\(\int x^3 (A+B x) (a^2+2 a b x+b^2 x^2)^3 \, dx\) [170]

Optimal result

Integrand size = 27, antiderivative size = 112 \[ \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=-\frac {a^3 (A b-a B) (a+b x)^7}{7 b^5}+\frac {a^2 (3 A b-4 a B) (a+b x)^8}{8 b^5}-\frac {a (A b-2 a B) (a+b x)^9}{3 b^5}+\frac {(A b-4 a B) (a+b x)^{10}}{10 b^5}+\frac {B (a+b x)^{11}}{11 b^5} \] Output:

-1/7*a^3*(A*b-B*a)*(b*x+a)^7/b^5+1/8*a^2*(3*A*b-4*B*a)*(b*x+a)^8/b^5-1/3*a 
*(A*b-2*B*a)*(b*x+a)^9/b^5+1/10*(A*b-4*B*a)*(b*x+a)^10/b^5+1/11*B*(b*x+a)^ 
11/b^5
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.28 \[ \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{4} a^6 A x^4+\frac {1}{5} a^5 (6 A b+a B) x^5+\frac {1}{2} a^4 b (5 A b+2 a B) x^6+\frac {5}{7} a^3 b^2 (4 A b+3 a B) x^7+\frac {5}{8} a^2 b^3 (3 A b+4 a B) x^8+\frac {1}{3} a b^4 (2 A b+5 a B) x^9+\frac {1}{10} b^5 (A b+6 a B) x^{10}+\frac {1}{11} b^6 B x^{11} \] Input:

Integrate[x^3*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
 

Output:

(a^6*A*x^4)/4 + (a^5*(6*A*b + a*B)*x^5)/5 + (a^4*b*(5*A*b + 2*a*B)*x^6)/2 
+ (5*a^3*b^2*(4*A*b + 3*a*B)*x^7)/7 + (5*a^2*b^3*(3*A*b + 4*a*B)*x^8)/8 + 
(a*b^4*(2*A*b + 5*a*B)*x^9)/3 + (b^5*(A*b + 6*a*B)*x^10)/10 + (b^6*B*x^11) 
/11
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1184, 27, 85, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle \frac {\int b^6 x^3 (a+b x)^6 (A+B x)dx}{b^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \int x^3 (a+b x)^6 (A+B x)dx\)

\(\Big \downarrow \) 85

\(\displaystyle \int \left (\frac {a^3 (a+b x)^6 (a B-A b)}{b^4}-\frac {a^2 (a+b x)^7 (4 a B-3 A b)}{b^4}+\frac {(a+b x)^9 (A b-4 a B)}{b^4}+\frac {3 a (a+b x)^8 (2 a B-A b)}{b^4}+\frac {B (a+b x)^{10}}{b^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^3 (a+b x)^7 (A b-a B)}{7 b^5}+\frac {a^2 (a+b x)^8 (3 A b-4 a B)}{8 b^5}+\frac {(a+b x)^{10} (A b-4 a B)}{10 b^5}-\frac {a (a+b x)^9 (A b-2 a B)}{3 b^5}+\frac {B (a+b x)^{11}}{11 b^5}\)

Input:

Int[x^3*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
 

Output:

-1/7*(a^3*(A*b - a*B)*(a + b*x)^7)/b^5 + (a^2*(3*A*b - 4*a*B)*(a + b*x)^8) 
/(8*b^5) - (a*(A*b - 2*a*B)*(a + b*x)^9)/(3*b^5) + ((A*b - 4*a*B)*(a + b*x 
)^10)/(10*b^5) + (B*(a + b*x)^11)/(11*b^5)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 85

