\(\int (a+b x)^4 (A+B x+C x^2+D x^3) \, dx\) [1]

Optimal result

Integrand size = 23, antiderivative size = 108 \[ \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (a+b x)^5}{5 b^4}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) (a+b x)^6}{6 b^4}+\frac {(b C-3 a D) (a+b x)^7}{7 b^4}+\frac {D (a+b x)^8}{8 b^4} \] Output:

1/5*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(b*x+a)^5/b^4+1/6*(B*b^2-2*C*a*b+3*D*a^2 
)*(b*x+a)^6/b^4+1/7*(C*b-3*D*a)*(b*x+a)^7/b^4+1/8*D*(b*x+a)^8/b^4
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.63 \[ \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx=a^4 A x+\frac {1}{2} a^3 (4 A b+a B) x^2+\frac {1}{3} a^2 \left (6 A b^2+4 a b B+a^2 C\right ) x^3+\frac {1}{4} a \left (4 A b^3+6 a b^2 B+4 a^2 b C+a^3 D\right ) x^4+\frac {1}{5} b \left (A b^3+4 a b^2 B+6 a^2 b C+4 a^3 D\right ) x^5+\frac {1}{6} b^2 \left (b^2 B+4 a b C+6 a^2 D\right ) x^6+\frac {1}{7} b^3 (b C+4 a D) x^7+\frac {1}{8} b^4 D x^8 \] Input:

Integrate[(a + b*x)^4*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

a^4*A*x + (a^3*(4*A*b + a*B)*x^2)/2 + (a^2*(6*A*b^2 + 4*a*b*B + a^2*C)*x^3 
)/3 + (a*(4*A*b^3 + 6*a*b^2*B + 4*a^2*b*C + a^3*D)*x^4)/4 + (b*(A*b^3 + 4* 
a*b^2*B + 6*a^2*b*C + 4*a^3*D)*x^5)/5 + (b^2*(b^2*B + 4*a*b*C + 6*a^2*D)*x 
^6)/6 + (b^3*(b*C + 4*a*D)*x^7)/7 + (b^4*D*x^8)/8
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2389, 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.

\(\displaystyle \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx\)

\(\Big \downarrow \) 2389

\(\displaystyle \int \left (\frac {(a+b x)^4 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^3}+\frac {(a+b x)^5 \left (3 a^2 D-2 a b C+b^2 B\right )}{b^3}+\frac {(a+b x)^6 (b C-3 a D)}{b^3}+\frac {D (a+b x)^7}{b^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(a+b x)^5 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{5 b^4}+\frac {(a+b x)^6 \left (3 a^2 D-2 a b C+b^2 B\right )}{6 b^4}+\frac {(a+b x)^7 (b C-3 a D)}{7 b^4}+\frac {D (a+b x)^8}{8 b^4}\)

Input:

Int[(a + b*x)^4*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

((A*b^3 - a*(b^2*B - a*b*C + a^2*D))*(a + b*x)^5)/(5*b^4) + ((b^2*B - 2*a* 
b*C + 3*a^2*D)*(a + b*x)^6)/(6*b^4) + ((b*C - 3*a*D)*(a + b*x)^7)/(7*b^4) 
+ (D*(a + b*x)^8)/(8*b^4)
 

Defintions of rubi rules used

rule 2009

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

rule 2389

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand 
[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p 
, 0] || EqQ[n, 1])
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.59

