Integrand size = 18, antiderivative size = 97 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx=-\frac {c^3}{2 a^2 x^2}+\frac {c^2 (2 b c-3 a d)}{a^3 x}+\frac {(b c-a d)^3}{a^3 b (a+b x)}+\frac {3 c (b c-a d)^2 \log (x)}{a^4}-\frac {3 c (b c-a d)^2 \log (a+b x)}{a^4} \] Output:
-1/2*c^3/a^2/x^2+c^2*(-3*a*d+2*b*c)/a^3/x+(-a*d+b*c)^3/a^3/b/(b*x+a)+3*c*( -a*d+b*c)^2*ln(x)/a^4-3*c*(-a*d+b*c)^2*ln(b*x+a)/a^4
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx=-\frac {\frac {a^2 c^3}{x^2}+\frac {2 a c^2 (-2 b c+3 a d)}{x}+\frac {2 a (-b c+a d)^3}{b (a+b x)}-6 c (b c-a d)^2 \log (x)+6 c (b c-a d)^2 \log (a+b x)}{2 a^4} \] Input:
Integrate[(c + d*x)^3/(x^3*(a + b*x)^2),x]
Output:
-1/2*((a^2*c^3)/x^2 + (2*a*c^2*(-2*b*c + 3*a*d))/x + (2*a*(-(b*c) + a*d)^3 )/(b*(a + b*x)) - 6*c*(b*c - a*d)^2*Log[x] + 6*c*(b*c - a*d)^2*Log[a + b*x ])/a^4
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 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.
\(\displaystyle \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {3 c (a d-b c)^2}{a^4 x}-\frac {3 b c (a d-b c)^2}{a^4 (a+b x)}+\frac {c^2 (3 a d-2 b c)}{a^3 x^2}+\frac {(a d-b c)^3}{a^3 (a+b x)^2}+\frac {c^3}{a^2 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 c \log (x) (b c-a d)^2}{a^4}-\frac {3 c (b c-a d)^2 \log (a+b x)}{a^4}+\frac {c^2 (2 b c-3 a d)}{a^3 x}+\frac {(b c-a d)^3}{a^3 b (a+b x)}-\frac {c^3}{2 a^2 x^2}\) |
Input:
Int[(c + d*x)^3/(x^3*(a + b*x)^2),x]
Output:
-1/2*c^3/(a^2*x^2) + (c^2*(2*b*c - 3*a*d))/(a^3*x) + (b*c - a*d)^3/(a^3*b* (a + b*x)) + (3*c*(b*c - a*d)^2*Log[x])/a^4 - (3*c*(b*c - a*d)^2*Log[a + b *x])/a^4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.51
method | result | size |
default | \(-\frac {c^{3}}{2 a^{2} x^{2}}-\frac {c^{2} \left (3 a d -2 b c \right )}{a^{3} x}+\frac {3 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4}}-\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{a^{3} b \left (b x +a \right )}-\frac {3 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{4}}\) | \(146\) |
norman | \(\frac {\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-6 a \,b^{2} c^{2} d +3 b^{3} c^{3}\right ) x^{2}}{a^{3} b}-\frac {c^{3}}{2 a}-\frac {3 c^{2} \left (2 a d -b c \right ) x}{2 a^{2}}}{x^{2} \left (b x +a \right )}+\frac {3 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x \right )}{a^{4}}-\frac {3 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{4}}\) | \(149\) |
