Integrand size = 18, antiderivative size = 199 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx=-\frac {c^3}{5 a^2 x^5}+\frac {c^2 (2 b c-3 a d)}{4 a^3 x^4}-\frac {c (b c-a d)^2}{a^4 x^3}+\frac {(b c-a d)^2 (4 b c-a d)}{2 a^5 x^2}-\frac {b (5 b c-2 a d) (b c-a d)^2}{a^6 x}-\frac {b^2 (b c-a d)^3}{a^6 (a+b x)}-\frac {3 b^2 (b c-a d)^2 (2 b c-a d) \log (x)}{a^7}+\frac {3 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{a^7} \] Output:
-1/5*c^3/a^2/x^5+1/4*c^2*(-3*a*d+2*b*c)/a^3/x^4-c*(-a*d+b*c)^2/a^4/x^3+1/2 *(-a*d+b*c)^2*(-a*d+4*b*c)/a^5/x^2-b*(-2*a*d+5*b*c)*(-a*d+b*c)^2/a^6/x-b^2 *(-a*d+b*c)^3/a^6/(b*x+a)-3*b^2*(-a*d+b*c)^2*(-a*d+2*b*c)*ln(x)/a^7+3*b^2* (-a*d+b*c)^2*(-a*d+2*b*c)*ln(b*x+a)/a^7
Time = 0.06 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx=-\frac {\frac {4 a^5 c^3}{x^5}+\frac {5 a^4 c^2 (-2 b c+3 a d)}{x^4}+\frac {20 a^3 c (b c-a d)^2}{x^3}+\frac {10 a^2 (b c-a d)^2 (-4 b c+a d)}{x^2}-\frac {20 a b (b c-a d)^2 (-5 b c+2 a d)}{x}-\frac {20 a b^2 (-b c+a d)^3}{a+b x}+60 b^2 (b c-a d)^2 (2 b c-a d) \log (x)-60 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{20 a^7} \] Input:
Integrate[(c + d*x)^3/(x^6*(a + b*x)^2),x]
Output:
-1/20*((4*a^5*c^3)/x^5 + (5*a^4*c^2*(-2*b*c + 3*a*d))/x^4 + (20*a^3*c*(b*c - a*d)^2)/x^3 + (10*a^2*(b*c - a*d)^2*(-4*b*c + a*d))/x^2 - (20*a*b*(b*c - a*d)^2*(-5*b*c + 2*a*d))/x - (20*a*b^2*(-(b*c) + a*d)^3)/(a + b*x) + 60* b^2*(b*c - a*d)^2*(2*b*c - a*d)*Log[x] - 60*b^2*(b*c - a*d)^2*(2*b*c - a*d )*Log[a + b*x])/a^7
Time = 0.40 (sec) , antiderivative size = 199, 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.
| \(\displaystyle \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {3 b^3 (a d-2 b c) (a d-b c)^2}{a^7 (a+b x)}+\frac {3 b^2 (a d-2 b c) (a d-b c)^2}{a^7 x}-\frac {b^3 (a d-b c)^3}{a^6 (a+b x)^2}-\frac {b (a d-b c)^2 (2 a d-5 b c)}{a^6 x^2}+\frac {(a d-4 b c) (a d-b c)^2}{a^5 x^3}+\frac {3 c (a d-b c)^2}{a^4 x^4}+\frac {c^2 (3 a d-2 b c)}{a^3 x^5}+\frac {c^3}{a^2 x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 b^2 \log (x) (b c-a d)^2 (2 b c-a d)}{a^7}+\frac {3 b^2 (b c-a d)^2 (2 b c-a d) \log (a+b x)}{a^7}-\frac {b^2 (b c-a d)^3}{a^6 (a+b x)}-\frac {b (5 b c-2 a d) (b c-a d)^2}{a^6 x}+\frac {(b c-a d)^2 (4 b c-a d)}{2 a^5 x^2}-\frac {c (b c-a d)^2}{a^4 x^3}+\frac {c^2 (2 b c-3 a d)}{4 a^3 x^4}-\frac {c^3}{5 a^2 x^5}\) |
Input:
Int[(c + d*x)^3/(x^6*(a + b*x)^2),x]
Output:
-1/5*c^3/(a^2*x^5) + (c^2*(2*b*c - 3*a*d))/(4*a^3*x^4) - (c*(b*c - a*d)^2) /(a^4*x^3) + ((b*c - a*d)^2*(4*b*c - a*d))/(2*a^5*x^2) - (b*(5*b*c - 2*a*d )*(b*c - a*d)^2)/(a^6*x) - (b^2*(b*c - a*d)^3)/(a^6*(a + b*x)) - (3*b^2*(b *c - a*d)^2*(2*b*c - a*d)*Log[x])/a^7 + (3*b^2*(b*c - a*d)^2*(2*b*c - a*d) *Log[a + b*x])/a^7
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 = 0.22 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.50
| method | result | size |
