Integrand size = 23, antiderivative size = 108 \[ \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) (a+b x)^4}{4 b^4}+\frac {\left (b^2 B-2 a b C+3 a^2 D\right ) (a+b x)^5}{5 b^4}+\frac {(b C-3 a D) (a+b x)^6}{6 b^4}+\frac {D (a+b x)^7}{7 b^4} \] Output:
1/4*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(b*x+a)^4/b^4+1/5*(B*b^2-2*C*a*b+3*D*a^2 )*(b*x+a)^5/b^4+1/6*(C*b-3*D*a)*(b*x+a)^6/b^4+1/7*D*(b*x+a)^7/b^4
Time = 0.01 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.26 \[ \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx=a^3 A x+\frac {1}{2} a^2 (3 A b+a B) x^2+\frac {1}{3} a \left (3 A b^2+3 a b B+a^2 C\right ) x^3+\frac {1}{4} \left (A b^3+3 a b^2 B+3 a^2 b C+a^3 D\right ) x^4+\frac {1}{5} b \left (b^2 B+3 a b C+3 a^2 D\right ) x^5+\frac {1}{6} b^2 (b C+3 a D) x^6+\frac {1}{7} b^3 D x^7 \] Input:
Integrate[(a + b*x)^3*(A + B*x + C*x^2 + D*x^3),x]
Output:
a^3*A*x + (a^2*(3*A*b + a*B)*x^2)/2 + (a*(3*A*b^2 + 3*a*b*B + a^2*C)*x^3)/ 3 + ((A*b^3 + 3*a*b^2*B + 3*a^2*b*C + a^3*D)*x^4)/4 + (b*(b^2*B + 3*a*b*C + 3*a^2*D)*x^5)/5 + (b^2*(b*C + 3*a*D)*x^6)/6 + (b^3*D*x^7)/7
Time = 0.33 (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.
\(\displaystyle \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {(a+b x)^3 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^3}+\frac {(a+b x)^4 \left (3 a^2 D-2 a b C+b^2 B\right )}{b^3}+\frac {(a+b x)^5 (b C-3 a D)}{b^3}+\frac {D (a+b x)^6}{b^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b x)^4 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{4 b^4}+\frac {(a+b x)^5 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 b^4}+\frac {(a+b x)^6 (b C-3 a D)}{6 b^4}+\frac {D (a+b x)^7}{7 b^4}\) |
Input:
Int[(a + b*x)^3*(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)^4)/(4*b^4) + ((b^2*B - 2*a* b*C + 3*a^2*D)*(a + b*x)^5)/(5*b^4) + ((b*C - 3*a*D)*(a + b*x)^6)/(6*b^4) + (D*(a + b*x)^7)/(7*b^4)
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])
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.22
method | result | size |
norman | \(\frac {b^{3} D x^{7}}{7}+\left (\frac {1}{6} b^{3} C +\frac {1}{2} a \,b^{2} D\right ) x^{6}+\left (\frac {1}{5} b^{3} B +\frac {3}{5} a \,b^{2} C +\frac {3}{5} a^{2} b D\right ) x^{5}+\left (\frac {1}{4} b^{3} A +\frac {3}{4} a \,b^{2} B +\frac {3}{4} a^{2} b C +\frac {1}{4} a^{3} D\right ) x^{4}+\left (a \,b^{2} A +a^{2} b B +\frac {1}{3} a^{3} C \right ) x^{3}+\left (\frac {3}{2} a^{2} b A +\frac {1}{2} a^{3} B \right ) x^{2}+a^{3} A x\) | \(132\) |
default | \(\frac {b^{3} D x^{7}}{7}+\frac {\left (b^{3} C +3 a \,b^{2} D\right ) x^{6}}{6}+\frac {\left (b^{3} B +3 a \,b^{2} C +3 a^{2} b D\right ) x^{5}}{5}+\frac {\left (b^{3} A +3 a \,b^{2} B +3 a^{2} b C +a^{3} D\right ) x^{4}}{4}+\frac {\left (3 a \,b^{2} A +3 a^{2} b B +a^{3} C \right ) x^{3}}{3}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{2}}{2}+a^{3} A x\) | \(133\) |
