Integrand size = 18, antiderivative size = 165 \[ \int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx=\frac {a^2 (3 b c-5 a d) (b c-a d) x}{b^6}-\frac {a (b c-2 a d) (b c-a d) x^2}{b^5}+\frac {(b c-3 a d) (b c-a d) x^3}{3 b^4}+\frac {d (b c-a d) x^4}{2 b^3}+\frac {d^2 x^5}{5 b^2}-\frac {a^4 (b c-a d)^2}{b^7 (a+b x)}-\frac {2 a^3 (2 b c-3 a d) (b c-a d) \log (a+b x)}{b^7} \]
a^2*(-5*a*d+3*b*c)*(-a*d+b*c)*x/b^6-a*(-2*a*d+b*c)*(-a*d+b*c)*x^2/b^5+1/3* (-3*a*d+b*c)*(-a*d+b*c)*x^3/b^4+1/2*d*(-a*d+b*c)*x^4/b^3+1/5*d^2*x^5/b^2-a ^4*(-a*d+b*c)^2/b^7/(b*x+a)-2*a^3*(-3*a*d+2*b*c)*(-a*d+b*c)*ln(b*x+a)/b^7
Time = 0.07 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.11 \[ \int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx=\frac {30 a^2 b \left (3 b^2 c^2-8 a b c d+5 a^2 d^2\right ) x-30 a b^2 \left (b^2 c^2-3 a b c d+2 a^2 d^2\right ) x^2+10 b^3 \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) x^3+15 b^4 d (b c-a d) x^4+6 b^5 d^2 x^5-\frac {30 a^4 (b c-a d)^2}{a+b x}-60 a^3 \left (2 b^2 c^2-5 a b c d+3 a^2 d^2\right ) \log (a+b x)}{30 b^7} \]
(30*a^2*b*(3*b^2*c^2 - 8*a*b*c*d + 5*a^2*d^2)*x - 30*a*b^2*(b^2*c^2 - 3*a* b*c*d + 2*a^2*d^2)*x^2 + 10*b^3*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*x^3 + 15 *b^4*d*(b*c - a*d)*x^4 + 6*b^5*d^2*x^5 - (30*a^4*(b*c - a*d)^2)/(a + b*x) - 60*a^3*(2*b^2*c^2 - 5*a*b*c*d + 3*a^2*d^2)*Log[a + b*x])/(30*b^7)
Time = 0.40 (sec) , antiderivative size = 165, 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.
| \(\displaystyle \int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (\frac {a^4 (a d-b c)^2}{b^6 (a+b x)^2}+\frac {2 a^3 (2 b c-3 a d) (a d-b c)}{b^6 (a+b x)}+\frac {a^2 (3 b c-5 a d) (b c-a d)}{b^6}+\frac {2 a x (b c-2 a d) (a d-b c)}{b^5}+\frac {x^2 (b c-3 a d) (b c-a d)}{b^4}+\frac {2 d x^3 (b c-a d)}{b^3}+\frac {d^2 x^4}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^4 (b c-a d)^2}{b^7 (a+b x)}-\frac {2 a^3 (2 b c-3 a d) (b c-a d) \log (a+b x)}{b^7}+\frac {a^2 x (3 b c-5 a d) (b c-a d)}{b^6}-\frac {a x^2 (b c-2 a d) (b c-a d)}{b^5}+\frac {x^3 (b c-3 a d) (b c-a d)}{3 b^4}+\frac {d x^4 (b c-a d)}{2 b^3}+\frac {d^2 x^5}{5 b^2}\) |
(a^2*(3*b*c - 5*a*d)*(b*c - a*d)*x)/b^6 - (a*(b*c - 2*a*d)*(b*c - a*d)*x^2 )/b^5 + ((b*c - 3*a*d)*(b*c - a*d)*x^3)/(3*b^4) + (d*(b*c - a*d)*x^4)/(2*b ^3) + (d^2*x^5)/(5*b^2) - (a^4*(b*c - a*d)^2)/(b^7*(a + b*x)) - (2*a^3*(2* b*c - 3*a*d)*(b*c - a*d)*Log[a + b*x])/b^7