| default | \(-\frac {c^{3}}{5 a^{2} x^{5}}-\frac {a^{3} d^{3}-6 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -4 b^{3} c^{3}}{2 a^{5} x^{2}}-\frac {c^{2} \left (3 a d -2 b c \right )}{4 a^{3} x^{4}}+\frac {b \left (2 a^{3} d^{3}-9 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right )}{a^{6} x}+\frac {3 b^{2} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{7}}-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{a^{4} x^{3}}-\frac {3 b^{2} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{7}}+\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2}}{a^{6} \left (b x +a \right )}\) | \(298\) |
| norman | \(\frac {\frac {b \left (-3 b^{2} d^{3} a^{3}+12 b^{3} c \,d^{2} a^{2}-15 c^{2} d \,b^{4} a +6 c^{3} b^{5}\right ) x^{6}}{a^{7}}-\frac {c^{3}}{5 a}-\frac {\left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{3}}{2 a^{4}}+\frac {3 b \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{4}}{2 a^{5}}-\frac {c \left (4 a^{2} d^{2}-5 a b c d +2 b^{2} c^{2}\right ) x^{2}}{4 a^{3}}-\frac {3 c^{2} \left (5 a d -2 b c \right ) x}{20 a^{2}}}{x^{5} \left (b x +a \right )}+\frac {3 b^{2} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{7}}-\frac {3 b^{2} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{7}}\) | \(307\) |
| risch | \(\frac {\frac {3 b^{2} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{5}}{a^{6}}+\frac {3 b \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{4}}{2 a^{5}}-\frac {\left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{3}}{2 a^{4}}-\frac {c \left (4 a^{2} d^{2}-5 a b c d +2 b^{2} c^{2}\right ) x^{2}}{4 a^{3}}-\frac {3 c^{2} \left (5 a d -2 b c \right ) x}{20 a^{2}}-\frac {c^{3}}{5 a}}{x^{5} \left (b x +a \right )}+\frac {3 b^{2} \ln \left (-x \right ) d^{3}}{a^{4}}-\frac {12 b^{3} \ln \left (-x \right ) c \,d^{2}}{a^{5}}+\frac {15 b^{4} \ln \left (-x \right ) c^{2} d}{a^{6}}-\frac {6 b^{5} \ln \left (-x \right ) c^{3}}{a^{7}}-\frac {3 b^{2} \ln \left (b x +a \right ) d^{3}}{a^{4}}+\frac {12 b^{3} \ln \left (b x +a \right ) c \,d^{2}}{a^{5}}-\frac {15 b^{4} \ln \left (b x +a \right ) c^{2} d}{a^{6}}+\frac {6 b^{5} \ln \left (b x +a \right ) c^{3}}{a^{7}}\) | \(340\) |
| parallelrisch | \(\frac {40 x^{3} a^{5} b c \,d^{2}-50 x^{3} a^{4} b^{2} c^{2} d +25 x^{2} a^{5} b \,c^{2} d +60 \ln \left (x \right ) x^{6} a^{3} b^{3} d^{3}-60 \ln \left (b x +a \right ) x^{6} a^{3} b^{3} d^{3}+60 \ln \left (x \right ) x^{5} a^{4} b^{2} d^{3}-120 \ln \left (x \right ) x^{5} a \,b^{5} c^{3}-60 \ln \left (b x +a \right ) x^{5} a^{4} b^{2} d^{3}+120 \ln \left (b x +a \right ) x^{5} a \,b^{5} c^{3}+240 x^{6} a^{2} b^{4} c \,d^{2}-300 x^{6} a \,b^{5} c^{2} d -120 x^{4} a^{4} b^{2} c \,d^{2}+150 x^{4} a^{3} b^{3} c^{2} d -240 \ln \left (x \right ) x^{6} a^{2} b^{4} c \,d^{2}+300 \ln \left (x \right ) x^{6} a \,b^{5} c^{2} d +240 \ln \left (b x +a \right ) x^{6} a^{2} b^{4} c \,d^{2}-300 \ln \left (b x +a \right ) x^{6} a \,b^{5} c^{2} d -240 \ln \left (x \right ) x^{5} a^{3} b^{3} c \,d^{2}+300 \ln \left (x \right ) x^{5} a^{2} b^{4} c^{2} d +240 \ln \left (b x +a \right ) x^{5} a^{3} b^{3} c \,d^{2}-300 \ln \left (b x +a \right ) x^{5} a^{2} b^{4} c^{2} d -4 c^{3} a^{6}+120 x^{6} b^{6} c^{3}-10 x^{3} a^{6} d^{3}-60 x^{6} a^{3} b^{3} d^{3}+30 x^{4} a^{5} b \,d^{3}-60 x^{4} a^{2} b^{4} c^{3}+20 x^{3} a^{3} b^{3} c^{3}-20 x^{2} a^{6} c \,d^{2}-10 x^{2} a^{4} b^{2} c^{3}-15 x \,a^{6} c^{2} d +6 x \,a^{5} b \,c^{3}-120 \ln \left (x \right ) x^{6} b^{6} c^{3}+120 \ln \left (b x +a \right ) x^{6} b^{6} c^{3}}{20 a^{7} x^{5} \left (b x +a \right )}\) | \(528\) |
Input:
int((d*x+c)^3/x^6/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/5*c^3/a^2/x^5-1/2*(a^3*d^3-6*a^2*b*c*d^2+9*a*b^2*c^2*d-4*b^3*c^3)/a^5/x ^2-1/4*c^2*(3*a*d-2*b*c)/a^3/x^4+b*(2*a^3*d^3-9*a^2*b*c*d^2+12*a*b^2*c^2*d -5*b^3*c^3)/a^6/x+3*b^2*(a^3*d^3-4*a^2*b*c*d^2+5*a*b^2*c^2*d-2*b^3*c^3)/a^ 7*ln(x)-c*(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^4/x^3-3*b^2*(a^3*d^3-4*a^2*b*c*d^2 +5*a*b^2*c^2*d-2*b^3*c^3)/a^7*ln(b*x+a)+(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2 *d-b^3*c^3)/a^6*b^2/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (193) = 386\).
Time = 0.08 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.20 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx=-\frac {4 \, a^{6} c^{3} + 60 \, {\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5} + 30 \, {\left (2 \, a^{2} b^{4} c^{3} - 5 \, a^{3} b^{3} c^{2} d + 4 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x^{4} - 10 \, {\left (2 \, a^{3} b^{3} c^{3} - 5 \, a^{4} b^{2} c^{2} d + 4 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} x^{3} + 5 \, {\left (2 \, a^{4} b^{2} c^{3} - 5 \, a^{5} b c^{2} d + 4 \, a^{6} c d^{2}\right )} x^{2} - 3 \, {\left (2 \, a^{5} b c^{3} - 5 \, a^{6} c^{2} d\right )} x - 60 \, {\left ({\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + {\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5}\right )} \log \left (b x + a\right ) + 60 \, {\left ({\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{6} + {\left (2 \, a b^{5} c^{3} - 5 \, a^{2} b^{4} c^{2} d + 4 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{5}\right )} \log \left (x\right )}{20 \, {\left (a^{7} b x^{6} + a^{8} x^{5}\right )}} \] Input:
integrate((d*x+c)^3/x^6/(b*x+a)^2,x, algorithm="fricas")
Output:
-1/20*(4*a^6*c^3 + 60*(2*a*b^5*c^3 - 5*a^2*b^4*c^2*d + 4*a^3*b^3*c*d^2 - a ^4*b^2*d^3)*x^5 + 30*(2*a^2*b^4*c^3 - 5*a^3*b^3*c^2*d + 4*a^4*b^2*c*d^2 - a^5*b*d^3)*x^4 - 10*(2*a^3*b^3*c^3 - 5*a^4*b^2*c^2*d + 4*a^5*b*c*d^2 - a^6 *d^3)*x^3 + 5*(2*a^4*b^2*c^3 - 5*a^5*b*c^2*d + 4*a^6*c*d^2)*x^2 - 3*(2*a^5 *b*c^3 - 5*a^6*c^2*d)*x - 60*((2*b^6*c^3 - 5*a*b^5*c^2*d + 4*a^2*b^4*c*d^2 - a^3*b^3*d^3)*x^6 + (2*a*b^5*c^3 - 5*a^2*b^4*c^2*d + 4*a^3*b^3*c*d^2 - a ^4*b^2*d^3)*x^5)*log(b*x + a) + 60*((2*b^6*c^3 - 5*a*b^5*c^2*d + 4*a^2*b^4 *c*d^2 - a^3*b^3*d^3)*x^6 + (2*a*b^5*c^3 - 5*a^2*b^4*c^2*d + 4*a^3*b^3*c*d ^2 - a^4*b^2*d^3)*x^5)*log(x))/(a^7*b*x^6 + a^8*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (182) = 364\).