gosper | \(\frac {1}{7} b^{3} D x^{7}+\frac {1}{6} x^{6} b^{3} C +\frac {1}{2} x^{6} a \,b^{2} D+\frac {1}{5} b^{3} B \,x^{5}+\frac {3}{5} x^{5} a \,b^{2} C +\frac {3}{5} x^{5} a^{2} b D+\frac {1}{4} x^{4} b^{3} A +\frac {3}{4} x^{4} a \,b^{2} B +\frac {3}{4} x^{4} a^{2} b C +\frac {1}{4} x^{4} a^{3} D+x^{3} a \,b^{2} A +x^{3} a^{2} b B +\frac {1}{3} x^{3} a^{3} C +\frac {3}{2} x^{2} a^{2} b A +\frac {1}{2} x^{2} a^{3} B +a^{3} A x\) | \(149\) |
parallelrisch | \(\frac {1}{7} b^{3} D x^{7}+\frac {1}{6} x^{6} b^{3} C +\frac {1}{2} x^{6} a \,b^{2} D+\frac {1}{5} b^{3} B \,x^{5}+\frac {3}{5} x^{5} a \,b^{2} C +\frac {3}{5} x^{5} a^{2} b D+\frac {1}{4} x^{4} b^{3} A +\frac {3}{4} x^{4} a \,b^{2} B +\frac {3}{4} x^{4} a^{2} b C +\frac {1}{4} x^{4} a^{3} D+x^{3} a \,b^{2} A +x^{3} a^{2} b B +\frac {1}{3} x^{3} a^{3} C +\frac {3}{2} x^{2} a^{2} b A +\frac {1}{2} x^{2} a^{3} B +a^{3} A x\) | \(149\) |
orering | \(\frac {x \left (60 b^{3} D x^{6}+70 C \,b^{3} x^{5}+210 D a \,b^{2} x^{5}+84 B \,b^{3} x^{4}+252 C a \,b^{2} x^{4}+252 D a^{2} b \,x^{4}+105 A \,b^{3} x^{3}+315 B a \,b^{2} x^{3}+315 C \,a^{2} b \,x^{3}+105 D a^{3} x^{3}+420 a A \,b^{2} x^{2}+420 B \,a^{2} b \,x^{2}+140 C \,a^{3} x^{2}+630 a^{2} A b x +210 B \,a^{3} x +420 a^{3} A \right )}{420}\) | \(150\) |
Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
Output:
1/7*b^3*D*x^7+(1/6*b^3*C+1/2*a*b^2*D)*x^6+(1/5*b^3*B+3/5*a*b^2*C+3/5*a^2*b *D)*x^5+(1/4*b^3*A+3/4*a*b^2*B+3/4*a^2*b*C+1/4*a^3*D)*x^4+(a*b^2*A+a^2*b*B +1/3*a^3*C)*x^3+(3/2*a^2*b*A+1/2*a^3*B)*x^2+a^3*A*x
Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.22 \[ \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b^{3} x^{7} + \frac {1}{6} \, {\left (3 \, D a b^{2} + C b^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} x^{4} + \frac {1}{3} \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
Output:
1/7*D*b^3*x^7 + 1/6*(3*D*a*b^2 + C*b^3)*x^6 + 1/5*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*x^5 + A*a^3*x + 1/4*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*x^4 + 1/3*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*x^3 + 1/2*(B*a^3 + 3*A*a^2*b)*x^2
Time = 0.03 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.35 \[ \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx=A a^{3} x + \frac {D b^{3} x^{7}}{7} + x^{6} \left (\frac {C b^{3}}{6} + \frac {D a b^{2}}{2}\right ) + x^{5} \left (\frac {B b^{3}}{5} + \frac {3 C a b^{2}}{5} + \frac {3 D a^{2} b}{5}\right ) + x^{4} \left (\frac {A b^{3}}{4} + \frac {3 B a b^{2}}{4} + \frac {3 C a^{2} b}{4} + \frac {D a^{3}}{4}\right ) + x^{3} \left (A a b^{2} + B a^{2} b + \frac {C a^{3}}{3}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} b}{2} + \frac {B a^{3}}{2}\right ) \] Input:
integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A),x)
Output:
A*a**3*x + D*b**3*x**7/7 + x**6*(C*b**3/6 + D*a*b**2/2) + x**5*(B*b**3/5 + 3*C*a*b**2/5 + 3*D*a**2*b/5) + x**4*(A*b**3/4 + 3*B*a*b**2/4 + 3*C*a**2*b /4 + D*a**3/4) + x**3*(A*a*b**2 + B*a**2*b + C*a**3/3) + x**2*(3*A*a**2*b/ 2 + B*a**3/2)