risch | \(\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -3 b^{3} c^{3}\right ) x^{2}}{a^{3} b}-\frac {3 c^{2} \left (2 a d -b c \right ) x}{2 a^{2}}-\frac {c^{3}}{2 a}}{x^{2} \left (b x +a \right )}-\frac {3 c \ln \left (b x +a \right ) d^{2}}{a^{2}}+\frac {6 c^{2} \ln \left (b x +a \right ) b d}{a^{3}}-\frac {3 c^{3} \ln \left (b x +a \right ) b^{2}}{a^{4}}+\frac {3 c \ln \left (-x \right ) d^{2}}{a^{2}}-\frac {6 c^{2} \ln \left (-x \right ) b d}{a^{3}}+\frac {3 c^{3} \ln \left (-x \right ) b^{2}}{a^{4}}\) | \(177\) |
parallelrisch | \(\frac {6 \ln \left (x \right ) x^{3} a^{2} b^{2} c \,d^{2}-12 \ln \left (x \right ) x^{3} a \,b^{3} c^{2} d +6 \ln \left (x \right ) x^{3} b^{4} c^{3}-6 \ln \left (b x +a \right ) x^{3} a^{2} b^{2} c \,d^{2}+12 \ln \left (b x +a \right ) x^{3} a \,b^{3} c^{2} d -6 \ln \left (b x +a \right ) x^{3} b^{4} c^{3}+6 \ln \left (x \right ) x^{2} a^{3} b c \,d^{2}-12 \ln \left (x \right ) x^{2} a^{2} b^{2} c^{2} d +6 \ln \left (x \right ) x^{2} a \,b^{3} c^{3}-6 \ln \left (b x +a \right ) x^{2} a^{3} b c \,d^{2}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} c^{2} d -6 \ln \left (b x +a \right ) x^{2} a \,b^{3} c^{3}-2 x^{2} a^{4} d^{3}+6 x^{2} a^{3} b c \,d^{2}-12 x^{2} a^{2} b^{2} c^{2} d +6 x^{2} a \,b^{3} c^{3}-6 x \,a^{3} b \,c^{2} d +3 x \,a^{2} b^{2} c^{3}-a^{3} c^{3} b}{2 a^{4} b \,x^{2} \left (b x +a \right )}\) | \(309\) |
Input:
int((d*x+c)^3/x^3/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/2*c^3/a^2/x^2-c^2*(3*a*d-2*b*c)/a^3/x+3*c*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a ^4*ln(x)-(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/a^3/b/(b*x+a)-3*c*( a^2*d^2-2*a*b*c*d+b^2*c^2)/a^4*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (95) = 190\).
Time = 0.13 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.57 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx=-\frac {a^{3} b c^{3} - 2 \, {\left (3 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2} - 3 \, {\left (a^{2} b^{2} c^{3} - 2 \, a^{3} b c^{2} d\right )} x + 6 \, {\left ({\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) - 6 \, {\left ({\left (b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2}\right )} x^{3} + {\left (a b^{3} c^{3} - 2 \, a^{2} b^{2} c^{2} d + a^{3} b c d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{4} b^{2} x^{3} + a^{5} b x^{2}\right )}} \] Input:
integrate((d*x+c)^3/x^3/(b*x+a)^2,x, algorithm="fricas")
Output:
-1/2*(a^3*b*c^3 - 2*(3*a*b^3*c^3 - 6*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d ^3)*x^2 - 3*(a^2*b^2*c^3 - 2*a^3*b*c^2*d)*x + 6*((b^4*c^3 - 2*a*b^3*c^2*d + a^2*b^2*c*d^2)*x^3 + (a*b^3*c^3 - 2*a^2*b^2*c^2*d + a^3*b*c*d^2)*x^2)*lo g(b*x + a) - 6*((b^4*c^3 - 2*a*b^3*c^2*d + a^2*b^2*c*d^2)*x^3 + (a*b^3*c^3 - 2*a^2*b^2*c^2*d + a^3*b*c*d^2)*x^2)*log(x))/(a^4*b^2*x^3 + a^5*b*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (88) = 176\).
