Integrand size = 23, antiderivative size = 108 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx=-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{6 b^4 (a+b x)^6}-\frac {b^2 B-2 a b C+3 a^2 D}{5 b^4 (a+b x)^5}-\frac {b C-3 a D}{4 b^4 (a+b x)^4}-\frac {D}{3 b^4 (a+b x)^3} \] Output:
-1/6*(A*b^3-a*(B*b^2-C*a*b+D*a^2))/b^4/(b*x+a)^6-1/5*(B*b^2-2*C*a*b+3*D*a^ 2)/b^4/(b*x+a)^5-1/4*(C*b-3*D*a)/b^4/(b*x+a)^4-1/3*D/b^4/(b*x+a)^3
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx=-\frac {10 A b^3+a^3 D+a^2 b (C+6 D x)+b^3 x (12 B+5 x (3 C+4 D x))+a b^2 (2 B+3 x (2 C+5 D x))}{60 b^4 (a+b x)^6} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x)^7,x]
Output:
-1/60*(10*A*b^3 + a^3*D + a^2*b*(C + 6*D*x) + b^3*x*(12*B + 5*x*(3*C + 4*D *x)) + a*b^2*(2*B + 3*x*(2*C + 5*D*x)))/(b^4*(a + b*x)^6)
Time = 0.30 (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.
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{b^3 (a+b x)^7}+\frac {3 a^2 D-2 a b C+b^2 B}{b^3 (a+b x)^6}+\frac {b C-3 a D}{b^3 (a+b x)^5}+\frac {D}{b^3 (a+b x)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{6 b^4 (a+b x)^6}-\frac {3 a^2 D-2 a b C+b^2 B}{5 b^4 (a+b x)^5}-\frac {b C-3 a D}{4 b^4 (a+b x)^4}-\frac {D}{3 b^4 (a+b x)^3}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(a + b*x)^7,x]
Output:
-1/6*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(b^4*(a + b*x)^6) - (b^2*B - 2*a* b*C + 3*a^2*D)/(5*b^4*(a + b*x)^5) - (b*C - 3*a*D)/(4*b^4*(a + b*x)^4) - D /(3*b^4*(a + b*x)^3)
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 = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {20 D x^{3} b^{3}+15 C \,x^{2} b^{3}+15 D x^{2} a \,b^{2}+12 b^{3} B x +6 C x a \,b^{2}+6 D x \,a^{2} b +10 b^{3} A +2 a \,b^{2} B +a^{2} b C +a^{3} D}{60 b^{4} \left (b x +a \right )^{6}}\) | \(89\) |
orering | \(-\frac {20 D x^{3} b^{3}+15 C \,x^{2} b^{3}+15 D x^{2} a \,b^{2}+12 b^{3} B x +6 C x a \,b^{2}+6 D x \,a^{2} b +10 b^{3} A +2 a \,b^{2} B +a^{2} b C +a^{3} D}{60 b^{4} \left (b x +a \right )^{6}}\) | \(89\) |
parallelrisch | \(-\frac {20 D x^{3} b^{5}+15 C \,b^{5} x^{2}+15 D a \,b^{4} x^{2}+12 b^{5} B x +6 C a \,b^{4} x +6 D a^{2} b^{3} x +10 b^{5} A +2 a \,b^{4} B +C \,a^{2} b^{3}+D a^{3} b^{2}}{60 b^{6} \left (b x +a \right )^{6}}\) | \(96\) |
default | \(-\frac {B \,b^{2}-2 C a b +3 D a^{2}}{5 b^{4} \left (b x +a \right )^{5}}-\frac {C b -3 D a}{4 b^{4} \left (b x +a \right )^{4}}-\frac {D}{3 b^{4} \left (b x +a \right )^{3}}-\frac {b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D}{6 b^{4} \left (b x +a \right )^{6}}\) | \(101\) |
norman | \(\frac {-\frac {D x^{3}}{3 b}-\frac {\left (b^{3} C +a \,b^{2} D\right ) x^{2}}{4 b^{4}}-\frac {\left (2 b^{4} B +a \,b^{3} C +a^{2} b^{2} D\right ) x}{10 b^{5}}-\frac {10 b^{5} A +2 a \,b^{4} B +C \,a^{2} b^{3}+D a^{3} b^{2}}{60 b^{6}}}{\left (b x +a \right )^{6}}\) | \(101\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(b*x+a)^7,x,method=_RETURNVERBOSE)
Output:
-1/60/b^4*(20*D*b^3*x^3+15*C*b^3*x^2+15*D*a*b^2*x^2+12*B*b^3*x+6*C*a*b^2*x +6*D*a^2*b*x+10*A*b^3+2*B*a*b^2+C*a^2*b+D*a^3)/(b*x+a)^6
Time = 0.06 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx=-\frac {20 \, D b^{3} x^{3} + D a^{3} + C a^{2} b + 2 \, B a b^{2} + 10 \, A b^{3} + 15 \, {\left (D a b^{2} + C b^{3}\right )} x^{2} + 6 \, {\left (D a^{2} b + C a b^{2} + 2 \, B b^{3}\right )} x}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^7,x, algorithm="fricas")
Output:
-1/60*(20*D*b^3*x^3 + D*a^3 + C*a^2*b + 2*B*a*b^2 + 10*A*b^3 + 15*(D*a*b^2 + C*b^3)*x^2 + 6*(D*a^2*b + C*a*b^2 + 2*B*b^3)*x)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3 + 15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^ 4)
Time = 15.66 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx=\frac {- 10 A b^{3} - 2 B a b^{2} - C a^{2} b - D a^{3} - 20 D b^{3} x^{3} + x^{2} \left (- 15 C b^{3} - 15 D a b^{2}\right ) + x \left (- 12 B b^{3} - 6 C a b^{2} - 6 D a^{2} b\right )}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)**7,x)
Output:
(-10*A*b**3 - 2*B*a*b**2 - C*a**2*b - D*a**3 - 20*D*b**3*x**3 + x**2*(-15* C*b**3 - 15*D*a*b**2) + x*(-12*B*b**3 - 6*C*a*b**2 - 6*D*a**2*b))/(60*a**6 *b**4 + 360*a**5*b**5*x + 900*a**4*b**6*x**2 + 1200*a**3*b**7*x**3 + 900*a **2*b**8*x**4 + 360*a*b**9*x**5 + 60*b**10*x**6)
Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx=-\frac {20 \, D b^{3} x^{3} + D a^{3} + C a^{2} b + 2 \, B a b^{2} + 10 \, A b^{3} + 15 \, {\left (D a b^{2} + C b^{3}\right )} x^{2} + 6 \, {\left (D a^{2} b + C a b^{2} + 2 \, B b^{3}\right )} x}{60 \, {\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^7,x, algorithm="maxima")
Output:
-1/60*(20*D*b^3*x^3 + D*a^3 + C*a^2*b + 2*B*a*b^2 + 10*A*b^3 + 15*(D*a*b^2 + C*b^3)*x^2 + 6*(D*a^2*b + C*a*b^2 + 2*B*b^3)*x)/(b^10*x^6 + 6*a*b^9*x^5 + 15*a^2*b^8*x^4 + 20*a^3*b^7*x^3 + 15*a^4*b^6*x^2 + 6*a^5*b^5*x + a^6*b^ 4)
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx=-\frac {20 \, D b^{3} x^{3} + 15 \, D a b^{2} x^{2} + 15 \, C b^{3} x^{2} + 6 \, D a^{2} b x + 6 \, C a b^{2} x + 12 \, B b^{3} x + D a^{3} + C a^{2} b + 2 \, B a b^{2} + 10 \, A b^{3}}{60 \, {\left (b x + a\right )}^{6} b^{4}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^7,x, algorithm="giac")
Output:
-1/60*(20*D*b^3*x^3 + 15*D*a*b^2*x^2 + 15*C*b^3*x^2 + 6*D*a^2*b*x + 6*C*a* b^2*x + 12*B*b^3*x + D*a^3 + C*a^2*b + 2*B*a*b^2 + 10*A*b^3)/((b*x + a)^6* b^4)
Time = 3.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.56 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx=\frac {D\,\left (\frac {3\,a}{4\,{\left (a+b\,x\right )}^4}-\frac {1}{3\,{\left (a+b\,x\right )}^3}-\frac {3\,a^2}{5\,{\left (a+b\,x\right )}^5}+\frac {a^3}{6\,{\left (a+b\,x\right )}^6}\right )}{b^4}-\frac {A}{6\,b\,\left (a^6+6\,a^5\,b\,x+15\,a^4\,b^2\,x^2+20\,a^3\,b^3\,x^3+15\,a^2\,b^4\,x^4+6\,a\,b^5\,x^5+b^6\,x^6\right )}-\frac {B\,\left (a+6\,b\,x\right )}{30\,b^2\,{\left (a+b\,x\right )}^6}-\frac {C\,\left (8\,a^2+48\,a\,b\,x+120\,b^2\,x^2\right )}{480\,b^3\,{\left (a+b\,x\right )}^6} \] Input:
int((A + B*x + C*x^2 + x^3*D)/(a + b*x)^7,x)
Output:
(D*((3*a)/(4*(a + b*x)^4) - 1/(3*(a + b*x)^3) - (3*a^2)/(5*(a + b*x)^5) + a^3/(6*(a + b*x)^6)))/b^4 - A/(6*b*(a^6 + b^6*x^6 + 6*a*b^5*x^5 + 15*a^4*b ^2*x^2 + 20*a^3*b^3*x^3 + 15*a^2*b^4*x^4 + 6*a^5*b*x)) - (B*(a + 6*b*x))/( 30*b^2*(a + b*x)^6) - (C*(8*a^2 + 120*b^2*x^2 + 48*a*b*x))/(480*b^3*(a + b *x)^6)
Time = 0.14 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^7} \, dx=\frac {-20 b^{3} d \,x^{3}-15 a \,b^{2} d \,x^{2}-15 b^{3} c \,x^{2}-6 a^{2} b d x -6 a \,b^{2} c x -12 b^{4} x -a^{3} d -a^{2} b c -12 a \,b^{3}}{60 b^{4} \left (b^{6} x^{6}+6 a \,b^{5} x^{5}+15 a^{2} b^{4} x^{4}+20 a^{3} b^{3} x^{3}+15 a^{4} b^{2} x^{2}+6 a^{5} b x +a^{6}\right )} \] Input:
int((D*x^3+C*x^2+B*x+A)/(b*x+a)^7,x)
Output:
( - a**3*d - a**2*b*c - 6*a**2*b*d*x - 12*a*b**3 - 6*a*b**2*c*x - 15*a*b** 2*d*x**2 - 12*b**4*x - 15*b**3*c*x**2 - 20*b**3*d*x**3)/(60*b**4*(a**6 + 6 *a**5*b*x + 15*a**4*b**2*x**2 + 20*a**3*b**3*x**3 + 15*a**2*b**4*x**4 + 6* a*b**5*x**5 + b**6*x**6))