3.3.59.3.1 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 = 0.45 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.25
| method | result | size |
| norman | \(\frac {\frac {a^{2} \left (3 a^{2} d^{2}-5 a b c d +2 b^{2} c^{2}\right ) x^{2}}{b^{5}}-\frac {a \left (6 a^{5} d^{2}-10 a^{4} b c d +4 a^{3} b^{2} c^{2}\right )}{b^{7}}+\frac {d^{2} x^{6}}{5 b}+\frac {\left (3 a^{2} d^{2}-5 a b c d +2 b^{2} c^{2}\right ) x^{4}}{6 b^{3}}-\frac {a \left (3 a^{2} d^{2}-5 a b c d +2 b^{2} c^{2}\right ) x^{3}}{3 b^{4}}-\frac {d \left (3 a d -5 b c \right ) x^{5}}{10 b^{2}}}{b x +a}-\frac {2 a^{3} \left (3 a^{2} d^{2}-5 a b c d +2 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{7}}\) | \(207\) |
| default | \(\frac {\frac {1}{5} d^{2} x^{5} b^{4}-\frac {1}{2} a \,b^{3} d^{2} x^{4}+\frac {1}{2} b^{4} c d \,x^{4}+a^{2} b^{2} d^{2} x^{3}-\frac {4}{3} a \,b^{3} c d \,x^{3}+\frac {1}{3} b^{4} c^{2} x^{3}-2 a^{3} b \,d^{2} x^{2}+3 a^{2} b^{2} c d \,x^{2}-a \,b^{3} c^{2} x^{2}+5 a^{4} d^{2} x -8 a^{3} b c d x +3 a^{2} b^{2} c^{2} x}{b^{6}}-\frac {2 a^{3} \left (3 a^{2} d^{2}-5 a b c d +2 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{7}}-\frac {a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{7} \left (b x +a \right )}\) | \(215\) |
| risch | \(\frac {d^{2} x^{5}}{5 b^{2}}-\frac {a \,d^{2} x^{4}}{2 b^{3}}+\frac {c d \,x^{4}}{2 b^{2}}+\frac {a^{2} d^{2} x^{3}}{b^{4}}-\frac {4 a c d \,x^{3}}{3 b^{3}}+\frac {c^{2} x^{3}}{3 b^{2}}-\frac {2 a^{3} d^{2} x^{2}}{b^{5}}+\frac {3 a^{2} c d \,x^{2}}{b^{4}}-\frac {a \,c^{2} x^{2}}{b^{3}}+\frac {5 a^{4} d^{2} x}{b^{6}}-\frac {8 a^{3} c d x}{b^{5}}+\frac {3 a^{2} c^{2} x}{b^{4}}-\frac {6 a^{5} \ln \left (b x +a \right ) d^{2}}{b^{7}}+\frac {10 a^{4} \ln \left (b x +a \right ) c d}{b^{6}}-\frac {4 a^{3} \ln \left (b x +a \right ) c^{2}}{b^{5}}-\frac {a^{6} d^{2}}{b^{7} \left (b x +a \right )}+\frac {2 a^{5} c d}{b^{6} \left (b x +a \right )}-\frac {a^{4} c^{2}}{b^{5} \left (b x +a \right )}\) | \(247\) |
| parallelrisch | \(-\frac {-6 x^{6} d^{2} b^{6}+9 x^{5} a \,b^{5} d^{2}-15 x^{5} b^{6} c d -15 x^{4} a^{2} b^{4} d^{2}+25 x^{4} a \,b^{5} c d -10 x^{4} b^{6} c^{2}+30 x^{3} a^{3} b^{3} d^{2}-50 x^{3} a^{2} b^{4} c d +20 x^{3} a \,b^{5} c^{2}+180 \ln \left (b x +a \right ) x \,a^{5} b \,d^{2}-300 \ln \left (b x +a \right ) x \,a^{4} b^{2} c d +120 \ln \left (b x +a \right ) x \,a^{3} b^{3} c^{2}-90 x^{2} a^{4} b^{2} d^{2}+150 x^{2} a^{3} b^{3} c d -60 x^{2} a^{2} b^{4} c^{2}+180 \ln \left (b x +a \right ) a^{6} d^{2}-300 \ln \left (b x +a \right ) a^{5} b c d +120 \ln \left (b x +a \right ) a^{4} b^{2} c^{2}+180 a^{6} d^{2}-300 a^{5} b c d +120 a^{4} b^{2} c^{2}}{30 b^{7} \left (b x +a \right )}\) | \(286\) |
