Integrand size = 18, antiderivative size = 200 \[ \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx=\frac {3 a^2 (b c-2 a d) (b c-a d)^2 x}{b^7}-\frac {a (2 b c-5 a d) (b c-a d)^2 x^2}{2 b^6}+\frac {(b c-4 a d) (b c-a d)^2 x^3}{3 b^5}+\frac {3 d (b c-a d)^2 x^4}{4 b^4}+\frac {d^2 (3 b c-2 a d) x^5}{5 b^3}+\frac {d^3 x^6}{6 b^2}-\frac {a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8} \]
3*a^2*(-2*a*d+b*c)*(-a*d+b*c)^2*x/b^7-1/2*a*(-5*a*d+2*b*c)*(-a*d+b*c)^2*x^ 2/b^6+1/3*(-4*a*d+b*c)*(-a*d+b*c)^2*x^3/b^5+3/4*d*(-a*d+b*c)^2*x^4/b^4+1/5 *d^2*(-2*a*d+3*b*c)*x^5/b^3+1/6*d^3*x^6/b^2-a^4*(-a*d+b*c)^3/b^8/(b*x+a)-a ^3*(-7*a*d+4*b*c)*(-a*d+b*c)^2*ln(b*x+a)/b^8
Time = 0.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.95 \[ \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx=\frac {-180 a^2 b (b c-a d)^2 (-b c+2 a d) x+30 a b^2 (b c-a d)^2 (-2 b c+5 a d) x^2+20 b^3 (b c-4 a d) (b c-a d)^2 x^3+45 b^4 d (b c-a d)^2 x^4+12 b^5 d^2 (3 b c-2 a d) x^5+10 b^6 d^3 x^6+\frac {60 a^4 (-b c+a d)^3}{a+b x}+60 a^3 (b c-a d)^2 (-4 b c+7 a d) \log (a+b x)}{60 b^8} \]
(-180*a^2*b*(b*c - a*d)^2*(-(b*c) + 2*a*d)*x + 30*a*b^2*(b*c - a*d)^2*(-2* b*c + 5*a*d)*x^2 + 20*b^3*(b*c - 4*a*d)*(b*c - a*d)^2*x^3 + 45*b^4*d*(b*c - a*d)^2*x^4 + 12*b^5*d^2*(3*b*c - 2*a*d)*x^5 + 10*b^6*d^3*x^6 + (60*a^4*( -(b*c) + a*d)^3)/(a + b*x) + 60*a^3*(b*c - a*d)^2*(-4*b*c + 7*a*d)*Log[a + b*x])/(60*b^8)
Time = 0.46 (sec) , antiderivative size = 200, 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.
| \(\displaystyle \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^4 (a d-b c)^3}{b^7 (a+b x)^2}+\frac {a^3 (a d-b c)^2 (7 a d-4 b c)}{b^7 (a+b x)}-\frac {3 a^2 (a d-b c)^2 (2 a d-b c)}{b^7}+\frac {a x (a d-b c)^2 (5 a d-2 b c)}{b^6}+\frac {x^2 (b c-4 a d) (b c-a d)^2}{b^5}+\frac {3 d x^3 (b c-a d)^2}{b^4}+\frac {d^2 x^4 (3 b c-2 a d)}{b^3}+\frac {d^3 x^5}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^4 (b c-a d)^3}{b^8 (a+b x)}-\frac {a^3 (4 b c-7 a d) (b c-a d)^2 \log (a+b x)}{b^8}+\frac {3 a^2 x (b c-2 a d) (b c-a d)^2}{b^7}-\frac {a x^2 (2 b c-5 a d) (b c-a d)^2}{2 b^6}+\frac {x^3 (b c-4 a d) (b c-a d)^2}{3 b^5}+\frac {3 d x^4 (b c-a d)^2}{4 b^4}+\frac {d^2 x^5 (3 b c-2 a d)}{5 b^3}+\frac {d^3 x^6}{6 b^2}\) |
