\(\int (a+b x)^m (A+B x+C x^2+D x^3) \, dx\) [30]

Optimal result

Integrand size = 23, antiderivative size = 124 \[ \int (a+b x)^m \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (a+b x)^{1+m}}{b^4 (1+m)}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) (a+b x)^{2+m}}{b^4 (2+m)}+\frac {(b C-3 a D) (a+b x)^{3+m}}{b^4 (3+m)}+\frac {D (a+b x)^{4+m}}{b^4 (4+m)} \] Output:

(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(b*x+a)^(1+m)/b^4/(1+m)+(B*b^2-2*C*a*b+3*D*a 
^2)*(b*x+a)^(2+m)/b^4/(2+m)+(C*b-3*D*a)*(b*x+a)^(3+m)/b^4/(3+m)+D*(b*x+a)^ 
(4+m)/b^4/(4+m)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.87 \[ \int (a+b x)^m \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {(a+b x)^{1+m} \left (\frac {A b^3-a b^2 B+a^2 b C-a^3 D}{1+m}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) (a+b x)}{2+m}+\frac {(b C-3 a D) (a+b x)^2}{3+m}+\frac {D (a+b x)^3}{4+m}\right )}{b^4} \] Input:

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

Output:

((a + b*x)^(1 + m)*((A*b^3 - a*b^2*B + a^2*b*C - a^3*D)/(1 + m) + ((b^2*B 
- 2*a*b*C + 3*a^2*D)*(a + b*x))/(2 + m) + ((b*C - 3*a*D)*(a + b*x)^2)/(3 + 
 m) + (D*(a + b*x)^3)/(4 + m)))/b^4
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 124, 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.

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

\(\Big \downarrow \) 2389

\(\displaystyle \int \left (\frac {(a+b x)^m \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^3}+\frac {(a+b x)^{m+1} \left (3 a^2 D-2 a b C+b^2 B\right )}{b^3}+\frac {(b C-3 a D) (a+b x)^{m+2}}{b^3}+\frac {D (a+b x)^{m+3}}{b^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(a+b x)^{m+1} \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^4 (m+1)}+\frac {(a+b x)^{m+2} \left (3 a^2 D-2 a b C+b^2 B\right )}{b^4 (m+2)}+\frac {(b C-3 a D) (a+b x)^{m+3}}{b^4 (m+3)}+\frac {D (a+b x)^{m+4}}{b^4 (m+4)}\)

Input:

Int[(a + b*x)^m*(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)^(1 + m))/(b^4*(1 + m)) + (( 
b^2*B - 2*a*b*C + 3*a^2*D)*(a + b*x)^(2 + m))/(b^4*(2 + m)) + ((b*C - 3*a* 
D)*(a + b*x)^(3 + m))/(b^4*(3 + m)) + (D*(a + b*x)^(4 + m))/(b^4*(4 + m))
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(124)=248\).

