d)*c^2+2*B*e*b*c)*x^6+1/5*(A*c^2*d+b^2*B*e+2*b *c*(A*e+B*d))*x^5+1/4*(2*A*b*c*d+(A*e+B*d)*b^2)*x^4+1/3*A*b^2*d*x^3
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.03 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, B c^{2} e x^{7} + \frac {1}{3} \, A b^{2} d x^{3} + \frac {1}{6} \, {\left (B c^{2} d + {\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d + {\left (B b^{2} + 2 \, A b c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left (A b^{2} e + {\left (B b^{2} + 2 \, A b c\right )} d\right )} x^{4} \] Input:
integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^2,x, algorithm="fricas")
Output:
1/7*B*c^2*e*x^7 + 1/3*A*b^2*d*x^3 + 1/6*(B*c^2*d + (2*B*b*c + A*c^2)*e)*x^ 6 + 1/5*((2*B*b*c + A*c^2)*d + (B*b^2 + 2*A*b*c)*e)*x^5 + 1/4*(A*b^2*e + ( B*b^2 + 2*A*b*c)*d)*x^4
Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.21 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {A b^{2} d x^{3}}{3} + \frac {B c^{2} e x^{7}}{7} + x^{6} \left (\frac {A c^{2} e}{6} + \frac {B b c e}{3} + \frac {B c^{2} d}{6}\right ) + x^{5} \cdot \left (\frac {2 A b c e}{5} + \frac {A c^{2} d}{5} + \frac {B b^{2} e}{5} + \frac {2 B b c d}{5}\right ) + x^{4} \left (\frac {A b^{2} e}{4} + \frac {A b c d}{2} + \frac {B b^{2} d}{4}\right ) \] Input:
integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**2,x)
Output:
A*b**2*d*x**3/3 + B*c**2*e*x**7/7 + x**6*(A*c**2*e/6 + B*b*c*e/3 + B*c**2* d/6) + x**5*(2*A*b*c*e/5 + A*c**2*d/5 + B*b**2*e/5 + 2*B*b*c*d/5) + x**4*( A*b**2*e/4 + A*b*c*d/2 + B*b**2*d/4)
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.03 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, B c^{2} e x^{7} + \frac {1}{3} \, A b^{2} d x^{3} + \frac {1}{6} \, {\left (B c^{2} d + {\left (2 \, B b c + A c^{2}\right )} e\right )} x^{6} + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d + {\left (B b^{2} + 2 \, A b c\right )} e\right )} x^{5} + \frac {1}{4} \, {\left (A b^{2} e + {\left (B b^{2} + 2 \, A b c\right )} d\right )} x^{4} \] Input:
integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^2,x, algorithm="maxima")
Output:
1/7*B*c^2*e*x^7 + 1/3*A*b^2*d*x^3 + 1/6*(B*c^2*d + (2*B*b*c + A*c^2)*e)*x^ 6 + 1/5*((2*B*b*c + A*c^2)*d + (B*b^2 + 2*A*b*c)*e)*x^5 + 1/4*(A*b^2*e + ( B*b^2 + 2*A*b*c)*d)*x^4
Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.17 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {1}{7} \, B c^{2} e x^{7} + \frac {1}{6} \, B c^{2} d x^{6} + \frac {1}{3} \, B b c e x^{6} + \frac {1}{6} \, A c^{2} e x^{6} + \frac {2}{5} \, B b c d x^{5} + \frac {1}{5} \, A c^{2} d x^{5} + \frac {1}{5} \, B b^{2} e x^{5} + \frac {2}{5} \, A b c e x^{5} + \frac {1}{4} \, B b^{2} d x^{4} + \frac {1}{2} \, A b c d x^{4} + \frac {1}{4} \, A b^{2} e x^{4} + \frac {1}{3} \, A b^{2} d x^{3} \] Input:
integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^2,x, algorithm="giac")
Output:
1/7*B*c^2*e*x^7 + 1/6*B*c^2*d*x^6 + 1/3*B*b*c*e*x^6 + 1/6*A*c^2*e*x^6 + 2/ 5*B*b*c*d*x^5 + 1/5*A*c^2*d*x^5 + 1/5*B*b^2*e*x^5 + 2/5*A*b*c*e*x^5 + 1/4* B*b^2*d*x^4 + 1/2*A*b*c*d*x^4 + 1/4*A*b^2*e*x^4 + 1/3*A*b^2*d*x^3
Time = 10.61 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx=x^5\,\left (\frac {A\,c^2\,d}{5}+\frac {B\,b^2\,e}{5}+\frac {2\,A\,b\,c\,e}{5}+\frac {2\,B\,b\,c\,d}{5}\right )+x^4\,\left (\frac {A\,b^2\,e}{4}+\frac {B\,b^2\,d}{4}+\frac {A\,b\,c\,d}{2}\right )+x^6\,\left (\frac {A\,c^2\,e}{6}+\frac {B\,c^2\,d}{6}+\frac {B\,b\,c\,e}{3}\right )+\frac {A\,b^2\,d\,x^3}{3}+\frac {B\,c^2\,e\,x^7}{7} \] Input:
int((b*x + c*x^2)^2*(A + B*x)*(d + e*x),x)
Output:
x^5*((A*c^2*d)/5 + (B*b^2*e)/5 + (2*A*b*c*e)/5 + (2*B*b*c*d)/5) + x^4*((A* b^2*e)/4 + (B*b^2*d)/4 + (A*b*c*d)/2) + x^6*((A*c^2*e)/6 + (B*c^2*d)/6 + ( B*b*c*e)/3) + (A*b^2*d*x^3)/3 + (B*c^2*e*x^7)/7
Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.13 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^2 \, dx=\frac {x^{3} \left (60 b \,c^{2} e \,x^{4}+70 a \,c^{2} e \,x^{3}+140 b^{2} c e \,x^{3}+70 b \,c^{2} d \,x^{3}+168 a b c e \,x^{2}+84 a \,c^{2} d \,x^{2}+84 b^{3} e \,x^{2}+168 b^{2} c d \,x^{2}+105 a \,b^{2} e x +210 a b c d x +105 b^{3} d x +140 a \,b^{2} d \right )}{420} \] Input:
int((B*x+A)*(e*x+d)*(c*x^2+b*x)^2,x)
Output:
(x**3*(140*a*b**2*d + 105*a*b**2*e*x + 210*a*b*c*d*x + 168*a*b*c*e*x**2 + 84*a*c**2*d*x**2 + 70*a*c**2*e*x**3 + 105*b**3*d*x + 84*b**3*e*x**2 + 168* b**2*c*d*x**2 + 140*b**2*c*e*x**3 + 70*b*c**2*d*x**3 + 60*b*c**2*e*x**4))/ 420