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : 
> Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, 
 d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && NeQ[b*e + a* 
f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n 
 + p + 2, 0] && RationalQ[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 
1])
 

rule 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 2009

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

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.28

method	result	size
norman	\(\frac {B \,b^{6} x^{11}}{11}+\left (\frac {1}{10} A \,b^{6}+\frac {3}{5} B a \,b^{5}\right ) x^{10}+\left (\frac {2}{3} A a \,b^{5}+\frac {5}{3} B \,a^{2} b^{4}\right ) x^{9}+\left (\frac {15}{8} A \,a^{2} b^{4}+\frac {5}{2} B \,a^{3} b^{3}\right ) x^{8}+\left (\frac {20}{7} A \,a^{3} b^{3}+\frac {15}{7} B \,a^{4} b^{2}\right ) x^{7}+\left (\frac {5}{2} A \,a^{4} b^{2}+B \,a^{5} b \right ) x^{6}+\left (\frac {6}{5} A \,a^{5} b +\frac {1}{5} B \,a^{6}\right ) x^{5}+\frac {A \,a^{6} x^{4}}{4}\)	\(143\)
gosper	\(\frac {x^{4} \left (840 b^{6} B \,x^{7}+924 A \,b^{6} x^{6}+5544 B a \,b^{5} x^{6}+6160 A a \,b^{5} x^{5}+15400 B \,a^{2} b^{4} x^{5}+17325 A \,a^{2} b^{4} x^{4}+23100 B \,a^{3} b^{3} x^{4}+26400 A \,a^{3} b^{3} x^{3}+19800 B \,a^{4} b^{2} x^{3}+23100 A \,a^{4} b^{2} x^{2}+9240 B \,a^{5} b \,x^{2}+11088 A \,a^{5} b x +1848 B \,a^{6} x +2310 A \,a^{6}\right )}{9240}\)	\(148\)
default	\(\frac {B \,b^{6} x^{11}}{11}+\frac {\left (A \,b^{6}+6 B a \,b^{5}\right ) x^{10}}{10}+\frac {\left (6 A a \,b^{5}+15 B \,a^{2} b^{4}\right ) x^{9}}{9}+\frac {\left (15 A \,a^{2} b^{4}+20 B \,a^{3} b^{3}\right ) x^{8}}{8}+\frac {\left (20 A \,a^{3} b^{3}+15 B \,a^{4} b^{2}\right ) x^{7}}{7}+\frac {\left (15 A \,a^{4} b^{2}+6 B \,a^{5} b \right ) x^{6}}{6}+\frac {\left (6 A \,a^{5} b +B \,a^{6}\right ) x^{5}}{5}+\frac {A \,a^{6} x^{4}}{4}\)	\(148\)
risch	\(\frac {1}{11} B \,b^{6} x^{11}+\frac {1}{10} x^{10} A \,b^{6}+\frac {3}{5} x^{10} B a \,b^{5}+\frac {2}{3} x^{9} A a \,b^{5}+\frac {5}{3} x^{9} B \,a^{2} b^{4}+\frac {15}{8} x^{8} A \,a^{2} b^{4}+\frac {5}{2} x^{8} B \,a^{3} b^{3}+\frac {20}{7} x^{7} A \,a^{3} b^{3}+\frac {15}{7} x^{7} B \,a^{4} b^{2}+\frac {5}{2} x^{6} A \,a^{4} b^{2}+x^{6} B \,a^{5} b +\frac {6}{5} x^{5} A \,a^{5} b +\frac {1}{5} x^{5} B \,a^{6}+\frac {1}{4} A \,a^{6} x^{4}\)	\(149\)
parallelrisch	\(\frac {1}{11} B \,b^{6} x^{11}+\frac {1}{10} x^{10} A \,b^{6}+\frac {3}{5} x^{10} B a \,b^{5}+\frac {2}{3} x^{9} A a \,b^{5}+\frac {5}{3} x^{9} B \,a^{2} b^{4}+\frac {15}{8} x^{8} A \,a^{2} b^{4}+\frac {5}{2} x^{8} B \,a^{3} b^{3}+\frac {20}{7} x^{7} A \,a^{3} b^{3}+\frac {15}{7} x^{7} B \,a^{4} b^{2}+\frac {5}{2} x^{6} A \,a^{4} b^{2}+x^{6} B \,a^{5} b +\frac {6}{5} x^{5} A \,a^{5} b +\frac {1}{5} x^{5} B \,a^{6}+\frac {1}{4} A \,a^{6} x^{4}\)	\(149\)
orering	\(\frac {x^{4} \left (840 b^{6} B \,x^{7}+924 A \,b^{6} x^{6}+5544 B a \,b^{5} x^{6}+6160 A a \,b^{5} x^{5}+15400 B \,a^{2} b^{4} x^{5}+17325 A \,a^{2} b^{4} x^{4}+23100 B \,a^{3} b^{3} x^{4}+26400 A \,a^{3} b^{3} x^{3}+19800 B \,a^{4} b^{2} x^{3}+23100 A \,a^{4} b^{2} x^{2}+9240 B \,a^{5} b \,x^{2}+11088 A \,a^{5} b x +1848 B \,a^{6} x +2310 A \,a^{6}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{3}}{9240 \left (b x +a \right )^{6}}\)	\(173\)