Time = 1.02 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.66 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx=\frac {- 4 a^{5} c^{3} + x^{5} \cdot \left (60 a^{3} b^{2} d^{3} - 240 a^{2} b^{3} c d^{2} + 300 a b^{4} c^{2} d - 120 b^{5} c^{3}\right ) + x^{4} \cdot \left (30 a^{4} b d^{3} - 120 a^{3} b^{2} c d^{2} + 150 a^{2} b^{3} c^{2} d - 60 a b^{4} c^{3}\right ) + x^{3} \left (- 10 a^{5} d^{3} + 40 a^{4} b c d^{2} - 50 a^{3} b^{2} c^{2} d + 20 a^{2} b^{3} c^{3}\right ) + x^{2} \left (- 20 a^{5} c d^{2} + 25 a^{4} b c^{2} d - 10 a^{3} b^{2} c^{3}\right ) + x \left (- 15 a^{5} c^{2} d + 6 a^{4} b c^{3}\right )}{20 a^{7} x^{5} + 20 a^{6} b x^{6}} + \frac {3 b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2} \log {\left (x + \frac {3 a^{4} b^{2} d^{3} - 12 a^{3} b^{3} c d^{2} + 15 a^{2} b^{4} c^{2} d - 6 a b^{5} c^{3} - 3 a b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2}}{6 a^{3} b^{3} d^{3} - 24 a^{2} b^{4} c d^{2} + 30 a b^{5} c^{2} d - 12 b^{6} c^{3}} \right )}}{a^{7}} - \frac {3 b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2} \log {\left (x + \frac {3 a^{4} b^{2} d^{3} - 12 a^{3} b^{3} c d^{2} + 15 a^{2} b^{4} c^{2} d - 6 a b^{5} c^{3} + 3 a b^{2} \left (a d - 2 b c\right ) \left (a d - b c\right )^{2}}{6 a^{3} b^{3} d^{3} - 24 a^{2} b^{4} c d^{2} + 30 a b^{5} c^{2} d - 12 b^{6} c^{3}} \right )}}{a^{7}} \] Input:
integrate((d*x+c)**3/x**6/(b*x+a)**2,x)
Output:
(-4*a**5*c**3 + x**5*(60*a**3*b**2*d**3 - 240*a**2*b**3*c*d**2 + 300*a*b** 4*c**2*d - 120*b**5*c**3) + x**4*(30*a**4*b*d**3 - 120*a**3*b**2*c*d**2 + 150*a**2*b**3*c**2*d - 60*a*b**4*c**3) + x**3*(-10*a**5*d**3 + 40*a**4*b*c *d**2 - 50*a**3*b**2*c**2*d + 20*a**2*b**3*c**3) + x**2*(-20*a**5*c*d**2 + 25*a**4*b*c**2*d - 10*a**3*b**2*c**3) + x*(-15*a**5*c**2*d + 6*a**4*b*c** 3))/(20*a**7*x**5 + 20*a**6*b*x**6) + 3*b**2*(a*d - 2*b*c)*(a*d - b*c)**2* log(x + (3*a**4*b**2*d**3 - 12*a**3*b**3*c*d**2 + 15*a**2*b**4*c**2*d - 6* a*b**5*c**3 - 3*a*b**2*(a*d - 2*b*c)*(a*d - b*c)**2)/(6*a**3*b**3*d**3 - 2 4*a**2*b**4*c*d**2 + 30*a*b**5*c**2*d - 12*b**6*c**3))/a**7 - 3*b**2*(a*d - 2*b*c)*(a*d - b*c)**2*log(x + (3*a**4*b**2*d**3 - 12*a**3*b**3*c*d**2 + 15*a**2*b**4*c**2*d - 6*a*b**5*c**3 + 3*a*b**2*(a*d - 2*b*c)*(a*d - b*c)** 2)/(6*a**3*b**3*d**3 - 24*a**2*b**4*c*d**2 + 30*a*b**5*c**2*d - 12*b**6*c* *3))/a**7