(3*a^2*(b*c - 2*a*d)*(b*c - a*d)^2*x)/b^7 - (a*(2*b*c - 5*a*d)*(b*c - a*d) ^2*x^2)/(2*b^6) + ((b*c - 4*a*d)*(b*c - a*d)^2*x^3)/(3*b^5) + (3*d*(b*c - a*d)^2*x^4)/(4*b^4) + (d^2*(3*b*c - 2*a*d)*x^5)/(5*b^3) + (d^3*x^6)/(6*b^2 ) - (a^4*(b*c - a*d)^3)/(b^8*(a + b*x)) - (a^3*(4*b*c - 7*a*d)*(b*c - a*d) ^2*Log[a + b*x])/b^8
3.3.69.3.1 Defintions of rubi rules used
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 = 0.46 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.55
| method | result | size |
| norman | \(\frac {\frac {a \left (7 a^{6} d^{3}-18 a^{5} b c \,d^{2}+15 a^{4} b^{2} c^{2} d -4 a^{3} b^{3} c^{3}\right )}{b^{8}}+\frac {d^{3} x^{7}}{6 b}-\frac {\left (7 a^{3} d^{3}-18 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{4}}{12 b^{4}}+\frac {a \left (7 a^{3} d^{3}-18 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{3}}{6 b^{5}}-\frac {a^{2} \left (7 a^{3} d^{3}-18 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{2}}{2 b^{6}}+\frac {d \left (7 a^{2} d^{2}-18 a b c d +15 b^{2} c^{2}\right ) x^{5}}{20 b^{3}}-\frac {d^{2} \left (7 a d -18 b c \right ) x^{6}}{30 b^{2}}}{b x +a}+\frac {a^{3} \left (7 a^{3} d^{3}-18 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(310\) |
| default | \(-\frac {-\frac {1}{6} d^{3} x^{6} b^{5}+\frac {2}{5} x^{5} a \,b^{4} d^{3}-\frac {3}{5} x^{5} b^{5} c \,d^{2}-\frac {3}{4} x^{4} a^{2} b^{3} d^{3}+\frac {3}{2} x^{4} a \,b^{4} c \,d^{2}-\frac {3}{4} x^{4} b^{5} c^{2} d +\frac {4}{3} x^{3} a^{3} b^{2} d^{3}-3 x^{3} a^{2} b^{3} c \,d^{2}+2 x^{3} a \,b^{4} c^{2} d -\frac {1}{3} x^{3} b^{5} c^{3}-\frac {5}{2} x^{2} a^{4} b \,d^{3}+6 x^{2} a^{3} b^{2} c \,d^{2}-\frac {9}{2} x^{2} a^{2} b^{3} c^{2} d +x^{2} a \,b^{4} c^{3}+6 a^{5} d^{3} x -15 a^{4} c \,d^{2} b x +12 a^{3} c^{2} d \,b^{2} x -3 a^{2} c^{3} b^{3} x}{b^{7}}+\frac {a^{3} \left (7 a^{3} d^{3}-18 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{8}}+\frac {a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{8} \left (b x +a \right )}\) | \(333\) |
| risch | \(\frac {7 a^{6} \ln \left (b x +a \right ) d^{3}}{b^{8}}+\frac {a^{7} d^{3}}{b^{8} \left (b x +a \right )}-\frac {a^{4} c^{3}}{b^{5} \left (b x +a \right )}+\frac {3 a^{2} c^{3} x}{b^{4}}-\frac {6 a^{5} d^{3} x}{b^{7}}-\frac {x^{2} a \,c^{3}}{b^{3}}-\frac {4 x^{3} a^{3} d^{3}}{3 b^{5}}+\frac {5 x^{2} a^{4} d^{3}}{2 b^{6}}-\frac {2 x^{5} a \,d^{3}}{5 