Input:

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

Output:

1/11*B*b^6*x^11+(1/10*A*b^6+3/5*B*a*b^5)*x^10+(2/3*A*a*b^5+5/3*B*a^2*b^4)* 
x^9+(15/8*A*a^2*b^4+5/2*B*a^3*b^3)*x^8+(20/7*A*a^3*b^3+15/7*B*a^4*b^2)*x^7 
+(5/2*A*a^4*b^2+B*a^5*b)*x^6+(6/5*A*a^5*b+1/5*B*a^6)*x^5+1/4*A*a^6*x^4
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.31 \[ \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, B b^{6} x^{11} + \frac {1}{4} \, A a^{6} x^{4} + \frac {1}{10} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{9} + \frac {5}{8} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{8} + \frac {5}{7} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{5} \] Input:

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

Output:

1/11*B*b^6*x^11 + 1/4*A*a^6*x^4 + 1/10*(6*B*a*b^5 + A*b^6)*x^10 + 1/3*(5*B 
*a^2*b^4 + 2*A*a*b^5)*x^9 + 5/8*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^8 + 5/7*(3*B 
*a^4*b^2 + 4*A*a^3*b^3)*x^7 + 1/2*(2*B*a^5*b + 5*A*a^4*b^2)*x^6 + 1/5*(B*a 
^6 + 6*A*a^5*b)*x^5
 

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.45 \[ \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {A a^{6} x^{4}}{4} + \frac {B b^{6} x^{11}}{11} + x^{10} \left (\frac {A b^{6}}{10} + \frac {3 B a b^{5}}{5}\right ) + x^{9} \cdot \left (\frac {2 A a b^{5}}{3} + \frac {5 B a^{2} b^{4}}{3}\right ) + x^{8} \cdot \left (\frac {15 A a^{2} b^{4}}{8} + \frac {5 B a^{3} b^{3}}{2}\right ) + x^{7} \cdot \left (\frac {20 A a^{3} b^{3}}{7} + \frac {15 B a^{4} b^{2}}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{4} b^{2}}{2} + B a^{5} b\right ) + x^{5} \cdot \left (\frac {6 A a^{5} b}{5} + \frac {B a^{6}}{5}\right ) \] Input:

integrate(x**3*(B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

A*a**6*x**4/4 + B*b**6*x**11/11 + x**10*(A*b**6/10 + 3*B*a*b**5/5) + x**9* 
(2*A*a*b**5/3 + 5*B*a**2*b**4/3) + x**8*(15*A*a**2*b**4/8 + 5*B*a**3*b**3/ 
2) + x**7*(20*A*a**3*b**3/7 + 15*B*a**4*b**2/7) + x**6*(5*A*a**4*b**2/2 + 
B*a**5*b) + x**5*(6*A*a**5*b/5 + B*a**6/5)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.31 \[ \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, B b^{6} x^{11} + \frac {1}{4} \, A a^{6} x^{4} + \frac {1}{10} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{10} + \frac {1}{3} \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{9} + \frac {5}{8} \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{8} + \frac {5}{7} \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{7} + \frac {1}{2} \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x^{5} \] Input:

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

Output:

