Integrand size = 29, antiderivative size = 122 \[ \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b (b c-a d)^4 x}{d^5}-\frac {(b c-a d)^3 (a+b x)^2}{2 d^4}+\frac {(b c-a d)^2 (a+b x)^3}{3 d^3}-\frac {(b c-a d) (a+b x)^4}{4 d^2}+\frac {(a+b x)^5}{5 d}-\frac {(b c-a d)^5 \log (c+d x)}{d^6} \] Output:
b*(-a*d+b*c)^4*x/d^5-1/2*(-a*d+b*c)^3*(b*x+a)^2/d^4+1/3*(-a*d+b*c)^2*(b*x+ a)^3/d^3-1/4*(-a*d+b*c)*(b*x+a)^4/d^2+1/5*(b*x+a)^5/d-(-a*d+b*c)^5*ln(d*x+ c)/d^6
Time = 0.05 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b d x \left (300 a^4 d^4+300 a^3 b d^3 (-2 c+d x)+100 a^2 b^2 d^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )+25 a b^3 d \left (-12 c^3+6 c^2 d x-4 c d^2 x^2+3 d^3 x^3\right )+b^4 \left (60 c^4-30 c^3 d x+20 c^2 d^2 x^2-15 c d^3 x^3+12 d^4 x^4\right )\right )-60 (b c-a d)^5 \log (c+d x)}{60 d^6} \] Input:
Integrate[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2),x]
Output:
(b*d*x*(300*a^4*d^4 + 300*a^3*b*d^3*(-2*c + d*x) + 100*a^2*b^2*d^2*(6*c^2 - 3*c*d*x + 2*d^2*x^2) + 25*a*b^3*d*(-12*c^3 + 6*c^2*d*x - 4*c*d^2*x^2 + 3 *d^3*x^3) + b^4*(60*c^4 - 30*c^3*d*x + 20*c^2*d^2*x^2 - 15*c*d^3*x^3 + 12* d^4*x^4)) - 60*(b*c - a*d)^5*Log[c + d*x])/(60*d^6)
Time = 0.48 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1121, 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.
\(\displaystyle \int \frac {(a+b x)^6}{x (a d+b c)+a c+b d x^2} \, dx\) |
\(\Big \downarrow \) 1121 |
\(\displaystyle \int \left (\frac {(a d-b c)^5}{d^5 (c+d x)}+\frac {b (b c-a d)^4}{d^5}-\frac {b (a+b x) (b c-a d)^3}{d^4}+\frac {b (a+b x)^2 (b c-a d)^2}{d^3}-\frac {b (a+b x)^3 (b c-a d)}{d^2}+\frac {b (a+b x)^4}{d}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(b c-a d)^5 \log (c+d x)}{d^6}+\frac {b x (b c-a d)^4}{d^5}-\frac {(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac {(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac {(a+b x)^4 (b c-a d)}{4 d^2}+\frac {(a+b x)^5}{5 d}\) |
Input:
Int[(a + b*x)^6/(a*c + (b*c + a*d)*x + b*d*x^2),x]
Output:
(b*(b*c - a*d)^4*x)/d^5 - ((b*c - a*d)^3*(a + b*x)^2)/(2*d^4) + ((b*c - a* d)^2*(a + b*x)^3)/(3*d^3) - ((b*c - a*d)*(a + b*x)^4)/(4*d^2) + (a + b*x)^ 5/(5*d) - ((b*c - a*d)^5*Log[c + d*x])/d^6
Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && (Int egerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && LtQ[c, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(243\) vs. \(2(114)=228\).