Time = 0.05 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.67 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx=-\frac {4 \, a^{5} c^{3} + 60 \, {\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 30 \, {\left (2 \, a b^{4} c^{3} - 5 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} - 10 \, {\left (2 \, a^{2} b^{3} c^{3} - 5 \, a^{3} b^{2} c^{2} d + 4 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 5 \, {\left (2 \, a^{3} b^{2} c^{3} - 5 \, a^{4} b c^{2} d + 4 \, a^{5} c d^{2}\right )} x^{2} - 3 \, {\left (2 \, a^{4} b c^{3} - 5 \, a^{5} c^{2} d\right )} x}{20 \, {\left (a^{6} b x^{6} + a^{7} x^{5}\right )}} + \frac {3 \, {\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (b x + a\right )}{a^{7}} - \frac {3 \, {\left (2 \, b^{5} c^{3} - 5 \, a b^{4} c^{2} d + 4 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (x\right )}{a^{7}} \] Input:
integrate((d*x+c)^3/x^6/(b*x+a)^2,x, algorithm="maxima")
Output:
-1/20*(4*a^5*c^3 + 60*(2*b^5*c^3 - 5*a*b^4*c^2*d + 4*a^2*b^3*c*d^2 - a^3*b ^2*d^3)*x^5 + 30*(2*a*b^4*c^3 - 5*a^2*b^3*c^2*d + 4*a^3*b^2*c*d^2 - a^4*b* d^3)*x^4 - 10*(2*a^2*b^3*c^3 - 5*a^3*b^2*c^2*d + 4*a^4*b*c*d^2 - a^5*d^3)* x^3 + 5*(2*a^3*b^2*c^3 - 5*a^4*b*c^2*d + 4*a^5*c*d^2)*x^2 - 3*(2*a^4*b*c^3 - 5*a^5*c^2*d)*x)/(a^6*b*x^6 + a^7*x^5) + 3*(2*b^5*c^3 - 5*a*b^4*c^2*d + 4*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(b*x + a)/a^7 - 3*(2*b^5*c^3 - 5*a*b^4*c ^2*d + 4*a^2*b^3*c*d^2 - a^3*b^2*d^3)*log(x)/a^7
Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (193) = 386\).
Time = 0.12 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.19 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx=-\frac {3 \, {\left (2 \, b^{6} c^{3} - 5 \, a b^{5} c^{2} d + 4 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{7} b} - \frac {\frac {b^{11} c^{3}}{b x + a} - \frac {3 \, a b^{10} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{9} c d^{2}}{b x + a} - \frac {a^{3} b^{8} d^{3}}{b x + a}}{a^{6} b^{6}} + \frac {174 \, b^{5} c^{3} - 385 \, a b^{4} c^{2} d + 260 \, a^{2} b^{3} c d^{2} - 50 \, a^{3} b^{2} d^{3} - \frac {5 \, {\left (154 \, a b^{6} c^{3} - 337 \, a^{2} b^{5} c^{2} d + 224 \, a^{3} b^{4} c d^{2} - 42 \, a^{4} b^{3} d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {10 \, {\left (130 \, a^{2} b^{7} c^{3} - 280 \, a^{3} b^{6} c^{2} d + 182 \, a^{4} b^{5} c d^{2} - 33 \, a^{5} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {10 \, {\left (100 \, a^{3} b^{8} c^{3} - 210 \, a^{4} b^{7} c^{2} d + 132 \, a^{5} b^{6} c d^{2} - 23 \, a^{6} b^{5} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {60 \, {\left (5 \, a^{4} b^{9} c^{3} - 10 \, a^{5} b^{8} c^{2} d + 6 \, a^{6} b^{7} c d^{2} - a^{7} b^{6} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}}}{20 \, a^{7} {\left (\frac {a}{b x + a} - 1\right )}^{5}} \] Input:
