Integrand size = 23, antiderivative size = 102 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx=-\frac {A b^3-a \left (b^2 B-a b C+a^2 D\right )}{3 b^4 (a+b x)^3}-\frac {b^2 B-2 a b C+3 a^2 D}{2 b^4 (a+b x)^2}-\frac {b C-3 a D}{b^4 (a+b x)}+\frac {D \log (a+b x)}{b^4} \] Output:
-1/3*(A*b^3-a*(B*b^2-C*a*b+D*a^2))/b^4/(b*x+a)^3-1/2*(B*b^2-2*C*a*b+3*D*a^ 2)/b^4/(b*x+a)^2-(C*b-3*D*a)/b^4/(b*x+a)+D*ln(b*x+a)/b^4
Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx=-\frac {2 A b^3+a b^2 B+2 a^2 b C-11 a^3 D+3 b^3 B x+6 a b^2 C x-27 a^2 b D x+6 b^3 C x^2-18 a b^2 D x^2-6 D (a+b x)^3 \log (a+b x)}{6 b^4 (a+b x)^3} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x)^4,x]
Output:
-1/6*(2*A*b^3 + a*b^2*B + 2*a^2*b*C - 11*a^3*D + 3*b^3*B*x + 6*a*b^2*C*x - 27*a^2*b*D*x + 6*b^3*C*x^2 - 18*a*b^2*D*x^2 - 6*D*(a + b*x)^3*Log[a + b*x ])/(b^4*(a + b*x)^3)
Time = 0.30 (sec) , antiderivative size = 102, 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.
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{b^3 (a+b x)^4}+\frac {3 a^2 D-2 a b C+b^2 B}{b^3 (a+b x)^3}+\frac {b C-3 a D}{b^3 (a+b x)^2}+\frac {D}{b^3 (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {A b^3-a \left (a^2 D-a b C+b^2 B\right )}{3 b^4 (a+b x)^3}-\frac {3 a^2 D-2 a b C+b^2 B}{2 b^4 (a+b x)^2}-\frac {b C-3 a D}{b^4 (a+b x)}+\frac {D \log (a+b x)}{b^4}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(a + b*x)^4,x]
Output:
-1/3*(A*b^3 - a*(b^2*B - a*b*C + a^2*D))/(b^4*(a + b*x)^3) - (b^2*B - 2*a* b*C + 3*a^2*D)/(2*b^4*(a + b*x)^2) - (b*C - 3*a*D)/(b^4*(a + b*x)) + (D*Lo g[a + b*x])/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 = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90
method | result | size |
norman | \(\frac {-\frac {2 b^{3} A +a \,b^{2} B +2 a^{2} b C -11 a^{3} D}{6 b^{4}}-\frac {\left (C b -3 D a \right ) x^{2}}{b^{2}}-\frac {\left (B \,b^{2}+2 C a b -9 D a^{2}\right ) x}{2 b^{3}}}{\left (b x +a \right )^{3}}+\frac {D \ln \left (b x +a \right )}{b^{4}}\) | \(92\) |
default | \(-\frac {B \,b^{2}-2 C a b +3 D a^{2}}{2 b^{4} \left (b x +a \right )^{2}}-\frac {C b -3 D a}{b^{4} \left (b x +a \right )}+\frac {D \ln \left (b x +a \right )}{b^{4}}-\frac {b^{3} A -a \,b^{2} B +a^{2} b C -a^{3} D}{3 b^{4} \left (b x +a \right )^{3}}\) | \(99\) |
parallelrisch | \(-\frac {-6 D \ln \left (b x +a \right ) x^{3} b^{3}-18 D \ln \left (b x +a \right ) x^{2} a \,b^{2}+6 C \,x^{2} b^{3}-18 D \ln \left (b x +a \right ) x \,a^{2} b -18 D x^{2} a \,b^{2}+3 b^{3} B x +6 C x a \,b^{2}-6 D \ln \left (b x +a \right ) a^{3}-27 D x \,a^{2} b +2 b^{3} A +a \,b^{2} B +2 a^{2} b C -11 a^{3} D}{6 b^{4} \left (b x +a \right )^{3}}\) | \(138\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
(-1/6*(2*A*b^3+B*a*b^2+2*C*a^2*b-11*D*a^3)/b^4-(C*b-3*D*a)/b^2*x^2-1/2*(B* b^2+2*C*a*b-9*D*a^2)/b^3*x)/(b*x+a)^3+D*ln(b*x+a)/b^4
Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx=\frac {11 \, D a^{3} - 2 \, C a^{2} b - B a b^{2} - 2 \, A b^{3} + 6 \, {\left (3 \, D a b^{2} - C b^{3}\right )} x^{2} + 3 \, {\left (9 \, D a^{2} b - 2 \, C a b^{2} - B b^{3}\right )} x + 6 \, {\left (D b^{3} x^{3} + 3 \, D a b^{2} x^{2} + 3 \, D a^{2} b x + D a^{3}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x, algorithm="fricas")
Output:
1/6*(11*D*a^3 - 2*C*a^2*b - B*a*b^2 - 2*A*b^3 + 6*(3*D*a*b^2 - C*b^3)*x^2 + 3*(9*D*a^2*b - 2*C*a*b^2 - B*b^3)*x + 6*(D*b^3*x^3 + 3*D*a*b^2*x^2 + 3*D *a^2*b*x + D*a^3)*log(b*x + a))/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3 *b^4)
