Integrand size = 15, antiderivative size = 86 \[ \int \frac {(c+d x)^3}{(a+b x)^4} \, dx=-\frac {(b c-a d)^3}{3 b^4 (a+b x)^3}-\frac {3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac {3 d^2 (b c-a d)}{b^4 (a+b x)}+\frac {d^3 \log (a+b x)}{b^4} \] Output:
-1/3*(-a*d+b*c)^3/b^4/(b*x+a)^3-3/2*d*(-a*d+b*c)^2/b^4/(b*x+a)^2-3*d^2*(-a *d+b*c)/b^4/(b*x+a)+d^3*ln(b*x+a)/b^4
Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^3}{(a+b x)^4} \, dx=\frac {-\frac {(b c-a d) \left (11 a^2 d^2+a b d (5 c+27 d x)+b^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )\right )}{(a+b x)^3}+6 d^3 \log (a+b x)}{6 b^4} \] Input:
Integrate[(c + d*x)^3/(a + b*x)^4,x]
Output:
(-(((b*c - a*d)*(11*a^2*d^2 + a*b*d*(5*c + 27*d*x) + b^2*(2*c^2 + 9*c*d*x + 18*d^2*x^2)))/(a + b*x)^3) + 6*d^3*Log[a + b*x])/(6*b^4)
Time = 0.22 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {49, 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.
\(\displaystyle \int \frac {(c+d x)^3}{(a+b x)^4} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {3 d^2 (b c-a d)}{b^3 (a+b x)^2}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^3}+\frac {(b c-a d)^3}{b^3 (a+b x)^4}+\frac {d^3}{b^3 (a+b x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 d^2 (b c-a d)}{b^4 (a+b x)}-\frac {3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac {(b c-a d)^3}{3 b^4 (a+b x)^3}+\frac {d^3 \log (a+b x)}{b^4}\) |
Input:
Int[(c + d*x)^3/(a + b*x)^4,x]
Output:
-1/3*(b*c - a*d)^3/(b^4*(a + b*x)^3) - (3*d*(b*c - a*d)^2)/(2*b^4*(a + b*x )^2) - (3*d^2*(b*c - a*d))/(b^4*(a + b*x)) + (d^3*Log[a + b*x])/b^4
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\frac {\frac {3 d^{2} \left (a d -b c \right ) x^{2}}{b^{2}}+\frac {3 d \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x}{2 b^{3}}+\frac {11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}}{6 b^{4}}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}\) | \(115\) |
norman | \(\frac {\frac {11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}}{6 b^{4}}+\frac {3 \left (a \,d^{3}-b c \,d^{2}\right ) x^{2}}{b^{2}}+\frac {3 \left (3 a^{2} d^{3}-2 a b c \,d^{2}-b^{2} c^{2} d \right ) x}{2 b^{3}}}{\left (b x +a \right )^{3}}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}\) | \(119\) |
default | \(-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{4} \left (b x +a \right )^{2}}+\frac {3 d^{2} \left (a d -b c \right )}{b^{4} \left (b x +a \right )}+\frac {d^{3} \ln \left (b x +a \right )}{b^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{3 b^{4} \left (b x +a \right )^{3}}\) | \(120\) |
parallelrisch | \(\frac {6 \ln \left (b x +a \right ) x^{3} b^{3} d^{3}+18 \ln \left (b x +a \right ) x^{2} a \,b^{2} d^{3}+18 \ln \left (b x +a \right ) x \,a^{2} b \,d^{3}+18 x^{2} a \,b^{2} d^{3}-18 x^{2} b^{3} c \,d^{2}+6 \ln \left (b x +a \right ) a^{3} d^{3}+27 x \,a^{2} b \,d^{3}-18 x a \,b^{2} c \,d^{2}-9 x \,b^{3} c^{2} d +11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}}{6 b^{4} \left (b x +a \right )^{3}}\) | \(170\) |
Input:
int((d*x+c)^3/(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
(3*d^2*(a*d-b*c)/b^2*x^2+3/2*d*(3*a^2*d^2-2*a*b*c*d-b^2*c^2)/b^3*x+1/6*(11 *a^3*d^3-6*a^2*b*c*d^2-3*a*b^2*c^2*d-2*b^3*c^3)/b^4)/(b*x+a)^3+d^3*ln(b*x+ a)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (82) = 164\).