method	result	size
norman	\(\frac {b^{4} D x^{8}}{8}+\left (\frac {1}{7} b^{4} C +\frac {4}{7} a \,b^{3} D\right ) x^{7}+\left (\frac {1}{6} b^{4} B +\frac {2}{3} a \,b^{3} C +a^{2} b^{2} D\right ) x^{6}+\left (\frac {1}{5} A \,b^{4}+\frac {4}{5} B a \,b^{3}+\frac {6}{5} a^{2} b^{2} C +\frac {4}{5} a^{3} b D\right ) x^{5}+\left (a \,b^{3} A +\frac {3}{2} a^{2} b^{2} B +a^{3} b C +\frac {1}{4} a^{4} D\right ) x^{4}+\left (2 A \,a^{2} b^{2}+\frac {4}{3} B \,a^{3} b +\frac {1}{3} a^{4} C \right ) x^{3}+\left (2 A \,a^{3} b +\frac {1}{2} B \,a^{4}\right ) x^{2}+A \,a^{4} x\)	\(172\)
default	\(\frac {b^{4} D x^{8}}{8}+\frac {\left (b^{4} C +4 a \,b^{3} D\right ) x^{7}}{7}+\frac {\left (b^{4} B +4 a \,b^{3} C +6 a^{2} b^{2} D\right ) x^{6}}{6}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 a^{2} b^{2} C +4 a^{3} b D\right ) x^{5}}{5}+\frac {\left (4 a \,b^{3} A +6 a^{2} b^{2} B +4 a^{3} b C +a^{4} D\right ) x^{4}}{4}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) x^{2}}{2}+A \,a^{4} x\)	\(175\)
gosper	\(\frac {1}{8} b^{4} D x^{8}+\frac {1}{7} x^{7} b^{4} C +\frac {4}{7} x^{7} a \,b^{3} D+\frac {1}{6} x^{6} b^{4} B +\frac {2}{3} x^{6} a \,b^{3} C +x^{6} a^{2} b^{2} D+\frac {1}{5} x^{5} A \,b^{4}+\frac {4}{5} x^{5} B a \,b^{3}+\frac {6}{5} x^{5} a^{2} b^{2} C +\frac {4}{5} x^{5} a^{3} b D+x^{4} a \,b^{3} A +\frac {3}{2} x^{4} a^{2} b^{2} B +x^{4} a^{3} b C +\frac {1}{4} x^{4} a^{4} D+2 x^{3} A \,a^{2} b^{2}+\frac {4}{3} x^{3} B \,a^{3} b +\frac {1}{3} x^{3} a^{4} C +2 x^{2} A \,a^{3} b +\frac {1}{2} x^{2} B \,a^{4}+A \,a^{4} x\)	\(196\)
parallelrisch	\(\frac {1}{8} b^{4} D x^{8}+\frac {1}{7} x^{7} b^{4} C +\frac {4}{7} x^{7} a \,b^{3} D+\frac {1}{6} x^{6} b^{4} B +\frac {2}{3} x^{6} a \,b^{3} C +x^{6} a^{2} b^{2} D+\frac {1}{5} x^{5} A \,b^{4}+\frac {4}{5} x^{5} B a \,b^{3}+\frac {6}{5} x^{5} a^{2} b^{2} C +\frac {4}{5} x^{5} a^{3} b D+x^{4} a \,b^{3} A +\frac {3}{2} x^{4} a^{2} b^{2} B +x^{4} a^{3} b C +\frac {1}{4} x^{4} a^{4} D+2 x^{3} A \,a^{2} b^{2}+\frac {4}{3} x^{3} B \,a^{3} b +\frac {1}{3} x^{3} a^{4} C +2 x^{2} A \,a^{3} b +\frac {1}{2} x^{2} B \,a^{4}+A \,a^{4} x\)	\(196\)
orering	\(\frac {x \left (105 b^{4} D x^{7}+120 C \,b^{4} x^{6}+480 D a \,b^{3} x^{6}+140 B \,b^{4} x^{5}+560 C a \,b^{3} x^{5}+840 D a^{2} b^{2} x^{5}+168 A \,b^{4} x^{4}+672 B a \,b^{3} x^{4}+1008 C \,a^{2} b^{2} x^{4}+672 D a^{3} b \,x^{4}+840 A a \,b^{3} x^{3}+1260 B \,a^{2} b^{2} x^{3}+840 C \,a^{3} b \,x^{3}+210 D a^{4} x^{3}+1680 A \,a^{2} b^{2} x^{2}+1120 B \,a^{3} b \,x^{2}+280 C \,a^{4} x^{2}+1680 A \,a^{3} b x +420 B \,a^{4} x +840 A \,a^{4}\right )}{840}\)	\(198\)

Input:

int((b*x+a)^4*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
 

Output:

1/8*b^4*D*x^8+(1/7*b^4*C+4/7*a*b^3*D)*x^7+(1/6*b^4*B+2/3*a*b^3*C+a^2*b^2*D 
)*x^6+(1/5*A*b^4+4/5*B*a*b^3+6/5*a^2*b^2*C+4/5*a^3*b*D)*x^5+(a*b^3*A+3/2*a 
^2*b^2*B+a^3*b*C+1/4*a^4*D)*x^4+(2*A*a^2*b^2+4/3*B*a^3*b+1/3*a^4*C)*x^3+(2 
*A*a^3*b+1/2*B*a^4)*x^2+A*a^4*x
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.61 \[ \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{8} \, D b^{4} x^{8} + \frac {1}{7} \, {\left (4 \, D a b^{3} + C b^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, D a^{2} b^{2} + 4 \, C a b^{3} + B b^{4}\right )} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (4 \, D a^{3} b + 6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} x^{5} + \frac {1}{4} \, {\left (D a^{4} + 4 \, C a^{3} b + 6 \, B a^{2} b^{2} + 4 \, A a b^{3}\right )} x^{4} + \frac {1}{3} \, {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{2} \] Input:

integrate((b*x+a)^4*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
 

Output:

1/8*D*b^4*x^8 + 1/7*(4*D*a*b^3 + C*b^4)*x^7 + 1/6*(6*D*a^2*b^2 + 4*C*a*b^3 
 + B*b^4)*x^6 + A*a^4*x + 1/5*(4*D*a^3*b + 6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4 
)*x^5 + 1/4*(D*a^4 + 4*C*a^3*b + 6*B*a^2*b^2 + 4*A*a*b^3)*x^4 + 1/3*(C*a^4 
 + 4*B*a^3*b + 6*A*a^2*b^2)*x^3 + 1/2*(B*a^4 + 4*A*a^3*b)*x^2
 

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.78 \[ \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx=A a^{4} x + \frac {D b^{4} x^{8}}{8} + x^{7} \left (\frac {C b^{4}}{7} + \frac {4 D a b^{3}}{7}\right ) + x^{6} \left (\frac {B b^{4}}{6} + \frac {2 C a b^{3}}{3} + D a^{2} b^{2}\right ) + x^{5} \left (\frac {A b^{4}}{5} + \frac {4 B a b^{3}}{5} + \frac {6 C a^{2} b^{2}}{5} + \frac {4 D a^{3} b}{5}\right ) + x^{4} \left (A a b^{3} + \frac {3 B a^{2} b^{2}}{2} + C a^{3} b + \frac {D a^{4}}{4}\right ) + x^{3} \cdot \left (2 A a^{2} b^{2} + \frac {4 B a^{3} b}{3} + \frac {C a^{4}}{3}\right ) + x^{2} \cdot \left (2 A a^{3} b + \frac {B a^{4}}{2}\right ) \] Input:

integrate((b*x+a)**4*(D*x**3+C*x**2+B*x+A),x)
                                                                                    
                                                                                    
 

Output:

A*a**4*x + D*b**4*x**8/8 + x**7*(C*b**4/7 + 4*D*a*b**3/7) + x**6*(B*b**4/6 
 + 2*C*a*b**3/3 + D*a**2*b**2) + x**5*(A*b**4/5 + 4*B*a*b**3/5 + 6*C*a**2* 
b**2/5 + 4*D*a**3*b/5) + x**4*(A*a*b**3 + 3*B*a**2*b**2/2 + C*a**3*b + D*a 
**4/4) + x**3*(2*A*a**2*b**2 + 4*B*a**3*b/3 + C*a**4/3) + x**2*(2*A*a**3*b 
 + B*a**4/2)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.61 \[ \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{8} \, D b^{4} x^{8} + \frac {1}{7} \, {\left (4 \, D a b^{3} + C b^{4}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, D a^{2} b^{2} + 4 \, C a b^{3} + B b^{4}\right )} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (4 \, D a^{3} b + 6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} x^{5} + \frac {1}{4} \, {\left (D a^{4} + 4 \, C a^{3} b + 6 \, B a^{2} b^{2} + 4 \, A a b^{3}\right )} x^{4} + \frac {1}{3} \, {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{2} \] Input:

integrate((b*x+a)^4*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
 

Output:

1/8*D*b^4*x^8 + 1/7*(4*D*a*b^3 + C*b^4)*x^7 + 1/6*(6*D*a^2*b^2 + 4*C*a*b^3 
 + B*b^4)*x^6 + A*a^4*x + 1/5*(4*D*a^3*b + 6*C*a^2*b^2 + 4*B*a*b^3 + A*b^4 
)*x^5 + 1/4*(D*a^4 + 4*C*a^3*b + 6*B*a^2*b^2 + 4*A*a*b^3)*x^4 + 1/3*(C*a^4 
 + 4*B*a^3*b + 6*A*a^2*b^2)*x^3 + 1/2*(B*a^4 + 4*A*a^3*b)*x^2
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.81 \[ \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{8} \, D b^{4} x^{8} + \frac {4}{7} \, D a b^{3} x^{7} + \frac {1}{7} \, C b^{4} x^{7} + D a^{2} b^{2} x^{6} + \frac {2}{3} \, C a b^{3} x^{6} + \frac {1}{6} \, B b^{4} x^{6} + \frac {4}{5} \, D a^{3} b x^{5} + \frac {6}{5} \, C a^{2} b^{2} x^{5} + \frac {4}{5} \, B a b^{3} x^{5} + \frac {1}{5} \, A b^{4} x^{5} + \frac {1}{4} \, D a^{4} x^{4} + C a^{3} b x^{4} + \frac {3}{2} \, B a^{2} b^{2} x^{4} + A a b^{3} x^{4} + \frac {1}{3} \, C a^{4} x^{3} + \frac {4}{3} \, B a^{3} b x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac {1}{2} \, B a^{4} x^{2} + 2 \, A a^{3} b x^{2} + A a^{4} x \] Input:

integrate((b*x+a)^4*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
 

Output:

1/8*D*b^4*x^8 + 4/7*D*a*b^3*x^7 + 1/7*C*b^4*x^7 + D*a^2*b^2*x^6 + 2/3*C*a* 
b^3*x^6 + 1/6*B*b^4*x^6 + 4/5*D*a^3*b*x^5 + 6/5*C*a^2*b^2*x^5 + 4/5*B*a*b^ 
3*x^5 + 1/5*A*b^4*x^5 + 1/4*D*a^4*x^4 + C*a^3*b*x^4 + 3/2*B*a^2*b^2*x^4 + 
A*a*b^3*x^4 + 1/3*C*a^4*x^3 + 4/3*B*a^3*b*x^3 + 2*A*a^2*b^2*x^3 + 1/2*B*a^ 
4*x^2 + 2*A*a^3*b*x^2 + A*a^4*x
 

Mupad [B] (verification not implemented)

Time = 3.36 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.81 \[ \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {B\,a^4\,x^2}{2}+\frac {A\,b^4\,x^5}{5}+\frac {C\,a^4\,x^3}{3}+\frac {B\,b^4\,x^6}{6}+\frac {C\,b^4\,x^7}{7}+\frac {a^4\,x^4\,D}{4}+\frac {b^4\,x^8\,D}{8}+A\,a^4\,x+\frac {4\,a^3\,b\,x^5\,D}{5}+\frac {4\,a\,b^3\,x^7\,D}{7}+a^2\,b^2\,x^6\,D+2\,A\,a^3\,b\,x^2+A\,a\,b^3\,x^4+\frac {4\,B\,a^3\,b\,x^3}{3}+\frac {4\,B\,a\,b^3\,x^5}{5}+C\,a^3\,b\,x^4+\frac {2\,C\,a\,b^3\,x^6}{3}+2\,A\,a^2\,b^2\,x^3+\frac {3\,B\,a^2\,b^2\,x^4}{2}+\frac {6\,C\,a^2\,b^2\,x^5}{5} \] Input:

int((a + b*x)^4*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

(B*a^4*x^2)/2 + (A*b^4*x^5)/5 + (C*a^4*x^3)/3 + (B*b^4*x^6)/6 + (C*b^4*x^7 
)/7 + (a^4*x^4*D)/4 + (b^4*x^8*D)/8 + A*a^4*x + (4*a^3*b*x^5*D)/5 + (4*a*b 
^3*x^7*D)/7 + a^2*b^2*x^6*D + 2*A*a^3*b*x^2 + A*a*b^3*x^4 + (4*B*a^3*b*x^3 
)/3 + (4*B*a*b^3*x^5)/5 + C*a^3*b*x^4 + (2*C*a*b^3*x^6)/3 + 2*A*a^2*b^2*x^ 
3 + (3*B*a^2*b^2*x^4)/2 + (6*C*a^2*b^2*x^5)/5
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.44 \[ \int (a+b x)^4 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (105 b^{4} d \,x^{7}+480 a \,b^{3} d \,x^{6}+120 b^{4} c \,x^{6}+840 a^{2} b^{2} d \,x^{5}+560 a \,b^{3} c \,x^{5}+140 b^{5} x^{5}+672 a^{3} b d \,x^{4}+1008 a^{2} b^{2} c \,x^{4}+840 a \,b^{4} x^{4}+210 a^{4} d \,x^{3}+840 a^{3} b c \,x^{3}+2100 a^{2} b^{3} x^{3}+280 a^{4} c \,x^{2}+2800 a^{3} b^{2} x^{2}+2100 a^{4} b x +840 a^{5}\right )}{840} \] Input:

int((b*x+a)^4*(D*x^3+C*x^2+B*x+A),x)
 

Output:

(x*(840*a**5 + 2100*a**4*b*x + 280*a**4*c*x**2 + 210*a**4*d*x**3 + 2800*a* 
*3*b**2*x**2 + 840*a**3*b*c*x**3 + 672*a**3*b*d*x**4 + 2100*a**2*b**3*x**3 
 + 1008*a**2*b**2*c*x**4 + 840*a**2*b**2*d*x**5 + 840*a*b**4*x**4 + 560*a* 
b**3*c*x**5 + 480*a*b**3*d*x**6 + 140*b**5*x**5 + 120*b**4*c*x**6 + 105*b* 
*4*d*x**7))/840