integrate((d*x+c)^3/x^6/(b*x+a)^2,x, algorithm="giac")
Output:
-3*(2*b^6*c^3 - 5*a*b^5*c^2*d + 4*a^2*b^4*c*d^2 - a^3*b^3*d^3)*log(abs(-a/ (b*x + a) + 1))/(a^7*b) - (b^11*c^3/(b*x + a) - 3*a*b^10*c^2*d/(b*x + a) + 3*a^2*b^9*c*d^2/(b*x + a) - a^3*b^8*d^3/(b*x + a))/(a^6*b^6) + 1/20*(174* b^5*c^3 - 385*a*b^4*c^2*d + 260*a^2*b^3*c*d^2 - 50*a^3*b^2*d^3 - 5*(154*a* b^6*c^3 - 337*a^2*b^5*c^2*d + 224*a^3*b^4*c*d^2 - 42*a^4*b^3*d^3)/((b*x + a)*b) + 10*(130*a^2*b^7*c^3 - 280*a^3*b^6*c^2*d + 182*a^4*b^5*c*d^2 - 33*a ^5*b^4*d^3)/((b*x + a)^2*b^2) - 10*(100*a^3*b^8*c^3 - 210*a^4*b^7*c^2*d + 132*a^5*b^6*c*d^2 - 23*a^6*b^5*d^3)/((b*x + a)^3*b^3) + 60*(5*a^4*b^9*c^3 - 10*a^5*b^8*c^2*d + 6*a^6*b^7*c*d^2 - a^7*b^6*d^3)/((b*x + a)^4*b^4))/(a^ 7*(a/(b*x + a) - 1)^5)
Time = 0.12 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx=\frac {6\,b^2\,\mathrm {atanh}\left (\frac {3\,b^2\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-2\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (-3\,a^3\,b^2\,d^3+12\,a^2\,b^3\,c\,d^2-15\,a\,b^4\,c^2\,d+6\,b^5\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (a\,d-2\,b\,c\right )}{a^7}-\frac {\frac {c^3}{5\,a}+\frac {x^3\,\left (a^3\,d^3-4\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^4}+\frac {3\,c^2\,x\,\left (5\,a\,d-2\,b\,c\right )}{20\,a^2}-\frac {3\,b^2\,x^5\,\left (a^3\,d^3-4\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{a^6}+\frac {c\,x^2\,\left (4\,a^2\,d^2-5\,a\,b\,c\,d+2\,b^2\,c^2\right )}{4\,a^3}-\frac {3\,b\,x^4\,\left (a^3\,d^3-4\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-2\,b^3\,c^3\right )}{2\,a^5}}{b\,x^6+a\,x^5} \] Input:
int((c + d*x)^3/(x^6*(a + b*x)^2),x)
Output:
(6*b^2*atanh((3*b^2*(a*d - b*c)^2*(a*d - 2*b*c)*(a + 2*b*x))/(a*(6*b^5*c^3 - 3*a^3*b^2*d^3 + 12*a^2*b^3*c*d^2 - 15*a*b^4*c^2*d)))*(a*d - b*c)^2*(a*d - 2*b*c))/a^7 - (c^3/(5*a) + (x^3*(a^3*d^3 - 2*b^3*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2))/(2*a^4) + (3*c^2*x*(5*a*d - 2*b*c))/(20*a^2) - (3*b^2*x^5* (a^3*d^3 - 2*b^3*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2))/a^6 + (c*x^2*(4*a^2 *d^2 + 2*b^2*c^2 - 5*a*b*c*d))/(4*a^3) - (3*b*x^4*(a^3*d^3 - 2*b^3*c^3 + 5 *a*b^2*c^2*d - 4*a^2*b*c*d^2))/(2*a^5))/(a*x^5 + b*x^6)