Time = 0.34 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.48

method	result	size
gosper	\(\frac {\left (b x +a \right )^{1+m} \left (D b^{3} m^{3} x^{3}+C \,b^{3} m^{3} x^{2}+6 D b^{3} m^{2} x^{3}+B \,b^{3} m^{3} x +7 C \,b^{3} m^{2} x^{2}-3 D a \,b^{2} m^{2} x^{2}+11 D b^{3} m \,x^{3}+A \,b^{3} m^{3}+8 B \,b^{3} m^{2} x -2 C a \,b^{2} m^{2} x +14 C \,b^{3} m \,x^{2}-9 D a \,b^{2} m \,x^{2}+6 D x^{3} b^{3}+9 A \,b^{3} m^{2}-B a \,b^{2} m^{2}+19 B \,b^{3} m x -10 C a \,b^{2} m x +8 C \,x^{2} b^{3}+6 D a^{2} b m x -6 D x^{2} a \,b^{2}+26 A \,b^{3} m -7 B a \,b^{2} m +12 b^{3} B x +2 C \,a^{2} b m -8 C x a \,b^{2}+6 D x \,a^{2} b +24 b^{3} A -12 a \,b^{2} B +8 a^{2} b C -6 a^{3} D\right )}{b^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(308\)
orering	\(\frac {\left (b x +a \right )^{m} \left (D b^{3} m^{3} x^{3}+C \,b^{3} m^{3} x^{2}+6 D b^{3} m^{2} x^{3}+B \,b^{3} m^{3} x +7 C \,b^{3} m^{2} x^{2}-3 D a \,b^{2} m^{2} x^{2}+11 D b^{3} m \,x^{3}+A \,b^{3} m^{3}+8 B \,b^{3} m^{2} x -2 C a \,b^{2} m^{2} x +14 C \,b^{3} m \,x^{2}-9 D a \,b^{2} m \,x^{2}+6 D x^{3} b^{3}+9 A \,b^{3} m^{2}-B a \,b^{2} m^{2}+19 B \,b^{3} m x -10 C a \,b^{2} m x +8 C \,x^{2} b^{3}+6 D a^{2} b m x -6 D x^{2} a \,b^{2}+26 A \,b^{3} m -7 B a \,b^{2} m +12 b^{3} B x +2 C \,a^{2} b m -8 C x a \,b^{2}+6 D x \,a^{2} b +24 b^{3} A -12 a \,b^{2} B +8 a^{2} b C -6 a^{3} D\right ) \left (b x +a \right )}{b^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(311\)
norman	\(\frac {D x^{4} {\mathrm e}^{m \ln \left (b x +a \right )}}{4+m}+\frac {a \left (A \,b^{3} m^{3}+9 A \,b^{3} m^{2}-B a \,b^{2} m^{2}+26 A \,b^{3} m -7 B a \,b^{2} m +2 C \,a^{2} b m +24 b^{3} A -12 a \,b^{2} B +8 a^{2} b C -6 a^{3} D\right ) {\mathrm e}^{m \ln \left (b x +a \right )}}{b^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (C b m +D a m +4 C b \right ) x^{3} {\mathrm e}^{m \ln \left (b x +a \right )}}{b \left (m^{2}+7 m +12\right )}+\frac {\left (B \,b^{2} m^{2}+C a b \,m^{2}+7 B \,b^{2} m +4 C a b m -3 D a^{2} m +12 B \,b^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (b x +a \right )}}{b^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (A \,b^{3} m^{3}+B a \,b^{2} m^{3}+9 A \,b^{3} m^{2}+7 B a \,b^{2} m^{2}-2 C \,a^{2} b \,m^{2}+26 A \,b^{3} m +12 B a \,b^{2} m -8 C \,a^{2} b m +6 D a^{3} m +24 b^{3} A \right ) x \,{\mathrm e}^{m \ln \left (b x +a \right )}}{b^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(361\)
parallelrisch	\(\frac {7 B x \left (b x +a \right )^{m} a^{2} b^{3} m^{2}+4 C \,x^{2} \left (b x +a \right )^{m} a^{2} b^{3} m -2 C x \left (b x +a \right )^{m} a^{3} b^{2} m^{2}-3 D x^{2} \left (b x +a \right )^{m} a^{3} b^{2} m +26 A x \left (b x +a \right )^{m} a \,b^{4} m +12 B x \left (b x +a \right )^{m} a^{2} b^{3} m -8 C x \left (b x +a \right )^{m} a^{3} b^{2} m +6 D x^{4} \left (b x +a \right )^{m} a \,b^{4}+A \left (b x +a \right )^{m} a^{2} b^{3} m^{3}+8 C \,x^{3} \left (b x +a \right )^{m} a \,b^{4}+9 A \left (b x +a \right )^{m} a^{2} b^{3} m^{2}+12 B \,x^{2} \left (b x +a \right )^{m} a \,b^{4}-B \left (b x +a \right )^{m} a^{3} b^{2} m^{2}+24 A x \left (b x +a \right )^{m} a \,b^{4}+26 A \left (b x +a \right )^{m} a^{2} b^{3} m -7 B \left (b x +a \right )^{m} a^{3} b^{2} m +2 C \left (b x +a \right )^{m} a^{4} b m +24 A \left (b x +a \right )^{m} a^{2} b^{3}-12 B \left (b x +a \right )^{m} a^{3} b^{2}+8 C \left (b x +a \right )^{m} a^{4} b +6 D x \left (b x +a \right )^{m} a^{4} b m +D x^{4} \left (b x +a \right )^{m} a \,b^{4} m^{3}+C \,x^{3} \left (b x +a \right )^{m} a \,b^{4} m^{3}+6 D x^{4} \left (b x +a \right )^{m} a \,b^{4} m^{2}+D x^{3} \left (b x +a \right )^{m} a^{2} b^{3} m^{3}+B \,x^{2} \left (b x +a \right )^{m} a \,b^{4} m^{3}+7 C \,x^{3} \left (b x +a \right )^{m} a \,b^{4} m^{2}+C \,x^{2} \left (b x +a \right )^{m} a^{2} b^{3} m^{3}+11 D x^{4} \left (b x +a \right )^{m} a \,b^{4} m +3 D x^{3} \left (b x +a \right )^{m} a^{2} b^{3} m^{2}+A x \left (b x +a \right )^{m} a \,b^{4} m^{3}+8 B \,x^{2} \left (b x +a \right )^{m} a \,b^{4} m^{2}+B x \left (b x +a \right )^{m} a^{2} b^{3} m^{3}+14 C \,x^{3} \left (b x +a \right )^{m} a \,b^{4} m +5 C \,x^{2} \left (b x +a \right )^{m} a^{2} b^{3} m^{2}+2 D x^{3} \left (b x +a \right )^{m} a^{2} b^{3} m -3 D x^{2} \left (b x +a \right )^{m} a^{3} b^{2} m^{2}+9 A x \left (b x +a \right )^{m} a \,b^{4} m^{2}+19 B \,x^{2} \left (b x +a \right )^{m} a \,b^{4} m -6 D \left (b x +a \right )^{m} a^{5}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) a \,b^{4}}\)	\(760\)