b^{3}}+\frac {3 x^{5} c \,d^{2}}{5 b^{2}}+\frac {3 x^{4} a^{2} d^{3}}{4 b^{4}}+\frac {3 x^{4} c^{2} d}{4 b^{2}}-\frac {3 x^{4} a c \,d^{2}}{2 b^{3}}+\frac {3 x^{3} a^{2} c \,d^{2}}{b^{4}}-\frac {4 a^{3} \ln \left (b x +a \right ) c^{3}}{b^{5}}+\frac {d^{3} x^{6}}{6 b^{2}}-\frac {12 a^{3} c^{2} d x}{b^{5}}-\frac {3 a^{6} c \,d^{2}}{b^{7} \left (b x +a \right )}+\frac {3 a^{5} c^{2} d}{b^{6} \left (b x +a \right )}-\frac {18 a^{5} \ln \left (b x +a \right ) c \,d^{2}}{b^{7}}+\frac {15 a^{4} \ln \left (b x +a \right ) c^{2} d}{b^{6}}-\frac {2 x^{3} a \,c^{2} d}{b^{3}}-\frac {6 x^{2} a^{3} c \,d^{2}}{b^{5}}+\frac {9 x^{2} a^{2} c^{2} d}{2 b^{4}}+\frac {15 a^{4} c \,d^{2} x}{b^{6}}+\frac {x^{3} c^{3}}{3 b^{2}}\) | \(378\) |
| parallelrisch | \(\frac {-240 \ln \left (b x +a \right ) a^{4} b^{3} c^{3}-210 x^{2} a^{5} b^{2} d^{3}-240 a^{4} b^{3} c^{3}-1080 \ln \left (b x +a \right ) x \,a^{5} b^{2} c \,d^{2}+900 \ln \left (b x +a \right ) x \,a^{4} b^{3} c^{2} d +420 a^{7} d^{3}-1080 a^{6} b c \,d^{2}+900 a^{5} b^{2} c^{2} d -54 x^{5} a \,b^{6} c \,d^{2}+90 x^{4} a^{2} b^{5} c \,d^{2}-75 x^{4} a \,b^{6} c^{2} d -180 x^{3} a^{3} b^{4} c \,d^{2}+150 x^{3} a^{2} b^{5} c^{2} d +540 x^{2} a^{4} b^{3} c \,d^{2}-450 x^{2} a^{3} b^{4} c^{2} d +420 \ln \left (b x +a \right ) x \,a^{6} b \,d^{3}-240 \ln \left (b x +a \right ) x \,a^{3} b^{4} c^{3}+120 x^{2} a^{2} b^{5} c^{3}+70 x^{3} a^{4} b^{3} d^{3}-40 x^{3} a \,b^{6} c^{3}+20 x^{4} b^{7} c^{3}+420 \ln \left (b x +a \right ) a^{7} d^{3}+10 x^{7} d^{3} b^{7}-1080 \ln \left (b x +a \right ) a^{6} b c \,d^{2}+900 \ln \left (b x +a \right ) a^{5} b^{2} c^{2} d -14 x^{6} a \,b^{6} d^{3}-35 x^{4} a^{3} b^{4} d^{3}+36 x^{6} b^{7} c \,d^{2}+21 x^{5} a^{2} b^{5} d^{3}+45 x^{5} b^{7} c^{2} d}{60 b^{8} \left (b x +a \right )}\) | \(433\) |
(a*(7*a^6*d^3-18*a^5*b*c*d^2+15*a^4*b^2*c^2*d-4*a^3*b^3*c^3)/b^8+1/6/b*d^3 *x^7-1/12*(7*a^3*d^3-18*a^2*b*c*d^2+15*a*b^2*c^2*d-4*b^3*c^3)/b^4*x^4+1/6* a*(7*a^3*d^3-18*a^2*b*c*d^2+15*a*b^2*c^2*d-4*b^3*c^3)/b^5*x^3-1/2*a^2*(7*a ^3*d^3-18*a^2*b*c*d^2+15*a*b^2*c^2*d-4*b^3*c^3)/b^6*x^2+1/20*d*(7*a^2*d^2- 18*a*b*c*d+15*b^2*c^2)/b^3*x^5-1/30*d^2*(7*a*d-18*b*c)/b^2*x^6)/(b*x+a)+a^ 3/b^8*(7*a^3*d^3-18*a^2*b*c*d^2+15*a*b^2*c^2*d-4*b^3*c^3)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (190) = 380\).