Time = 0.19 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.65 \[ \int \frac {(c+d x)^3}{x^6 (a+b x)^2} \, dx=\frac {120 \,\mathrm {log}\left (b x +a \right ) b^{6} c^{3} x^{6}-120 \,\mathrm {log}\left (x \right ) b^{6} c^{3} x^{6}-15 a^{6} c^{2} d x -20 a^{6} c \,d^{2} x^{2}+6 a^{5} b \,c^{3} x +30 a^{5} b \,d^{3} x^{4}-10 a^{4} b^{2} c^{3} x^{2}+20 a^{3} b^{3} c^{3} x^{3}-60 a^{3} b^{3} d^{3} x^{6}-60 a^{2} b^{4} c^{3} x^{4}-4 a^{6} c^{3}-10 a^{6} d^{3} x^{3}+120 b^{6} c^{3} x^{6}+240 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} c \,d^{2} x^{5}-300 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{4} c^{2} d \,x^{5}+240 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{4} c \,d^{2} x^{6}-300 \,\mathrm {log}\left (b x +a \right ) a \,b^{5} c^{2} d \,x^{6}-240 \,\mathrm {log}\left (x \right ) a^{3} b^{3} c \,d^{2} x^{5}+300 \,\mathrm {log}\left (x \right ) a^{2} b^{4} c^{2} d \,x^{5}-240 \,\mathrm {log}\left (x \right ) a^{2} b^{4} c \,d^{2} x^{6}+300 \,\mathrm {log}\left (x \right ) a \,b^{5} c^{2} d \,x^{6}-60 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} d^{3} x^{5}-60 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} d^{3} x^{6}+120 \,\mathrm {log}\left (b x +a \right ) a \,b^{5} c^{3} x^{5}+60 \,\mathrm {log}\left (x \right ) a^{4} b^{2} d^{3} x^{5}+60 \,\mathrm {log}\left (x \right ) a^{3} b^{3} d^{3} x^{6}-120 \,\mathrm {log}\left (x \right ) a \,b^{5} c^{3} x^{5}+25 a^{5} b \,c^{2} d \,x^{2}+40 a^{5} b c \,d^{2} x^{3}-50 a^{4} b^{2} c^{2} d \,x^{3}-120 a^{4} b^{2} c \,d^{2} x^{4}+150 a^{3} b^{3} c^{2} d \,x^{4}+240 a^{2} b^{4} c \,d^{2} x^{6}-300 a \,b^{5} c^{2} d \,x^{6}}{20 a^{7} x^{5} \left (b x +a \right )} \] Input:
int((d*x+c)^3/x^6/(b*x+a)^2,x)
Output:
( - 60*log(a + b*x)*a**4*b**2*d**3*x**5 + 240*log(a + b*x)*a**3*b**3*c*d** 2*x**5 - 60*log(a + b*x)*a**3*b**3*d**3*x**6 - 300*log(a + b*x)*a**2*b**4* c**2*d*x**5 + 240*log(a + b*x)*a**2*b**4*c*d**2*x**6 + 120*log(a + b*x)*a* b**5*c**3*x**5 - 300*log(a + b*x)*a*b**5*c**2*d*x**6 + 120*log(a + b*x)*b* *6*c**3*x**6 + 60*log(x)*a**4*b**2*d**3*x**5 - 240*log(x)*a**3*b**3*c*d**2 *x**5 + 60*log(x)*a**3*b**3*d**3*x**6 + 300*log(x)*a**2*b**4*c**2*d*x**5 - 240*log(x)*a**2*b**4*c*d**2*x**6 - 120*log(x)*a*b**5*c**3*x**5 + 300*log( x)*a*b**5*c**2*d*x**6 - 120*log(x)*b**6*c**3*x**6 - 4*a**6*c**3 - 15*a**6* c**2*d*x - 20*a**6*c*d**2*x**2 - 10*a**6*d**3*x**3 + 6*a**5*b*c**3*x + 25* a**5*b*c**2*d*x**2 + 40*a**5*b*c*d**2*x**3 + 30*a**5*b*d**3*x**4 - 10*a**4 *b**2*c**3*x**2 - 50*a**4*b**2*c**2*d*x**3 - 120*a**4*b**2*c*d**2*x**4 + 2 0*a**3*b**3*c**3*x**3 + 150*a**3*b**3*c**2*d*x**4 - 60*a**3*b**3*d**3*x**6 - 60*a**2*b**4*c**3*x**4 + 240*a**2*b**4*c*d**2*x**6 - 300*a*b**5*c**2*d* x**6 + 120*b**6*c**3*x**6)/(20*a**7*x**5*(a + b*x))