(a^2*(3*a^2*d^2-5*a*b*c*d+2*b^2*c^2)/b^5*x^2-a*(6*a^5*d^2-10*a^4*b*c*d+4*a ^3*b^2*c^2)/b^7+1/5/b*d^2*x^6+1/6*(3*a^2*d^2-5*a*b*c*d+2*b^2*c^2)/b^3*x^4- 1/3*a*(3*a^2*d^2-5*a*b*c*d+2*b^2*c^2)/b^4*x^3-1/10*d*(3*a*d-5*b*c)/b^2*x^5 )/(b*x+a)-2*a^3/b^7*(3*a^2*d^2-5*a*b*c*d+2*b^2*c^2)*ln(b*x+a)
Time = 0.22 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.73 \[ \int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx=\frac {6 \, b^{6} d^{2} x^{6} - 30 \, a^{4} b^{2} c^{2} + 60 \, a^{5} b c d - 30 \, a^{6} d^{2} + 3 \, {\left (5 \, b^{6} c d - 3 \, a b^{5} d^{2}\right )} x^{5} + 5 \, {\left (2 \, b^{6} c^{2} - 5 \, a b^{5} c d + 3 \, a^{2} b^{4} d^{2}\right )} x^{4} - 10 \, {\left (2 \, a b^{5} c^{2} - 5 \, a^{2} b^{4} c d + 3 \, a^{3} b^{3} d^{2}\right )} x^{3} + 30 \, {\left (2 \, a^{2} b^{4} c^{2} - 5 \, a^{3} b^{3} c d + 3 \, a^{4} b^{2} d^{2}\right )} x^{2} + 30 \, {\left (3 \, a^{3} b^{3} c^{2} - 8 \, a^{4} b^{2} c d + 5 \, a^{5} b d^{2}\right )} x - 60 \, {\left (2 \, a^{4} b^{2} c^{2} - 5 \, a^{5} b c d + 3 \, a^{6} d^{2} + {\left (2 \, a^{3} b^{3} c^{2} - 5 \, a^{4} b^{2} c d + 3 \, a^{5} b d^{2}\right )} x\right )} \log \left (b x + a\right )}{30 \, {\left (b^{8} x + a b^{7}\right )}} \]
1/30*(6*b^6*d^2*x^6 - 30*a^4*b^2*c^2 + 60*a^5*b*c*d - 30*a^6*d^2 + 3*(5*b^ 6*c*d - 3*a*b^5*d^2)*x^5 + 5*(2*b^6*c^2 - 5*a*b^5*c*d + 3*a^2*b^4*d^2)*x^4 - 10*(2*a*b^5*c^2 - 5*a^2*b^4*c*d + 3*a^3*b^3*d^2)*x^3 + 30*(2*a^2*b^4*c^ 2 - 5*a^3*b^3*c*d + 3*a^4*b^2*d^2)*x^2 + 30*(3*a^3*b^3*c^2 - 8*a^4*b^2*c*d + 5*a^5*b*d^2)*x - 60*(2*a^4*b^2*c^2 - 5*a^5*b*c*d + 3*a^6*d^2 + (2*a^3*b ^3*c^2 - 5*a^4*b^2*c*d + 3*a^5*b*d^2)*x)*log(b*x + a))/(b^8*x + a*b^7)
Time = 0.37 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.27 \[ \int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx=- \frac {2 a^{3} \left (a d - b c\right ) \left (3 a d - 2 b c\right ) \log {\left (a + b x \right )}}{b^{7}} + x^{4} \left (- \frac {a d^{2}}{2 b^{3}} + \frac {c d}{2 b^{2}}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{b^{4}} - \frac {4 a c d}{3 b^{3}} + \frac {c^{2}}{3 b^{2}}\right ) + x^{2} \left (- \frac {2 a^{3} d^{2}}{b^{5}} + \frac {3 a^{2} c d}{b^{4}} - \frac {a c^{2}}{b^{3}}\right ) + x \left (\frac {5 a^{4} d^{2}}{b^{6}} - \frac {8 a^{3} c d}{b^{5}} + \frac {3 a^{2} c^{2}}{b^{4}}\right ) + \frac {- a^{6} d^{2} + 2 a^{5} b c d - a^{4} b^{2} c^{2}}{a b^{7} + b^{8} x} + \frac {d^{2} x^{5}}{5 b^{2}} \]