Time = 0.72 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.00 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx=\frac {- a^{2} b c^{3} + x^{2} \left (- 2 a^{3} d^{3} + 6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d + 6 b^{3} c^{3}\right ) + x \left (- 6 a^{2} b c^{2} d + 3 a b^{2} c^{3}\right )}{2 a^{4} b x^{2} + 2 a^{3} b^{2} x^{3}} + \frac {3 c \left (a d - b c\right )^{2} \log {\left (x + \frac {3 a^{3} c d^{2} - 6 a^{2} b c^{2} d + 3 a b^{2} c^{3} - 3 a c \left (a d - b c\right )^{2}}{6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d + 6 b^{3} c^{3}} \right )}}{a^{4}} - \frac {3 c \left (a d - b c\right )^{2} \log {\left (x + \frac {3 a^{3} c d^{2} - 6 a^{2} b c^{2} d + 3 a b^{2} c^{3} + 3 a c \left (a d - b c\right )^{2}}{6 a^{2} b c d^{2} - 12 a b^{2} c^{2} d + 6 b^{3} c^{3}} \right )}}{a^{4}} \] Input:
integrate((d*x+c)**3/x**3/(b*x+a)**2,x)
Output:
(-a**2*b*c**3 + x**2*(-2*a**3*d**3 + 6*a**2*b*c*d**2 - 12*a*b**2*c**2*d + 6*b**3*c**3) + x*(-6*a**2*b*c**2*d + 3*a*b**2*c**3))/(2*a**4*b*x**2 + 2*a* *3*b**2*x**3) + 3*c*(a*d - b*c)**2*log(x + (3*a**3*c*d**2 - 6*a**2*b*c**2* d + 3*a*b**2*c**3 - 3*a*c*(a*d - b*c)**2)/(6*a**2*b*c*d**2 - 12*a*b**2*c** 2*d + 6*b**3*c**3))/a**4 - 3*c*(a*d - b*c)**2*log(x + (3*a**3*c*d**2 - 6*a **2*b*c**2*d + 3*a*b**2*c**3 + 3*a*c*(a*d - b*c)**2)/(6*a**2*b*c*d**2 - 12 *a*b**2*c**2*d + 6*b**3*c**3))/a**4
Time = 0.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.68 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx=-\frac {a^{2} b c^{3} - 2 \, {\left (3 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} - 3 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d\right )} x}{2 \, {\left (a^{3} b^{2} x^{3} + a^{4} b x^{2}\right )}} - \frac {3 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (b x + a\right )}{a^{4}} + \frac {3 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \log \left (x\right )}{a^{4}} \] Input:
integrate((d*x+c)^3/x^3/(b*x+a)^2,x, algorithm="maxima")
Output:
-1/2*(a^2*b*c^3 - 2*(3*b^3*c^3 - 6*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)* x^2 - 3*(a*b^2*c^3 - 2*a^2*b*c^2*d)*x)/(a^3*b^2*x^3 + a^4*b*x^2) - 3*(b^2* c^3 - 2*a*b*c^2*d + a^2*c*d^2)*log(b*x + a)/a^4 + 3*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*log(x)/a^4
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (95) = 190\).
Time = 0.12 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx=\frac {3 \, {\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{4} b} + \frac {\frac {b^{5} c^{3}}{b x + a} - \frac {3 \, a b^{4} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac {a^{3} b^{2} d^{3}}{b x + a}}{a^{3} b^{3}} + \frac {5 \, b^{2} c^{3} - 6 \, a b c^{2} d - \frac {6 \, {\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )}}{{\left (b x + a\right )} b}}{2 \, a^{4} {\left (\frac {a}{b x + a} - 1\right )}^{2}} \] Input:
integrate((d*x+c)^3/x^3/(b*x+a)^2,x, algorithm="giac")
Output:
3*(b^3*c^3 - 2*a*b^2*c^2*d + a^2*b*c*d^2)*log(abs(-a/(b*x + a) + 1))/(a^4* b) + (b^5*c^3/(b*x + a) - 3*a*b^4*c^2*d/(b*x + a) + 3*a^2*b^3*c*d^2/(b*x + a) - a^3*b^2*d^3/(b*x + a))/(a^3*b^3) + 1/2*(5*b^2*c^3 - 6*a*b*c^2*d - 6* (a*b^3*c^3 - a^2*b^2*c^2*d)/((b*x + a)*b))/(a^4*(a/(b*x + a) - 1)^2)
Time = 0.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.61 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx=-\frac {\frac {c^3}{2\,a}+\frac {3\,c^2\,x\,\left (2\,a\,d-b\,c\right )}{2\,a^2}+\frac {x^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-3\,b^3\,c^3\right )}{a^3\,b}}{b\,x^3+a\,x^2}-\frac {6\,c\,\mathrm {atanh}\left (\frac {3\,c\,{\left (a\,d-b\,c\right )}^2\,\left (a+2\,b\,x\right )}{a\,\left (3\,a^2\,c\,d^2-6\,a\,b\,c^2\,d+3\,b^2\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^4} \] Input:
int((c + d*x)^3/(x^3*(a + b*x)^2),x)
Output:
- (c^3/(2*a) + (3*c^2*x*(2*a*d - b*c))/(2*a^2) + (x^2*(a^3*d^3 - 3*b^3*c^3 + 6*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a^3*b))/(a*x^2 + b*x^3) - (6*c*atanh(( 3*c*(a*d - b*c)^2*(a + 2*b*x))/(a*(3*b^2*c^3 + 3*a^2*c*d^2 - 6*a*b*c^2*d)) )*(a*d - b*c)^2)/a^4
Time = 0.17 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.97 \[ \int \frac {(c+d x)^3}{x^3 (a+b x)^2} \, dx=\frac {-6 \,\mathrm {log}\left (b x +a \right ) a^{3} c \,d^{2} x^{2}+12 \,\mathrm {log}\left (b x +a \right ) a^{2} b \,c^{2} d \,x^{2}-6 \,\mathrm {log}\left (b x +a \right ) a^{2} b c \,d^{2} x^{3}-6 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c^{3} x^{2}+12 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} c^{2} d \,x^{3}-6 \,\mathrm {log}\left (b x +a \right ) b^{3} c^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) a^{3} c \,d^{2} x^{2}-12 \,\mathrm {log}\left (x \right ) a^{2} b \,c^{2} d \,x^{2}+6 \,\mathrm {log}\left (x \right ) a^{2} b c \,d^{2} x^{3}+6 \,\mathrm {log}\left (x \right ) a \,b^{2} c^{3} x^{2}-12 \,\mathrm {log}\left (x \right ) a \,b^{2} c^{2} d \,x^{3}+6 \,\mathrm {log}\left (x \right ) b^{3} c^{3} x^{3}-a^{3} c^{3}-6 a^{3} c^{2} d x +2 a^{3} d^{3} x^{3}+3 a^{2} b \,c^{3} x -6 a^{2} b c \,d^{2} x^{3}+12 a \,b^{2} c^{2} d \,x^{3}-6 b^{3} c^{3} x^{3}}{2 a^{4} x^{2} \left (b x +a \right )} \] Input:
int((d*x+c)^3/x^3/(b*x+a)^2,x)
Output:
( - 6*log(a + b*x)*a**3*c*d**2*x**2 + 12*log(a + b*x)*a**2*b*c**2*d*x**2 - 6*log(a + b*x)*a**2*b*c*d**2*x**3 - 6*log(a + b*x)*a*b**2*c**3*x**2 + 12* log(a + b*x)*a*b**2*c**2*d*x**3 - 6*log(a + b*x)*b**3*c**3*x**3 + 6*log(x) *a**3*c*d**2*x**2 - 12*log(x)*a**2*b*c**2*d*x**2 + 6*log(x)*a**2*b*c*d**2* x**3 + 6*log(x)*a*b**2*c**3*x**2 - 12*log(x)*a*b**2*c**2*d*x**3 + 6*log(x) *b**3*c**3*x**3 - a**3*c**3 - 6*a**3*c**2*d*x + 2*a**3*d**3*x**3 + 3*a**2* b*c**3*x - 6*a**2*b*c*d**2*x**3 + 12*a*b**2*c**2*d*x**3 - 6*b**3*c**3*x**3 )/(2*a**4*x**2*(a + b*x))