Input:

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

Output:

1/b^4*(b*x+a)^(1+m)/(m^4+10*m^3+35*m^2+50*m+24)*(D*b^3*m^3*x^3+C*b^3*m^3*x 
^2+6*D*b^3*m^2*x^3+B*b^3*m^3*x+7*C*b^3*m^2*x^2-3*D*a*b^2*m^2*x^2+11*D*b^3* 
m*x^3+A*b^3*m^3+8*B*b^3*m^2*x-2*C*a*b^2*m^2*x+14*C*b^3*m*x^2-9*D*a*b^2*m*x 
^2+6*D*b^3*x^3+9*A*b^3*m^2-B*a*b^2*m^2+19*B*b^3*m*x-10*C*a*b^2*m*x+8*C*b^3 
*x^2+6*D*a^2*b*m*x-6*D*a*b^2*x^2+26*A*b^3*m-7*B*a*b^2*m+12*B*b^3*x+2*C*a^2 
*b*m-8*C*a*b^2*x+6*D*a^2*b*x+24*A*b^3-12*B*a*b^2+8*C*a^2*b-6*D*a^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (126) = 252\).

Time = 0.08 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.18 \[ \int (a+b x)^m \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {{\left (A a b^{3} m^{3} - 6 \, D a^{4} + 8 \, C a^{3} b - 12 \, B a^{2} b^{2} + 24 \, A a b^{3} + {\left (D b^{4} m^{3} + 6 \, D b^{4} m^{2} + 11 \, D b^{4} m + 6 \, D b^{4}\right )} x^{4} + {\left (8 \, C b^{4} + {\left (D a b^{3} + C b^{4}\right )} m^{3} + {\left (3 \, D a b^{3} + 7 \, C b^{4}\right )} m^{2} + 2 \, {\left (D a b^{3} + 7 \, C b^{4}\right )} m\right )} x^{3} - {\left (B a^{2} b^{2} - 9 \, A a b^{3}\right )} m^{2} + {\left (12 \, B b^{4} + {\left (C a b^{3} + B b^{4}\right )} m^{3} - {\left (3 \, D a^{2} b^{2} - 5 \, C a b^{3} - 8 \, B b^{4}\right )} m^{2} - {\left (3 \, D a^{2} b^{2} - 4 \, C a b^{3} - 19 \, B b^{4}\right )} m\right )} x^{2} + {\left (2 \, C a^{3} b - 7 \, B a^{2} b^{2} + 26 \, A a b^{3}\right )} m + {\left (24 \, A b^{4} + {\left (B a b^{3} + A b^{4}\right )} m^{3} - {\left (2 \, C a^{2} b^{2} - 7 \, B a b^{3} - 9 \, A b^{4}\right )} m^{2} + 2 \, {\left (3 \, D a^{3} b - 4 \, C a^{2} b^{2} + 6 \, B a b^{3} + 13 \, A b^{4}\right )} m\right )} x\right )} {\left (b x + a\right )}^{m}}{b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}} \] Input:

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

Output:

(A*a*b^3*m^3 - 6*D*a^4 + 8*C*a^3*b - 12*B*a^2*b^2 + 24*A*a*b^3 + (D*b^4*m^ 
3 + 6*D*b^4*m^2 + 11*D*b^4*m + 6*D*b^4)*x^4 + (8*C*b^4 + (D*a*b^3 + C*b^4) 
*m^3 + (3*D*a*b^3 + 7*C*b^4)*m^2 + 2*(D*a*b^3 + 7*C*b^4)*m)*x^3 - (B*a^2*b 
^2 - 9*A*a*b^3)*m^2 + (12*B*b^4 + (C*a*b^3 + B*b^4)*m^3 - (3*D*a^2*b^2 - 5 
*C*a*b^3 - 8*B*b^4)*m^2 - (3*D*a^2*b^2 - 4*C*a*b^3 - 19*B*b^4)*m)*x^2 + (2 
*C*a^3*b - 7*B*a^2*b^2 + 26*A*a*b^3)*m + (24*A*b^4 + (B*a*b^3 + A*b^4)*m^3 
 - (2*C*a^2*b^2 - 7*B*a*b^3 - 9*A*b^4)*m^2 + 2*(3*D*a^3*b - 4*C*a^2*b^2 + 
6*B*a*b^3 + 13*A*b^4)*m)*x)*(b*x + a)^m/(b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 
 + 50*b^4*m + 24*b^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3798 vs. \(2 (110) = 220\).

Time = 1.07 (sec) , antiderivative size = 3798, normalized size of antiderivative = 30.63 \[ \int (a+b x)^m \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((a**m*(A*x + B*x**2/2 + C*x**3/3 + D*x**4/4), Eq(b, 0)), (-2*A*b 
**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - B*a*b* 
*2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 3*B*b** 
3*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*C*a* 
*2*b/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 6*C*a 
*b**2*x/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 6* 
C*b**3*x**2/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) 
+ 6*D*a**3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6 
*b**7*x**3) + 11*D*a**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6 
*b**7*x**3) + 18*D*a**2*b*x*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 1 
8*a*b**6*x**2 + 6*b**7*x**3) + 27*D*a**2*b*x/(6*a**3*b**4 + 18*a**2*b**5*x 
 + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*D*a*b**2*x**2*log(a/b + x)/(6*a**3*b 
**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*D*a*b**2*x**2/(6 
*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*D*b**3*x** 
3*log(a/b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x** 
3), Eq(m, -4)), (-A*b**3/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - B*a*b* 
*2/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 2*B*b**3*x/(2*a**2*b**4 + 4* 
a*b**5*x + 2*b**6*x**2) + 2*C*a**2*b*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5* 
x + 2*b**6*x**2) + 3*C*a**2*b/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 4 
*C*a*b**2*x*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 4*C...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.89 \[ \int (a+b x)^m \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {{\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} B}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{m + 1} A}{b {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} C}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a b^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b m x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{m} D}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{4}} \] Input:

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

Output:

(b^2*(m + 1)*x^2 + a*b*m*x - a^2)*(b*x + a)^m*B/((m^2 + 3*m + 2)*b^2) + (b 
*x + a)^(m + 1)*A/(b*(m + 1)) + ((m^2 + 3*m + 2)*b^3*x^3 + (m^2 + m)*a*b^2 
*x^2 - 2*a^2*b*m*x + 2*a^3)*(b*x + a)^m*C/((m^3 + 6*m^2 + 11*m + 6)*b^3) + 
 ((m^3 + 6*m^2 + 11*m + 6)*b^4*x^4 + (m^3 + 3*m^2 + 2*m)*a*b^3*x^3 - 3*(m^ 
2 + m)*a^2*b^2*x^2 + 6*a^3*b*m*x - 6*a^4)*(b*x + a)^m*D/((m^4 + 10*m^3 + 3 
5*m^2 + 50*m + 24)*b^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (126) = 252\).

Time = 0.13 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.87 \[ \int (a+b x)^m \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