Time = 1.53 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.00
method | result | size |
norman | \(\frac {b \left (5 a^{4} d^{4}-10 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}-5 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) x}{d^{5}}+\frac {b^{5} x^{5}}{5 d}+\frac {b^{2} \left (10 a^{3} d^{3}-10 a^{2} b c \,d^{2}+5 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 d^{4}}+\frac {b^{3} \left (10 a^{2} d^{2}-5 a b c d +b^{2} c^{2}\right ) x^{3}}{3 d^{3}}+\frac {b^{4} \left (5 a d -b c \right ) x^{4}}{4 d^{2}}+\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \ln \left (d x +c \right )}{d^{6}}\) | \(244\) |
default | \(\frac {b \left (\frac {1}{5} b^{4} x^{5} d^{4}+\frac {5}{4} a \,b^{3} d^{4} x^{4}-\frac {1}{4} b^{4} c \,x^{4} d^{3}+\frac {10}{3} x^{3} a^{2} b^{2} d^{4}-\frac {5}{3} a \,b^{3} c \,d^{3} x^{3}+\frac {1}{3} x^{3} b^{4} c^{2} d^{2}+5 a^{3} b \,d^{4} x^{2}-5 x^{2} a^{2} b^{2} c \,d^{3}+\frac {5}{2} a \,b^{3} c^{2} d^{2} x^{2}-\frac {1}{2} x^{2} b^{4} c^{3} d +5 a^{4} d^{4} x -10 a^{3} b c \,d^{3} x +10 a^{2} b^{2} c^{2} d^{2} x -5 a \,b^{3} c^{3} d x +c^{4} b^{4} x \right )}{d^{5}}+\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \ln \left (d x +c \right )}{d^{6}}\) | \(266\) |
risch | \(\frac {b^{5} x^{5}}{5 d}+\frac {5 b^{4} a \,x^{4}}{4 d}-\frac {b^{5} c \,x^{4}}{4 d^{2}}+\frac {10 b^{3} x^{3} a^{2}}{3 d}-\frac {5 b^{4} a c \,x^{3}}{3 d^{2}}+\frac {b^{5} x^{3} c^{2}}{3 d^{3}}+\frac {5 b^{2} a^{3} x^{2}}{d}-\frac {5 b^{3} x^{2} a^{2} c}{d^{2}}+\frac {5 b^{4} a \,c^{2} x^{2}}{2 d^{3}}-\frac {b^{5} x^{2} c^{3}}{2 d^{4}}+\frac {5 b \,a^{4} x}{d}-\frac {10 b^{2} a^{3} c x}{d^{2}}+\frac {10 b^{3} a^{2} c^{2} x}{d^{3}}-\frac {5 b^{4} a \,c^{3} x}{d^{4}}+\frac {b^{5} c^{4} x}{d^{5}}+\frac {\ln \left (d x +c \right ) a^{5}}{d}-\frac {5 \ln \left (d x +c \right ) a^{4} b c}{d^{2}}+\frac {10 \ln \left (d x +c \right ) a^{3} b^{2} c^{2}}{d^{3}}-\frac {10 \ln \left (d x +c \right ) a^{2} b^{3} c^{3}}{d^{4}}+\frac {5 \ln \left (d x +c \right ) a \,b^{4} c^{4}}{d^{5}}-\frac {\ln \left (d x +c \right ) b^{5} c^{5}}{d^{6}}\) | \(302\) |
parallelrisch | \(\frac {12 b^{5} d^{5} x^{5}+75 a \,b^{4} d^{5} x^{4}-15 b^{5} c \,d^{4} x^{4}+200 a^{2} b^{3} d^{5} x^{3}-100 a \,b^{4} c \,d^{4} x^{3}+20 b^{5} c^{2} d^{3} x^{3}+300 a^{3} b^{2} d^{5} x^{2}-300 a^{2} b^{3} c \,d^{4} x^{2}+150 a \,b^{4} c^{2} d^{3} x^{2}-30 b^{5} c^{3} d^{2} x^{2}+60 \ln \left (d x +c \right ) a^{5} d^{5}-300 \ln \left (d x +c \right ) a^{4} b c \,d^{4}+600 \ln \left (d x +c \right ) a^{3} b^{2} c^{2} d^{3}-600 \ln \left (d x +c \right ) a^{2} b^{3} c^{3} d^{2}+300 \ln \left (d x +c \right ) a \,b^{4} c^{4} d -60 \ln \left (d x +c \right ) b^{5} c^{5}+300 a^{4} b \,d^{5} x -600 a^{3} b^{2} c \,d^{4} x +600 a^{2} b^{3} c^{2} d^{3} x -300 a \,b^{4} c^{3} d^{2} x +60 b^{5} c^{4} d x}{60 d^{6}}\) | \(302\) |
Input:
int((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x,method=_RETURNVERBOSE)
Output:
b*(5*a^4*d^4-10*a^3*b*c*d^3+10*a^2*b^2*c^2*d^2-5*a*b^3*c^3*d+b^4*c^4)/d^5* x+1/5*b^5/d*x^5+1/2*b^2/d^4*(10*a^3*d^3-10*a^2*b*c*d^2+5*a*b^2*c^2*d-b^3*c ^3)*x^2+1/3*b^3/d^3*(10*a^2*d^2-5*a*b*c*d+b^2*c^2)*x^3+1/4*b^4/d^2*(5*a*d- b*c)*x^4+(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5*a* b^4*c^4*d-b^5*c^5)/d^6*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (114) = 228\).