Time = 0.23 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.10 \[ \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx=\frac {10 \, b^{7} d^{3} x^{7} - 60 \, a^{4} b^{3} c^{3} + 180 \, a^{5} b^{2} c^{2} d - 180 \, a^{6} b c d^{2} + 60 \, a^{7} d^{3} + 2 \, {\left (18 \, b^{7} c d^{2} - 7 \, a b^{6} d^{3}\right )} x^{6} + 3 \, {\left (15 \, b^{7} c^{2} d - 18 \, a b^{6} c d^{2} + 7 \, a^{2} b^{5} d^{3}\right )} x^{5} + 5 \, {\left (4 \, b^{7} c^{3} - 15 \, a b^{6} c^{2} d + 18 \, a^{2} b^{5} c d^{2} - 7 \, a^{3} b^{4} d^{3}\right )} x^{4} - 10 \, {\left (4 \, a b^{6} c^{3} - 15 \, a^{2} b^{5} c^{2} d + 18 \, a^{3} b^{4} c d^{2} - 7 \, a^{4} b^{3} d^{3}\right )} x^{3} + 30 \, {\left (4 \, a^{2} b^{5} c^{3} - 15 \, a^{3} b^{4} c^{2} d + 18 \, a^{4} b^{3} c d^{2} - 7 \, a^{5} b^{2} d^{3}\right )} x^{2} + 180 \, {\left (a^{3} b^{4} c^{3} - 4 \, a^{4} b^{3} c^{2} d + 5 \, a^{5} b^{2} c d^{2} - 2 \, a^{6} b d^{3}\right )} x - 60 \, {\left (4 \, a^{4} b^{3} c^{3} - 15 \, a^{5} b^{2} c^{2} d + 18 \, a^{6} b c d^{2} - 7 \, a^{7} d^{3} + {\left (4 \, a^{3} b^{4} c^{3} - 15 \, a^{4} b^{3} c^{2} d + 18 \, a^{5} b^{2} c d^{2} - 7 \, a^{6} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{9} x + a b^{8}\right )}} \]
1/60*(10*b^7*d^3*x^7 - 60*a^4*b^3*c^3 + 180*a^5*b^2*c^2*d - 180*a^6*b*c*d^ 2 + 60*a^7*d^3 + 2*(18*b^7*c*d^2 - 7*a*b^6*d^3)*x^6 + 3*(15*b^7*c^2*d - 18 *a*b^6*c*d^2 + 7*a^2*b^5*d^3)*x^5 + 5*(4*b^7*c^3 - 15*a*b^6*c^2*d + 18*a^2 *b^5*c*d^2 - 7*a^3*b^4*d^3)*x^4 - 10*(4*a*b^6*c^3 - 15*a^2*b^5*c^2*d + 18* a^3*b^4*c*d^2 - 7*a^4*b^3*d^3)*x^3 + 30*(4*a^2*b^5*c^3 - 15*a^3*b^4*c^2*d + 18*a^4*b^3*c*d^2 - 7*a^5*b^2*d^3)*x^2 + 180*(a^3*b^4*c^3 - 4*a^4*b^3*c^2 *d + 5*a^5*b^2*c*d^2 - 2*a^6*b*d^3)*x - 60*(4*a^4*b^3*c^3 - 15*a^5*b^2*c^2 *d + 18*a^6*b*c*d^2 - 7*a^7*d^3 + (4*a^3*b^4*c^3 - 15*a^4*b^3*c^2*d + 18*a ^5*b^2*c*d^2 - 7*a^6*b*d^3)*x)*log(b*x + a))/(b^9*x + a*b^8)
Time = 0.52 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.62 \[ \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx=\frac {a^{3} \left (a d - b c\right )^{2} \cdot \left (7 a d - 4 b c\right ) \log {\left (a + b x \right )}}{b^{8}} + x^{5} \left (- \frac {2 a d^{3}}{5 b^{3}} + \frac {3 c d^{2}}{5 b^{2}}\right ) + x^{4} \cdot \left (\frac {3 a^{2} d^{3}}{4 b^{4}} - \frac {3 a c d^{2}}{2 b^{3}} + \frac {3 c^{2} d}{4 b^{2}}\right ) + x^{3} \left (- \frac {4 a^{3} d^{3}}{3 b^{5}} + \frac {3 a^{2} c d^{2}}{b^{4}} - \frac {2 a c^{2} d}{b^{3}} + \frac {c^{3}}{3 b^{2}}\right ) + x^{2} \cdot \left (\frac {5 a^{4} d^{3}}{2 b^{6}} - \frac {6 a^{3} c d^{2}}{b^{5}} + \frac {9 a^{2} c^{2} d}{2 b^{4}} - \frac {a c^{3}}{b^{3}}\right ) + x \left (- \frac {6 a^{5} d^{3}}{b^{7}} + \frac {15 a^{4} c d^{2}}{b^{6}} - \frac {12 a^{3} c^{2} d}{b^{5}} + \frac {3 a^{2} c^{3}}{b^{4}}\right ) + \frac {a^{7} d^{3} - 3 a^{6} b c d^{2} + 3 a^{5} b^{2} c^{2} d - a^{4} b^{3} c^{3}}{a b^{8} + b^{9} x} + \frac {d^{3} x^{6}}{6 b^{2}} \]