-2*a**3*(a*d - b*c)*(3*a*d - 2*b*c)*log(a + b*x)/b**7 + x**4*(-a*d**2/(2*b **3) + c*d/(2*b**2)) + x**3*(a**2*d**2/b**4 - 4*a*c*d/(3*b**3) + c**2/(3*b **2)) + x**2*(-2*a**3*d**2/b**5 + 3*a**2*c*d/b**4 - a*c**2/b**3) + x*(5*a* *4*d**2/b**6 - 8*a**3*c*d/b**5 + 3*a**2*c**2/b**4) + (-a**6*d**2 + 2*a**5* b*c*d - a**4*b**2*c**2)/(a*b**7 + b**8*x) + d**2*x**5/(5*b**2)
Time = 0.20 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.30 \[ \int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx=-\frac {a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2}}{b^{8} x + a b^{7}} + \frac {6 \, b^{4} d^{2} x^{5} + 15 \, {\left (b^{4} c d - a b^{3} d^{2}\right )} x^{4} + 10 \, {\left (b^{4} c^{2} - 4 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{3} - 30 \, {\left (a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} x^{2} + 30 \, {\left (3 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 5 \, a^{4} d^{2}\right )} x}{30 \, b^{6}} - \frac {2 \, {\left (2 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left (b x + a\right )}{b^{7}} \]
-(a^4*b^2*c^2 - 2*a^5*b*c*d + a^6*d^2)/(b^8*x + a*b^7) + 1/30*(6*b^4*d^2*x ^5 + 15*(b^4*c*d - a*b^3*d^2)*x^4 + 10*(b^4*c^2 - 4*a*b^3*c*d + 3*a^2*b^2* d^2)*x^3 - 30*(a*b^3*c^2 - 3*a^2*b^2*c*d + 2*a^3*b*d^2)*x^2 + 30*(3*a^2*b^ 2*c^2 - 8*a^3*b*c*d + 5*a^4*d^2)*x)/b^6 - 2*(2*a^3*b^2*c^2 - 5*a^4*b*c*d + 3*a^5*d^2)*log(b*x + a)/b^7
Time = 0.28 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.70 \[ \int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx=\frac {{\left (6 \, d^{2} + \frac {15 \, {\left (b^{2} c d - 3 \, a b d^{2}\right )}}{{\left (b x + a\right )} b} + \frac {10 \, {\left (b^{4} c^{2} - 10 \, a b^{3} c d + 15 \, a^{2} b^{2} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {60 \, {\left (a b^{5} c^{2} - 5 \, a^{2} b^{4} c d + 5 \, a^{3} b^{3} d^{2}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac {30 \, {\left (6 \, a^{2} b^{6} c^{2} - 20 \, a^{3} b^{5} c d + 15 \, a^{4} b^{4} d^{2}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )} {\left (b x + a\right )}^{5}}{30 \, b^{7}} + \frac {2 \, {\left (2 \, a^{3} b^{2} c^{2} - 5 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{7}} - \frac {\frac {a^{4} b^{7} c^{2}}{b x + a} - \frac {2 \, a^{5} b^{6} c d}{b x + a} + \frac {a^{6} b^{5} d^{2}}{b x + a}}{b^{12}} \]
