\(\int (A+B x) (d+e x) (b x+c x^2)^3 \, dx\) [25]

Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d x^4+\frac {1}{5} b^2 (b B d+3 A c d+A b e) x^5+\frac {1}{6} b \left (3 A c^2 d+b^2 B e+3 b c (B d+A e)\right ) x^6+\frac {1}{7} c \left (A c^2 d+3 b^2 B e+3 b c (B d+A e)\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \] Output:

1/4*A*b^3*d*x^4+1/5*b^2*(A*b*e+3*A*c*d+B*b*d)*x^5+1/6*b*(3*A*c^2*d+b^2*B*e 
+3*b*c*(A*e+B*d))*x^6+1/7*c*(A*c^2*d+3*b^2*B*e+3*b*c*(A*e+B*d))*x^7+1/8*c^ 
2*(A*c*e+3*B*b*e+B*c*d)*x^8+1/9*B*c^3*e*x^9
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{4} A b^3 d x^4+\frac {1}{5} b^2 (b B d+3 A c d+A b e) x^5+\frac {1}{6} b \left (3 b B c d+3 A c^2 d+b^2 B e+3 A b c e\right ) x^6+\frac {1}{7} c \left (3 b B c d+A c^2 d+3 b^2 B e+3 A b c e\right ) x^7+\frac {1}{8} c^2 (B c d+3 b B e+A c e) x^8+\frac {1}{9} B c^3 e x^9 \] Input:

Integrate[(A + B*x)*(d + e*x)*(b*x + c*x^2)^3,x]
 

Output:

(A*b^3*d*x^4)/4 + (b^2*(b*B*d + 3*A*c*d + A*b*e)*x^5)/5 + (b*(3*b*B*c*d + 
3*A*c^2*d + b^2*B*e + 3*A*b*c*e)*x^6)/6 + (c*(3*b*B*c*d + A*c^2*d + 3*b^2* 
B*e + 3*A*b*c*e)*x^7)/7 + (c^2*(B*c*d + 3*b*B*e + A*c*e)*x^8)/8 + (B*c^3*e 
*x^9)/9
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 139, 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.

\(\displaystyle \int (A+B x) \left (b x+c x^2\right )^3 (d+e x) \, dx\)

\(\Big \downarrow \) 1195

\(\displaystyle \int \left (A b^3 d x^3+c x^6 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+b x^5 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+b^2 x^4 (A b e+3 A c d+b B d)+c^2 x^7 (A c e+3 b B e+B c d)+B c^3 e x^8\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} A b^3 d x^4+\frac {1}{7} c x^7 \left (3 b c (A e+B d)+A c^2 d+3 b^2 B e\right )+\frac {1}{6} b x^6 \left (3 b c (A e+B d)+3 A c^2 d+b^2 B e\right )+\frac {1}{5} b^2 x^5 (A b e+3 A c d+b B d)+\frac {1}{8} c^2 x^8 (A c e+3 b B e+B c d)+\frac {1}{9} B c^3 e x^9\)

Input:

Int[(A + B*x)*(d + e*x)*(b*x + c*x^2)^3,x]
 

Output:

(A*b^3*d*x^4)/4 + (b^2*(b*B*d + 3*A*c*d + A*b*e)*x^5)/5 + (b*(3*A*c^2*d + 
b^2*B*e + 3*b*c*(B*d + A*e))*x^6)/6 + (c*(A*c^2*d + 3*b^2*B*e + 3*b*c*(B*d 
 + A*e))*x^7)/7 + (c^2*(B*c*d + 3*b*B*e + A*c*e)*x^8)/8 + (B*c^3*e*x^9)/9
 