a**3*(a*d - b*c)**2*(7*a*d - 4*b*c)*log(a + b*x)/b**8 + x**5*(-2*a*d**3/(5 *b**3) + 3*c*d**2/(5*b**2)) + x**4*(3*a**2*d**3/(4*b**4) - 3*a*c*d**2/(2*b **3) + 3*c**2*d/(4*b**2)) + x**3*(-4*a**3*d**3/(3*b**5) + 3*a**2*c*d**2/b* *4 - 2*a*c**2*d/b**3 + c**3/(3*b**2)) + x**2*(5*a**4*d**3/(2*b**6) - 6*a** 3*c*d**2/b**5 + 9*a**2*c**2*d/(2*b**4) - a*c**3/b**3) + x*(-6*a**5*d**3/b* *7 + 15*a**4*c*d**2/b**6 - 12*a**3*c**2*d/b**5 + 3*a**2*c**3/b**4) + (a**7 *d**3 - 3*a**6*b*c*d**2 + 3*a**5*b**2*c**2*d - a**4*b**3*c**3)/(a*b**8 + b **9*x) + d**3*x**6/(6*b**2)
Time = 0.20 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.62 \[ \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx=-\frac {a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3}}{b^{9} x + a b^{8}} + \frac {10 \, b^{5} d^{3} x^{6} + 12 \, {\left (3 \, b^{5} c d^{2} - 2 \, a b^{4} d^{3}\right )} x^{5} + 45 \, {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{4} + 20 \, {\left (b^{5} c^{3} - 6 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 4 \, a^{3} b^{2} d^{3}\right )} x^{3} - 30 \, {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{2} + 180 \, {\left (a^{2} b^{3} c^{3} - 4 \, a^{3} b^{2} c^{2} d + 5 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} x}{60 \, b^{7}} - \frac {{\left (4 \, a^{3} b^{3} c^{3} - 15 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} - 7 \, a^{6} d^{3}\right )} \log \left (b x + a\right )}{b^{8}} \]
-(a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)/(b^9*x + a*b^8) + 1/60*(10*b^5*d^3*x^6 + 12*(3*b^5*c*d^2 - 2*a*b^4*d^3)*x^5 + 45*(b^5*c^2 *d - 2*a*b^4*c*d^2 + a^2*b^3*d^3)*x^4 + 20*(b^5*c^3 - 6*a*b^4*c^2*d + 9*a^ 2*b^3*c*d^2 - 4*a^3*b^2*d^3)*x^3 - 30*(2*a*b^4*c^3 - 9*a^2*b^3*c^2*d + 12* a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^2 + 180*(a^2*b^3*c^3 - 4*a^3*b^2*c^2*d + 5* a^4*b*c*d^2 - 2*a^5*d^3)*x)/b^7 - (4*a^3*b^3*c^3 - 15*a^4*b^2*c^2*d + 18*a ^5*b*c*d^2 - 7*a^6*d^3)*log(b*x + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (190) = 380\).
Time = 0.28 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.02 \[ \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx=\frac {{\left (10 \, d^{3} + \frac {12 \, {\left (3 \, b^{2} c d^{2} - 7 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {45 \, {\left (b^{4} c^{2} d - 6 \, a b^{3} c d^{2} + 7 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {20 \, {\left (b^{6} c^{3} - 15 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} - 35 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}} - \frac {30 \, {\left (4 \, a b^{7} c^{3} - 30 \, a^{2} b^{6} c^{2} d + 60 \, a^{3} b^{5} c d^{2} - 35 \, a^{4} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{4} b^{4}} + \frac {180 \, {\left (2 \, a^{2} b^{8} c^{3} - 10 \, a^{3} b^{7} c^{2} d + 15 \, a^{4} b^{6} c d^{2} - 7 \, a^{5} b^{5} d^{3}\right )}}{{\left (b x + a\right )}^{5} b^{5}}\right )} {\left (b x + a\right )}^{6}}{60 \, b^{8}} + \frac {{\left (4 \, a^{3} b^{3} c^{3} - 15 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} - 7 \, a^{6} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{8}} - \frac {\frac {a^{4} b^{9} c^{3}}{b x + a} - \frac {3 \, a^{5} b^{8} c^{2} d}{b x + a} + \frac {3 \, a^{6} b^{7} c d^{2}}{b x + a} - \frac {a^{7} b^{6} d^{3}}{b x + a}}{b^{14}} \]