1/30*(6*d^2 + 15*(b^2*c*d - 3*a*b*d^2)/((b*x + a)*b) + 10*(b^4*c^2 - 10*a* b^3*c*d + 15*a^2*b^2*d^2)/((b*x + a)^2*b^2) - 60*(a*b^5*c^2 - 5*a^2*b^4*c* d + 5*a^3*b^3*d^2)/((b*x + a)^3*b^3) + 30*(6*a^2*b^6*c^2 - 20*a^3*b^5*c*d + 15*a^4*b^4*d^2)/((b*x + a)^4*b^4))*(b*x + a)^5/b^7 + 2*(2*a^3*b^2*c^2 - 5*a^4*b*c*d + 3*a^5*d^2)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^7 - (a^4 *b^7*c^2/(b*x + a) - 2*a^5*b^6*c*d/(b*x + a) + a^6*b^5*d^2/(b*x + a))/b^12
Time = 0.08 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.27 \[ \int \frac {x^4 (c+d x)^2}{(a+b x)^2} \, dx=x^3\,\left (\frac {c^2}{3\,b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{3\,b}-\frac {a^2\,d^2}{3\,b^4}\right )+x^2\,\left (\frac {a^2\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{2\,b^2}-\frac {a\,\left (\frac {c^2}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{b^4}\right )}{b}\right )-x^4\,\left (\frac {a\,d^2}{2\,b^3}-\frac {c\,d}{2\,b^2}\right )-x\,\left (\frac {2\,a\,\left (\frac {a^2\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b^2}-\frac {2\,a\,\left (\frac {c^2}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{b^4}\right )}{b}\right )}{b}+\frac {a^2\,\left (\frac {c^2}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^2}{b^3}-\frac {2\,c\,d}{b^2}\right )}{b}-\frac {a^2\,d^2}{b^4}\right )}{b^2}\right )-\frac {a^6\,d^2-2\,a^5\,b\,c\,d+a^4\,b^2\,c^2}{b\,\left (x\,b^7+a\,b^6\right )}-\frac {\ln \left (a+b\,x\right )\,\left (6\,a^5\,d^2-10\,a^4\,b\,c\,d+4\,a^3\,b^2\,c^2\right )}{b^7}+\frac {d^2\,x^5}{5\,b^2} \]
x^3*(c^2/(3*b^2) + (2*a*((2*a*d^2)/b^3 - (2*c*d)/b^2))/(3*b) - (a^2*d^2)/( 3*b^4)) + x^2*((a^2*((2*a*d^2)/b^3 - (2*c*d)/b^2))/(2*b^2) - (a*(c^2/b^2 + (2*a*((2*a*d^2)/b^3 - (2*c*d)/b^2))/b - (a^2*d^2)/b^4))/b) - x^4*((a*d^2) /(2*b^3) - (c*d)/(2*b^2)) - x*((2*a*((a^2*((2*a*d^2)/b^3 - (2*c*d)/b^2))/b ^2 - (2*a*(c^2/b^2 + (2*a*((2*a*d^2)/b^3 - (2*c*d)/b^2))/b - (a^2*d^2)/b^4 ))/b))/b + (a^2*(c^2/b^2 + (2*a*((2*a*d^2)/b^3 - (2*c*d)/b^2))/b - (a^2*d^ 2)/b^4))/b^2) - (a^6*d^2 + a^4*b^2*c^2 - 2*a^5*b*c*d)/(b*(a*b^6 + b^7*x)) - (log(a + b*x)*(6*a^5*d^2 + 4*a^3*b^2*c^2 - 10*a^4*b*c*d))/b^7 + (d^2*x^5 )/(5*b^2)