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.22 \[ \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b^{3} x^{7} + \frac {1}{6} \, {\left (3 \, D a b^{2} + C b^{3}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} x^{4} + \frac {1}{3} \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
Output:
1/7*D*b^3*x^7 + 1/6*(3*D*a*b^2 + C*b^3)*x^6 + 1/5*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*x^5 + A*a^3*x + 1/4*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*x^4 + 1/3*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*x^3 + 1/2*(B*a^3 + 3*A*a^2*b)*x^2
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.37 \[ \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b^{3} x^{7} + \frac {1}{2} \, D a b^{2} x^{6} + \frac {1}{6} \, C b^{3} x^{6} + \frac {3}{5} \, D a^{2} b x^{5} + \frac {3}{5} \, C a b^{2} x^{5} + \frac {1}{5} \, B b^{3} x^{5} + \frac {1}{4} \, D a^{3} x^{4} + \frac {3}{4} \, C a^{2} b x^{4} + \frac {3}{4} \, B a b^{2} x^{4} + \frac {1}{4} \, A b^{3} x^{4} + \frac {1}{3} \, C a^{3} x^{3} + B a^{2} b x^{3} + A a b^{2} x^{3} + \frac {1}{2} \, B a^{3} x^{2} + \frac {3}{2} \, A a^{2} b x^{2} + A a^{3} x \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
Output:
1/7*D*b^3*x^7 + 1/2*D*a*b^2*x^6 + 1/6*C*b^3*x^6 + 3/5*D*a^2*b*x^5 + 3/5*C* a*b^2*x^5 + 1/5*B*b^3*x^5 + 1/4*D*a^3*x^4 + 3/4*C*a^2*b*x^4 + 3/4*B*a*b^2* x^4 + 1/4*A*b^3*x^4 + 1/3*C*a^3*x^3 + B*a^2*b*x^3 + A*a*b^2*x^3 + 1/2*B*a^ 3*x^2 + 3/2*A*a^2*b*x^2 + A*a^3*x
Time = 3.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.37 \[ \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {B\,a^3\,x^2}{2}+\frac {A\,b^3\,x^4}{4}+\frac {C\,a^3\,x^3}{3}+\frac {B\,b^3\,x^5}{5}+\frac {C\,b^3\,x^6}{6}+\frac {a^3\,x^4\,D}{4}+\frac {b^3\,x^7\,D}{7}+A\,a^3\,x+\frac {3\,a^2\,b\,x^5\,D}{5}+\frac {a\,b^2\,x^6\,D}{2}+\frac {3\,A\,a^2\,b\,x^2}{2}+A\,a\,b^2\,x^3+B\,a^2\,b\,x^3+\frac {3\,B\,a\,b^2\,x^4}{4}+\frac {3\,C\,a^2\,b\,x^4}{4}+\frac {3\,C\,a\,b^2\,x^5}{5} \] Input:
int((a + b*x)^3*(A + B*x + C*x^2 + x^3*D),x)
Output:
(B*a^3*x^2)/2 + (A*b^3*x^4)/4 + (C*a^3*x^3)/3 + (B*b^3*x^5)/5 + (C*b^3*x^6 )/6 + (a^3*x^4*D)/4 + (b^3*x^7*D)/7 + A*a^3*x + (3*a^2*b*x^5*D)/5 + (a*b^2 *x^6*D)/2 + (3*A*a^2*b*x^2)/2 + A*a*b^2*x^3 + B*a^2*b*x^3 + (3*B*a*b^2*x^4 )/4 + (3*C*a^2*b*x^4)/4 + (3*C*a*b^2*x^5)/5
Time = 0.14 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11 \[ \int (a+b x)^3 \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (60 b^{3} d \,x^{6}+210 a \,b^{2} d \,x^{5}+70 b^{3} c \,x^{5}+252 a^{2} b d \,x^{4}+252 a \,b^{2} c \,x^{4}+84 b^{4} x^{4}+105 a^{3} d \,x^{3}+315 a^{2} b c \,x^{3}+420 a \,b^{3} x^{3}+140 a^{3} c \,x^{2}+840 a^{2} b^{2} x^{2}+840 a^{3} b x +420 a^{4}\right )}{420} \] Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A),x)
Output:
(x*(420*a**4 + 840*a**3*b*x + 140*a**3*c*x**2 + 105*a**3*d*x**3 + 840*a**2 *b**2*x**2 + 315*a**2*b*c*x**3 + 252*a**2*b*d*x**4 + 420*a*b**3*x**3 + 252 *a*b**2*c*x**4 + 210*a*b**2*d*x**5 + 84*b**4*x**4 + 70*b**3*c*x**5 + 60*b* *3*d*x**6))/420