1/60*(10*d^3 + 12*(3*b^2*c*d^2 - 7*a*b*d^3)/((b*x + a)*b) + 45*(b^4*c^2*d - 6*a*b^3*c*d^2 + 7*a^2*b^2*d^3)/((b*x + a)^2*b^2) + 20*(b^6*c^3 - 15*a*b^ 5*c^2*d + 45*a^2*b^4*c*d^2 - 35*a^3*b^3*d^3)/((b*x + a)^3*b^3) - 30*(4*a*b ^7*c^3 - 30*a^2*b^6*c^2*d + 60*a^3*b^5*c*d^2 - 35*a^4*b^4*d^3)/((b*x + a)^ 4*b^4) + 180*(2*a^2*b^8*c^3 - 10*a^3*b^7*c^2*d + 15*a^4*b^6*c*d^2 - 7*a^5* b^5*d^3)/((b*x + a)^5*b^5))*(b*x + a)^6/b^8 + (4*a^3*b^3*c^3 - 15*a^4*b^2* c^2*d + 18*a^5*b*c*d^2 - 7*a^6*d^3)*log(abs(b*x + a)/((b*x + a)^2*abs(b))) /b^8 - (a^4*b^9*c^3/(b*x + a) - 3*a^5*b^8*c^2*d/(b*x + a) + 3*a^6*b^7*c*d^ 2/(b*x + a) - a^7*b^6*d^3/(b*x + a))/b^14
Time = 0.15 (sec) , antiderivative size = 688, normalized size of antiderivative = 3.44 \[ \int \frac {x^4 (c+d x)^3}{(a+b x)^2} \, dx=x^3\,\left (\frac {c^3}{3\,b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{3\,b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{3\,b^2}\right )-x^5\,\left (\frac {2\,a\,d^3}{5\,b^3}-\frac {3\,c\,d^2}{5\,b^2}\right )-x\,\left (\frac {a^2\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b^2}-\frac {2\,a\,\left (\frac {a^2\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b^2}+\frac {2\,a\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b}\right )}{b}\right )+x^4\,\left (\frac {3\,c^2\,d}{4\,b^2}+\frac {a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{2\,b}-\frac {a^2\,d^3}{4\,b^4}\right )-x^2\,\left (\frac {a^2\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{2\,b^2}+\frac {a\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )}{b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (7\,a^6\,d^3-18\,a^5\,b\,c\,d^2+15\,a^4\,b^2\,c^2\,d-4\,a^3\,b^3\,c^3\right )}{b^8}+\frac {a^7\,d^3-3\,a^6\,b\,c\,d^2+3\,a^5\,b^2\,c^2\,d-a^4\,b^3\,c^3}{b\,\left (x\,b^8+a\,b^7\right )}+\frac {d^3\,x^6}{6\,b^2} \]
x^3*(c^3/(3*b^2) - (2*a*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b ^2))/b - (a^2*d^3)/b^4))/(3*b) + (a^2*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/(3* b^2)) - x^5*((2*a*d^3)/(5*b^3) - (3*c*d^2)/(5*b^2)) - x*((a^2*(c^3/b^2 - ( 2*a*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b ^4))/b + (a^2*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b^2))/b^2 - (2*a*((a^2*((3* c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b^4))/b^2 + (2*a*(c^3/b^2 - (2*a*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b ^2))/b - (a^2*d^3)/b^4))/b + (a^2*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b^2))/b ))/b) + x^4*((3*c^2*d)/(4*b^2) + (a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/(2*b) - (a^2*d^3)/(4*b^4)) - x^2*((a^2*((3*c^2*d)/b^2 + (2*a*((2*a*d^3)/b^3 - ( 3*c*d^2)/b^2))/b - (a^2*d^3)/b^4))/(2*b^2) + (a*(c^3/b^2 - (2*a*((3*c^2*d) /b^2 + (2*a*((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/b^4))/b + (a^2* ((2*a*d^3)/b^3 - (3*c*d^2)/b^2))/b^2))/b) + (log(a + b*x)*(7*a^6*d^3 - 4*a ^3*b^3*c^3 + 15*a^4*b^2*c^2*d - 18*a^5*b*c*d^2))/b^8 + (a^7*d^3 - a^4*b^3* c^3 + 3*a^5*b^2*c^2*d - 3*a^6*b*c*d^2)/(b*(a*b^7 + b^8*x)) + (d^3*x^6)/(6* b^2)