Defintions of rubi rules used

rule 1195

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99

method	result	size
default	\(\frac {B \,c^{3} e \,x^{9}}{9}+\frac {\left (\left (A e +B d \right ) c^{3}+3 B e b \,c^{2}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d +3 \left (A e +B d \right ) b \,c^{2}+3 B e \,b^{2} c \right ) x^{7}}{7}+\frac {\left (3 A b \,c^{2} d +3 \left (A e +B d \right ) b^{2} c +B e \,b^{3}\right ) x^{6}}{6}+\frac {\left (3 A \,b^{2} c d +\left (A e +B d \right ) b^{3}\right ) x^{5}}{5}+\frac {A \,b^{3} d \,x^{4}}{4}\)	\(138\)
norman	\(\frac {B \,c^{3} e \,x^{9}}{9}+\left (\frac {1}{8} A \,c^{3} e +\frac {3}{8} B e b \,c^{2}+\frac {1}{8} B \,c^{3} d \right ) x^{8}+\left (\frac {3}{7} A b \,c^{2} e +\frac {1}{7} A \,c^{3} d +\frac {3}{7} B e \,b^{2} c +\frac {3}{7} B b \,c^{2} d \right ) x^{7}+\left (\frac {1}{2} A \,b^{2} c e +\frac {1}{2} A b \,c^{2} d +\frac {1}{6} B e \,b^{3}+\frac {1}{2} B \,b^{2} c d \right ) x^{6}+\left (\frac {1}{5} A \,b^{3} e +\frac {3}{5} A \,b^{2} c d +\frac {1}{5} B \,b^{3} d \right ) x^{5}+\frac {A \,b^{3} d \,x^{4}}{4}\)	\(148\)
gosper	\(\frac {x^{4} \left (280 B \,c^{3} e \,x^{5}+315 x^{4} A \,c^{3} e +945 x^{4} B e b \,c^{2}+315 x^{4} B \,c^{3} d +1080 x^{3} A b \,c^{2} e +360 x^{3} A \,c^{3} d +1080 x^{3} B e \,b^{2} c +1080 x^{3} B b \,c^{2} d +1260 x^{2} A \,b^{2} c e +1260 x^{2} A b \,c^{2} d +420 x^{2} B e \,b^{3}+1260 x^{2} B \,b^{2} c d +504 x A \,b^{3} e +1512 x A \,b^{2} c d +504 x B \,b^{3} d +630 A d \,b^{3}\right )}{2520}\)	\(166\)
risch	\(\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {1}{8} x^{8} A \,c^{3} e +\frac {3}{8} x^{8} B e b \,c^{2}+\frac {1}{8} x^{8} B \,c^{3} d +\frac {3}{7} x^{7} A b \,c^{2} e +\frac {1}{7} x^{7} A \,c^{3} d +\frac {3}{7} x^{7} B e \,b^{2} c +\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A b \,c^{2} d +\frac {1}{6} x^{6} B e \,b^{3}+\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {3}{5} x^{5} A \,b^{2} c d +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{4} A \,b^{3} d \,x^{4}\)	\(170\)
parallelrisch	\(\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {1}{8} x^{8} A \,c^{3} e +\frac {3}{8} x^{8} B e b \,c^{2}+\frac {1}{8} x^{8} B \,c^{3} d +\frac {3}{7} x^{7} A b \,c^{2} e +\frac {1}{7} x^{7} A \,c^{3} d +\frac {3}{7} x^{7} B e \,b^{2} c +\frac {3}{7} x^{7} B b \,c^{2} d +\frac {1}{2} x^{6} A \,b^{2} c e +\frac {1}{2} x^{6} A b \,c^{2} d +\frac {1}{6} x^{6} B e \,b^{3}+\frac {1}{2} x^{6} B \,b^{2} c d +\frac {1}{5} x^{5} A \,b^{3} e +\frac {3}{5} x^{5} A \,b^{2} c d +\frac {1}{5} x^{5} B \,b^{3} d +\frac {1}{4} A \,b^{3} d \,x^{4}\)	\(170\)
orering	\(\frac {x \left (280 B \,c^{3} e \,x^{5}+315 x^{4} A \,c^{3} e +945 x^{4} B e b \,c^{2}+315 x^{4} B \,c^{3} d +1080 x^{3} A b \,c^{2} e +360 x^{3} A \,c^{3} d +1080 x^{3} B e \,b^{2} c +1080 x^{3} B b \,c^{2} d +1260 x^{2} A \,b^{2} c e +1260 x^{2} A b \,c^{2} d +420 x^{2} B e \,b^{3}+1260 x^{2} B \,b^{2} c d +504 x A \,b^{3} e +1512 x A \,b^{2} c d +504 x B \,b^{3} d +630 A d \,b^{3}\right ) \left (c \,x^{2}+b x \right )^{3}}{2520 \left (c x +b \right )^{3}}\)	\(182\)

Input:

int((B*x+A)*(e*x+d)*(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
 

Output:

1/9*B*c^3*e*x^9+1/8*((A*e+B*d)*c^3+3*B*e*b*c^2)*x^8+1/7*(A*c^3*d+3*(A*e+B* 
d)*b*c^2+3*B*e*b^2*c)*x^7+1/6*(3*A*b*c^2*d+3*(A*e+B*d)*b^2*c+B*e*b^3)*x^6+ 
1/5*(3*A*b^2*c*d+(A*e+B*d)*b^3)*x^5+1/4*A*b^3*d*x^4
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.07 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + A b c^{2}\right )} d + {\left (B b^{3} + 3 \, A b^{2} c\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (A b^{3} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d\right )} x^{5} \] Input:

integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^3,x, algorithm="fricas")
 

Output:

1/9*B*c^3*e*x^9 + 1/4*A*b^3*d*x^4 + 1/8*(B*c^3*d + (3*B*b*c^2 + A*c^3)*e)* 
x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d + 3*(B*b^2*c + A*b*c^2)*e)*x^7 + 1/6*(3*( 
B*b^2*c + A*b*c^2)*d + (B*b^3 + 3*A*b^2*c)*e)*x^6 + 1/5*(A*b^3*e + (B*b^3 
+ 3*A*b^2*c)*d)*x^5
 

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.27 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {A b^{3} d x^{4}}{4} + \frac {B c^{3} e x^{9}}{9} + x^{8} \left (\frac {A c^{3} e}{8} + \frac {3 B b c^{2} e}{8} + \frac {B c^{3} d}{8}\right ) + x^{7} \cdot \left (\frac {3 A b c^{2} e}{7} + \frac {A c^{3} d}{7} + \frac {3 B b^{2} c e}{7} + \frac {3 B b c^{2} d}{7}\right ) + x^{6} \left (\frac {A b^{2} c e}{2} + \frac {A b c^{2} d}{2} + \frac {B b^{3} e}{6} + \frac {B b^{2} c d}{2}\right ) + x^{5} \left (\frac {A b^{3} e}{5} + \frac {3 A b^{2} c d}{5} + \frac {B b^{3} d}{5}\right ) \] Input:

integrate((B*x+A)*(e*x+d)*(c*x**2+b*x)**3,x)
 

Output:

A*b**3*d*x**4/4 + B*c**3*e*x**9/9 + x**8*(A*c**3*e/8 + 3*B*b*c**2*e/8 + B* 
c**3*d/8) + x**7*(3*A*b*c**2*e/7 + A*c**3*d/7 + 3*B*b**2*c*e/7 + 3*B*b*c** 
2*d/7) + x**6*(A*b**2*c*e/2 + A*b*c**2*d/2 + B*b**3*e/6 + B*b**2*c*d/2) + 
x**5*(A*b**3*e/5 + 3*A*b**2*c*d/5 + B*b**3*d/5)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.07 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{4} \, A b^{3} d x^{4} + \frac {1}{8} \, {\left (B c^{3} d + {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (B b^{2} c + A b c^{2}\right )} d + {\left (B b^{3} + 3 \, A b^{2} c\right )} e\right )} x^{6} + \frac {1}{5} \, {\left (A b^{3} e + {\left (B b^{3} + 3 \, A b^{2} c\right )} d\right )} x^{5} \] Input:

integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^3,x, algorithm="maxima")
 

Output:

1/9*B*c^3*e*x^9 + 1/4*A*b^3*d*x^4 + 1/8*(B*c^3*d + (3*B*b*c^2 + A*c^3)*e)* 
x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d + 3*(B*b^2*c + A*b*c^2)*e)*x^7 + 1/6*(3*( 
B*b^2*c + A*b*c^2)*d + (B*b^3 + 3*A*b^2*c)*e)*x^6 + 1/5*(A*b^3*e + (B*b^3 
+ 3*A*b^2*c)*d)*x^5
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.22 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, B c^{3} d x^{8} + \frac {3}{8} \, B b c^{2} e x^{8} + \frac {1}{8} \, A c^{3} e x^{8} + \frac {3}{7} \, B b c^{2} d x^{7} + \frac {1}{7} \, A c^{3} d x^{7} + \frac {3}{7} \, B b^{2} c e x^{7} + \frac {3}{7} \, A b c^{2} e x^{7} + \frac {1}{2} \, B b^{2} c d x^{6} + \frac {1}{2} \, A b c^{2} d x^{6} + \frac {1}{6} \, B b^{3} e x^{6} + \frac {1}{2} \, A b^{2} c e x^{6} + \frac {1}{5} \, B b^{3} d x^{5} + \frac {3}{5} \, A b^{2} c d x^{5} + \frac {1}{5} \, A b^{3} e x^{5} + \frac {1}{4} \, A b^{3} d x^{4} \] Input:

integrate((B*x+A)*(e*x+d)*(c*x^2+b*x)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/9*B*c^3*e*x^9 + 1/8*B*c^3*d*x^8 + 3/8*B*b*c^2*e*x^8 + 1/8*A*c^3*e*x^8 + 
3/7*B*b*c^2*d*x^7 + 1/7*A*c^3*d*x^7 + 3/7*B*b^2*c*e*x^7 + 3/7*A*b*c^2*e*x^ 
7 + 1/2*B*b^2*c*d*x^6 + 1/2*A*b*c^2*d*x^6 + 1/6*B*b^3*e*x^6 + 1/2*A*b^2*c* 
e*x^6 + 1/5*B*b^3*d*x^5 + 3/5*A*b^2*c*d*x^5 + 1/5*A*b^3*e*x^5 + 1/4*A*b^3* 
d*x^4
 

Mupad [B] (verification not implemented)

Time = 10.80 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.06 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=x^5\,\left (\frac {A\,b^3\,e}{5}+\frac {B\,b^3\,d}{5}+\frac {3\,A\,b^2\,c\,d}{5}\right )+x^8\,\left (\frac {A\,c^3\,e}{8}+\frac {B\,c^3\,d}{8}+\frac {3\,B\,b\,c^2\,e}{8}\right )+x^6\,\left (\frac {B\,b^3\,e}{6}+\frac {A\,b\,c^2\,d}{2}+\frac {A\,b^2\,c\,e}{2}+\frac {B\,b^2\,c\,d}{2}\right )+x^7\,\left (\frac {A\,c^3\,d}{7}+\frac {3\,A\,b\,c^2\,e}{7}+\frac {3\,B\,b\,c^2\,d}{7}+\frac {3\,B\,b^2\,c\,e}{7}\right )+\frac {A\,b^3\,d\,x^4}{4}+\frac {B\,c^3\,e\,x^9}{9} \] Input:

int((b*x + c*x^2)^3*(A + B*x)*(d + e*x),x)
 

Output:

x^5*((A*b^3*e)/5 + (B*b^3*d)/5 + (3*A*b^2*c*d)/5) + x^8*((A*c^3*e)/8 + (B* 
c^3*d)/8 + (3*B*b*c^2*e)/8) + x^6*((B*b^3*e)/6 + (A*b*c^2*d)/2 + (A*b^2*c* 
e)/2 + (B*b^2*c*d)/2) + x^7*((A*c^3*d)/7 + (3*A*b*c^2*e)/7 + (3*B*b*c^2*d) 
/7 + (3*B*b^2*c*e)/7) + (A*b^3*d*x^4)/4 + (B*c^3*e*x^9)/9
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.17 \[ \int (A+B x) (d+e x) \left (b x+c x^2\right )^3 \, dx=\frac {x^{4} \left (280 b \,c^{3} e \,x^{5}+315 a \,c^{3} e \,x^{4}+945 b^{2} c^{2} e \,x^{4}+315 b \,c^{3} d \,x^{4}+1080 a b \,c^{2} e \,x^{3}+360 a \,c^{3} d \,x^{3}+1080 b^{3} c e \,x^{3}+1080 b^{2} c^{2} d \,x^{3}+1260 a \,b^{2} c e \,x^{2}+1260 a b \,c^{2} d \,x^{2}+420 b^{4} e \,x^{2}+1260 b^{3} c d \,x^{2}+504 a \,b^{3} e x +1512 a \,b^{2} c d x +504 b^{4} d x +630 a \,b^{3} d \right )}{2520} \] Input:

int((B*x+A)*(e*x+d)*(c*x^2+b*x)^3,x)
 

Output:

(x**4*(630*a*b**3*d + 504*a*b**3*e*x + 1512*a*b**2*c*d*x + 1260*a*b**2*c*e 
*x**2 + 1260*a*b*c**2*d*x**2 + 1080*a*b*c**2*e*x**3 + 360*a*c**3*d*x**3 + 
315*a*c**3*e*x**4 + 504*b**4*d*x + 420*b**4*e*x**2 + 1260*b**3*c*d*x**2 + 
1080*b**3*c*e*x**3 + 1080*b**2*c**2*d*x**3 + 945*b**2*c**2*e*x**4 + 315*b* 
c**3*d*x**4 + 280*b*c**3*e*x**5))/2520