Time = 1.61 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx=\frac {D \log {\left (a + b x \right )}}{b^{4}} + \frac {- 2 A b^{3} - B a b^{2} - 2 C a^{2} b + 11 D a^{3} + x^{2} \left (- 6 C b^{3} + 18 D a b^{2}\right ) + x \left (- 3 B b^{3} - 6 C a b^{2} + 27 D a^{2} b\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(b*x+a)**4,x)
Output:
D*log(a + b*x)/b**4 + (-2*A*b**3 - B*a*b**2 - 2*C*a**2*b + 11*D*a**3 + x** 2*(-6*C*b**3 + 18*D*a*b**2) + x*(-3*B*b**3 - 6*C*a*b**2 + 27*D*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)
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx=\frac {11 \, D a^{3} - 2 \, C a^{2} b - B a b^{2} - 2 \, A b^{3} + 6 \, {\left (3 \, D a b^{2} - C b^{3}\right )} x^{2} + 3 \, {\left (9 \, D a^{2} b - 2 \, C a b^{2} - B b^{3}\right )} x}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {D \log \left (b x + a\right )}{b^{4}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x, algorithm="maxima")
Output:
1/6*(11*D*a^3 - 2*C*a^2*b - B*a*b^2 - 2*A*b^3 + 6*(3*D*a*b^2 - C*b^3)*x^2 + 3*(9*D*a^2*b - 2*C*a*b^2 - B*b^3)*x)/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5* x + a^3*b^4) + D*log(b*x + a)/b^4
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx=\frac {D \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {6 \, {\left (3 \, D a b - C b^{2}\right )} x^{2} + 3 \, {\left (9 \, D a^{2} - 2 \, C a b - B b^{2}\right )} x + \frac {11 \, D a^{3} - 2 \, C a^{2} b - B a b^{2} - 2 \, A b^{3}}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3}} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x, algorithm="giac")
Output:
D*log(abs(b*x + a))/b^4 + 1/6*(6*(3*D*a*b - C*b^2)*x^2 + 3*(9*D*a^2 - 2*C* a*b - B*b^2)*x + (11*D*a^3 - 2*C*a^2*b - B*a*b^2 - 2*A*b^3)/b)/((b*x + a)^ 3*b^3)
Time = 3.18 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx=\frac {D\,\left (\ln \left (a+b\,x\right )+\frac {3\,a}{a+b\,x}-\frac {3\,a^2}{2\,{\left (a+b\,x\right )}^2}+\frac {a^3}{3\,{\left (a+b\,x\right )}^3}\right )}{b^4}-\frac {C\,a^2+3\,C\,a\,b\,x+3\,C\,b^2\,x^2}{3\,a^3\,b^3+9\,a^2\,b^4\,x+9\,a\,b^5\,x^2+3\,b^6\,x^3}-\frac {A}{3\,b\,\left (a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3\right )}-\frac {\frac {B\,a}{6\,b^2}+\frac {B\,x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \] Input:
int((A + B*x + C*x^2 + x^3*D)/(a + b*x)^4,x)
Output:
(D*(log(a + b*x) + (3*a)/(a + b*x) - (3*a^2)/(2*(a + b*x)^2) + a^3/(3*(a + b*x)^3)))/b^4 - (C*a^2 + 3*C*b^2*x^2 + 3*C*a*b*x)/(3*a^3*b^3 + 3*b^6*x^3 + 9*a^2*b^4*x + 9*a*b^5*x^2) - A/(3*b*(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2 *b*x)) - ((B*a)/(6*b^2) + (B*x)/(2*b))/(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^ 2*b*x)
Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int \frac {A+B x+C x^2+D x^3}{(a+b x)^4} \, dx=\frac {6 \,\mathrm {log}\left (b x +a \right ) a^{4} d +18 \,\mathrm {log}\left (b x +a \right ) a^{3} b d x +18 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} d \,x^{2}+6 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} d \,x^{3}+5 a^{4} d +9 a^{3} b d x -3 a^{2} b^{3}-3 a \,b^{4} x -6 a \,b^{3} d \,x^{3}+2 b^{4} c \,x^{3}}{6 a \,b^{4} \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )} \] Input:
int((D*x^3+C*x^2+B*x+A)/(b*x+a)^4,x)
Output:
(6*log(a + b*x)*a**4*d + 18*log(a + b*x)*a**3*b*d*x + 18*log(a + b*x)*a**2 *b**2*d*x**2 + 6*log(a + b*x)*a*b**3*d*x**3 + 5*a**4*d + 9*a**3*b*d*x - 3* a**2*b**3 - 3*a*b**4*x - 6*a*b**3*d*x**3 + 2*b**4*c*x**3)/(6*a*b**4*(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))