((b*x + a)^m*D*b^4*m^3*x^4 + (b*x + a)^m*D*a*b^3*m^3*x^3 + (b*x + a)^m*C*b 
^4*m^3*x^3 + 6*(b*x + a)^m*D*b^4*m^2*x^4 + (b*x + a)^m*C*a*b^3*m^3*x^2 + ( 
b*x + a)^m*B*b^4*m^3*x^2 + 3*(b*x + a)^m*D*a*b^3*m^2*x^3 + 7*(b*x + a)^m*C 
*b^4*m^2*x^3 + 11*(b*x + a)^m*D*b^4*m*x^4 + (b*x + a)^m*B*a*b^3*m^3*x + (b 
*x + a)^m*A*b^4*m^3*x - 3*(b*x + a)^m*D*a^2*b^2*m^2*x^2 + 5*(b*x + a)^m*C* 
a*b^3*m^2*x^2 + 8*(b*x + a)^m*B*b^4*m^2*x^2 + 2*(b*x + a)^m*D*a*b^3*m*x^3 
+ 14*(b*x + a)^m*C*b^4*m*x^3 + 6*(b*x + a)^m*D*b^4*x^4 + (b*x + a)^m*A*a*b 
^3*m^3 - 2*(b*x + a)^m*C*a^2*b^2*m^2*x + 7*(b*x + a)^m*B*a*b^3*m^2*x + 9*( 
b*x + a)^m*A*b^4*m^2*x - 3*(b*x + a)^m*D*a^2*b^2*m*x^2 + 4*(b*x + a)^m*C*a 
*b^3*m*x^2 + 19*(b*x + a)^m*B*b^4*m*x^2 + 8*(b*x + a)^m*C*b^4*x^3 - (b*x + 
 a)^m*B*a^2*b^2*m^2 + 9*(b*x + a)^m*A*a*b^3*m^2 + 6*(b*x + a)^m*D*a^3*b*m* 
x - 8*(b*x + a)^m*C*a^2*b^2*m*x + 12*(b*x + a)^m*B*a*b^3*m*x + 26*(b*x + a 
)^m*A*b^4*m*x + 12*(b*x + a)^m*B*b^4*x^2 + 2*(b*x + a)^m*C*a^3*b*m - 7*(b* 
x + a)^m*B*a^2*b^2*m + 26*(b*x + a)^m*A*a*b^3*m + 24*(b*x + a)^m*A*b^4*x - 
 6*(b*x + a)^m*D*a^4 + 8*(b*x + a)^m*C*a^3*b - 12*(b*x + a)^m*B*a^2*b^2 + 
24*(b*x + a)^m*A*a*b^3)/(b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24 
*b^4)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^m \left (A+B x+C x^2+D x^3\right ) \, dx=\int {\left (a+b\,x\right )}^m\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.07 \[ \int (a+b x)^m \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\left (b x +a \right )^{m} \left (b^{4} d \,m^{3} x^{4}+a \,b^{3} d \,m^{3} x^{3}+b^{4} c \,m^{3} x^{3}+6 b^{4} d \,m^{2} x^{4}+a \,b^{3} c \,m^{3} x^{2}+3 a \,b^{3} d \,m^{2} x^{3}+b^{5} m^{3} x^{2}+7 b^{4} c \,m^{2} x^{3}+11 b^{4} d m \,x^{4}-3 a^{2} b^{2} d \,m^{2} x^{2}+2 a \,b^{4} m^{3} x +5 a \,b^{3} c \,m^{2} x^{2}+2 a \,b^{3} d m \,x^{3}+8 b^{5} m^{2} x^{2}+14 b^{4} c m \,x^{3}+6 b^{4} d \,x^{4}+a^{2} b^{3} m^{3}-2 a^{2} b^{2} c \,m^{2} x -3 a^{2} b^{2} d m \,x^{2}+16 a \,b^{4} m^{2} x +4 a \,b^{3} c m \,x^{2}+19 b^{5} m \,x^{2}+8 b^{4} c \,x^{3}+6 a^{3} b d m x +8 a^{2} b^{3} m^{2}-8 a^{2} b^{2} c m x +38 a \,b^{4} m x +12 b^{5} x^{2}+2 a^{3} b c m +19 a^{2} b^{3} m +24 a \,b^{4} x -6 a^{4} d +8 a^{3} b c +12 a^{2} b^{3}\right )}{b^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )} \] Input:

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

Output:

((a + b*x)**m*( - 6*a**4*d + 2*a**3*b*c*m + 8*a**3*b*c + 6*a**3*b*d*m*x + 
a**2*b**3*m**3 + 8*a**2*b**3*m**2 + 19*a**2*b**3*m + 12*a**2*b**3 - 2*a**2 
*b**2*c*m**2*x - 8*a**2*b**2*c*m*x - 3*a**2*b**2*d*m**2*x**2 - 3*a**2*b**2 
*d*m*x**2 + 2*a*b**4*m**3*x + 16*a*b**4*m**2*x + 38*a*b**4*m*x + 24*a*b**4 
*x + a*b**3*c*m**3*x**2 + 5*a*b**3*c*m**2*x**2 + 4*a*b**3*c*m*x**2 + a*b** 
3*d*m**3*x**3 + 3*a*b**3*d*m**2*x**3 + 2*a*b**3*d*m*x**3 + b**5*m**3*x**2 
+ 8*b**5*m**2*x**2 + 19*b**5*m*x**2 + 12*b**5*x**2 + b**4*c*m**3*x**3 + 7* 
b**4*c*m**2*x**3 + 14*b**4*c*m*x**3 + 8*b**4*c*x**3 + b**4*d*m**3*x**4 + 6 
*b**4*d*m**2*x**4 + 11*b**4*d*m*x**4 + 6*b**4*d*x**4))/(b**4*(m**4 + 10*m* 
*3 + 35*m**2 + 50*m + 24))