Time = 0.07 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.05 \[ \int \frac {(c+d x)^3}{(a+b x)^4} \, dx=-\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 18 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \] Input:
integrate((d*x+c)^3/(b*x+a)^4,x, algorithm="fricas")
Output:
-1/6*(2*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3 + 18*(b^3*c*d ^2 - a*b^2*d^3)*x^2 + 9*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b ^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^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 = 0.51 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.72 \[ \int \frac {(c+d x)^3}{(a+b x)^4} \, dx=\frac {11 a^{3} d^{3} - 6 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 2 b^{3} c^{3} + x^{2} \cdot \left (18 a b^{2} d^{3} - 18 b^{3} c d^{2}\right ) + x \left (27 a^{2} b d^{3} - 18 a b^{2} c d^{2} - 9 b^{3} c^{2} d\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac {d^{3} \log {\left (a + b x \right )}}{b^{4}} \] Input:
integrate((d*x+c)**3/(b*x+a)**4,x)
Output:
(11*a**3*d**3 - 6*a**2*b*c*d**2 - 3*a*b**2*c**2*d - 2*b**3*c**3 + x**2*(18 *a*b**2*d**3 - 18*b**3*c*d**2) + x*(27*a**2*b*d**3 - 18*a*b**2*c*d**2 - 9* b**3*c**2*d))/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3 ) + d**3*log(a + b*x)/b**4
Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^3}{(a+b x)^4} \, dx=-\frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 18 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {d^{3} \log \left (b x + a\right )}{b^{4}} \] Input:
integrate((d*x+c)^3/(b*x+a)^4,x, algorithm="maxima")
Output:
-1/6*(2*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3 + 18*(b^3*c*d ^2 - a*b^2*d^3)*x^2 + 9*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x)/(b^7* x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4) + d^3*log(b*x + a)/b^4
Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.37 \[ \int \frac {(c+d x)^3}{(a+b x)^4} \, dx=\frac {d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {18 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 9 \, {\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x + \frac {2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3}}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3}} \] Input:
integrate((d*x+c)^3/(b*x+a)^4,x, algorithm="giac")
Output:
d^3*log(abs(b*x + a))/b^4 - 1/6*(18*(b^2*c*d^2 - a*b*d^3)*x^2 + 9*(b^2*c^2 *d + 2*a*b*c*d^2 - 3*a^2*d^3)*x + (2*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d ^2 - 11*a^3*d^3)/b)/((b*x + a)^3*b^3)
Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.60 \[ \int \frac {(c+d x)^3}{(a+b x)^4} \, dx=\frac {d^3\,\ln \left (a+b\,x\right )}{b^4}-\frac {\frac {-11\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+2\,b^3\,c^3}{6\,b^4}+\frac {3\,x\,\left (-3\,a^2\,d^3+2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )}{2\,b^3}-\frac {3\,d^2\,x^2\,\left (a\,d-b\,c\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \] Input:
int((c + d*x)^3/(a + b*x)^4,x)
Output:
(d^3*log(a + b*x))/b^4 - ((2*b^3*c^3 - 11*a^3*d^3 + 3*a*b^2*c^2*d + 6*a^2* b*c*d^2)/(6*b^4) + (3*x*(b^2*c^2*d - 3*a^2*d^3 + 2*a*b*c*d^2))/(2*b^3) - ( 3*d^2*x^2*(a*d - b*c))/b^2)/(a^3 + b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x)
Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.09 \[ \int \frac {(c+d x)^3}{(a+b x)^4} \, dx=\frac {6 \,\mathrm {log}\left (b x +a \right ) a^{4} d^{3}+18 \,\mathrm {log}\left (b x +a \right ) a^{3} b \,d^{3} x +18 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} d^{3} x^{2}+6 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} d^{3} x^{3}+5 a^{4} d^{3}+9 a^{3} b \,d^{3} x -3 a^{2} b^{2} c^{2} d -2 a \,b^{3} c^{3}-9 a \,b^{3} c^{2} d x -6 a \,b^{3} d^{3} x^{3}+6 b^{4} c \,d^{2} x^{3}}{6 a \,b^{4} \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )} \] Input:
int((d*x+c)^3/(b*x+a)^4,x)
Output:
(6*log(a + b*x)*a**4*d**3 + 18*log(a + b*x)*a**3*b*d**3*x + 18*log(a + b*x )*a**2*b**2*d**3*x**2 + 6*log(a + b*x)*a*b**3*d**3*x**3 + 5*a**4*d**3 + 9* a**3*b*d**3*x - 3*a**2*b**2*c**2*d - 2*a*b**3*c**3 - 9*a*b**3*c**2*d*x - 6 *a*b**3*d**3*x**3 + 6*b**4*c*d**2*x**3)/(6*a*b**4*(a**3 + 3*a**2*b*x + 3*a *b**2*x**2 + b**3*x**3))