1/11*B*b^6*x^11 + 1/4*A*a^6*x^4 + 1/10*(6*B*a*b^5 + A*b^6)*x^10 + 1/3*(5*B 
*a^2*b^4 + 2*A*a*b^5)*x^9 + 5/8*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^8 + 5/7*(3*B 
*a^4*b^2 + 4*A*a^3*b^3)*x^7 + 1/2*(2*B*a^5*b + 5*A*a^4*b^2)*x^6 + 1/5*(B*a 
^6 + 6*A*a^5*b)*x^5
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.32 \[ \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {1}{11} \, B b^{6} x^{11} + \frac {3}{5} \, B a b^{5} x^{10} + \frac {1}{10} \, A b^{6} x^{10} + \frac {5}{3} \, B a^{2} b^{4} x^{9} + \frac {2}{3} \, A a b^{5} x^{9} + \frac {5}{2} \, B a^{3} b^{3} x^{8} + \frac {15}{8} \, A a^{2} b^{4} x^{8} + \frac {15}{7} \, B a^{4} b^{2} x^{7} + \frac {20}{7} \, A a^{3} b^{3} x^{7} + B a^{5} b x^{6} + \frac {5}{2} \, A a^{4} b^{2} x^{6} + \frac {1}{5} \, B a^{6} x^{5} + \frac {6}{5} \, A a^{5} b x^{5} + \frac {1}{4} \, A a^{6} x^{4} \] Input:

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

Output:

1/11*B*b^6*x^11 + 3/5*B*a*b^5*x^10 + 1/10*A*b^6*x^10 + 5/3*B*a^2*b^4*x^9 + 
 2/3*A*a*b^5*x^9 + 5/2*B*a^3*b^3*x^8 + 15/8*A*a^2*b^4*x^8 + 15/7*B*a^4*b^2 
*x^7 + 20/7*A*a^3*b^3*x^7 + B*a^5*b*x^6 + 5/2*A*a^4*b^2*x^6 + 1/5*B*a^6*x^ 
5 + 6/5*A*a^5*b*x^5 + 1/4*A*a^6*x^4
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.17 \[ \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=x^5\,\left (\frac {B\,a^6}{5}+\frac {6\,A\,b\,a^5}{5}\right )+x^{10}\,\left (\frac {A\,b^6}{10}+\frac {3\,B\,a\,b^5}{5}\right )+\frac {A\,a^6\,x^4}{4}+\frac {B\,b^6\,x^{11}}{11}+\frac {5\,a^3\,b^2\,x^7\,\left (4\,A\,b+3\,B\,a\right )}{7}+\frac {5\,a^2\,b^3\,x^8\,\left (3\,A\,b+4\,B\,a\right )}{8}+\frac {a^4\,b\,x^6\,\left (5\,A\,b+2\,B\,a\right )}{2}+\frac {a\,b^4\,x^9\,\left (2\,A\,b+5\,B\,a\right )}{3} \] Input:

int(x^3*(A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
 

Output:

x^5*((B*a^6)/5 + (6*A*a^5*b)/5) + x^10*((A*b^6)/10 + (3*B*a*b^5)/5) + (A*a 
^6*x^4)/4 + (B*b^6*x^11)/11 + (5*a^3*b^2*x^7*(4*A*b + 3*B*a))/7 + (5*a^2*b 
^3*x^8*(3*A*b + 4*B*a))/8 + (a^4*b*x^6*(5*A*b + 2*B*a))/2 + (a*b^4*x^9*(2* 
A*b + 5*B*a))/3
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.71 \[ \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx=\frac {x^{4} \left (120 b^{7} x^{7}+924 a \,b^{6} x^{6}+3080 a^{2} b^{5} x^{5}+5775 a^{3} b^{4} x^{4}+6600 a^{4} b^{3} x^{3}+4620 a^{5} b^{2} x^{2}+1848 a^{6} b x +330 a^{7}\right )}{1320} \] Input:

int(x^3*(B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

(x**4*(330*a**7 + 1848*a**6*b*x + 4620*a**5*b**2*x**2 + 6600*a**4*b**3*x** 
3 + 5775*a**3*b**4*x**4 + 3080*a**2*b**5*x**5 + 924*a*b**6*x**6 + 120*b**7 
*x**7))/1320