Time = 0.09 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx=\frac {12 \, b^{5} d^{5} x^{5} - 15 \, {\left (b^{5} c d^{4} - 5 \, a b^{4} d^{5}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{3} - 5 \, a b^{4} c d^{4} + 10 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} + 10 \, a^{2} b^{3} c d^{4} - 10 \, a^{3} b^{2} d^{5}\right )} x^{2} + 60 \, {\left (b^{5} c^{4} d - 5 \, a b^{4} c^{3} d^{2} + 10 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x - 60 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{60 \, d^{6}} \] Input:
integrate((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")
Output:
1/60*(12*b^5*d^5*x^5 - 15*(b^5*c*d^4 - 5*a*b^4*d^5)*x^4 + 20*(b^5*c^2*d^3 - 5*a*b^4*c*d^4 + 10*a^2*b^3*d^5)*x^3 - 30*(b^5*c^3*d^2 - 5*a*b^4*c^2*d^3 + 10*a^2*b^3*c*d^4 - 10*a^3*b^2*d^5)*x^2 + 60*(b^5*c^4*d - 5*a*b^4*c^3*d^2 + 10*a^2*b^3*c^2*d^3 - 10*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x - 60*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*log(d*x + c))/d^6
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (104) = 208\).
Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b^{5} x^{5}}{5 d} + x^{4} \cdot \left (\frac {5 a b^{4}}{4 d} - \frac {b^{5} c}{4 d^{2}}\right ) + x^{3} \cdot \left (\frac {10 a^{2} b^{3}}{3 d} - \frac {5 a b^{4} c}{3 d^{2}} + \frac {b^{5} c^{2}}{3 d^{3}}\right ) + x^{2} \cdot \left (\frac {5 a^{3} b^{2}}{d} - \frac {5 a^{2} b^{3} c}{d^{2}} + \frac {5 a b^{4} c^{2}}{2 d^{3}} - \frac {b^{5} c^{3}}{2 d^{4}}\right ) + x \left (\frac {5 a^{4} b}{d} - \frac {10 a^{3} b^{2} c}{d^{2}} + \frac {10 a^{2} b^{3} c^{2}}{d^{3}} - \frac {5 a b^{4} c^{3}}{d^{4}} + \frac {b^{5} c^{4}}{d^{5}}\right ) + \frac {\left (a d - b c\right )^{5} \log {\left (c + d x \right )}}{d^{6}} \] Input:
integrate((b*x+a)**6/(a*c+(a*d+b*c)*x+b*d*x**2),x)
Output:
b**5*x**5/(5*d) + x**4*(5*a*b**4/(4*d) - b**5*c/(4*d**2)) + x**3*(10*a**2* b**3/(3*d) - 5*a*b**4*c/(3*d**2) + b**5*c**2/(3*d**3)) + x**2*(5*a**3*b**2 /d - 5*a**2*b**3*c/d**2 + 5*a*b**4*c**2/(2*d**3) - b**5*c**3/(2*d**4)) + x *(5*a**4*b/d - 10*a**3*b**2*c/d**2 + 10*a**2*b**3*c**2/d**3 - 5*a*b**4*c** 3/d**4 + b**5*c**4/d**5) + (a*d - b*c)**5*log(c + d*x)/d**6
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (114) = 228\).
Time = 0.04 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.11 \[ \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx=\frac {12 \, b^{5} d^{4} x^{5} - 15 \, {\left (b^{5} c d^{3} - 5 \, a b^{4} d^{4}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{2} - 5 \, a b^{4} c d^{3} + 10 \, a^{2} b^{3} d^{4}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d - 5 \, a b^{4} c^{2} d^{2} + 10 \, a^{2} b^{3} c d^{3} - 10 \, a^{3} b^{2} d^{4}\right )} x^{2} + 60 \, {\left (b^{5} c^{4} - 5 \, a b^{4} c^{3} d + 10 \, a^{2} b^{3} c^{2} d^{2} - 10 \, a^{3} b^{2} c d^{3} + 5 \, a^{4} b d^{4}\right )} x}{60 \, d^{5}} - \frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left (d x + c\right )}{d^{6}} \] Input:
integrate((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="maxima")
Output:
1/60*(12*b^5*d^4*x^5 - 15*(b^5*c*d^3 - 5*a*b^4*d^4)*x^4 + 20*(b^5*c^2*d^2 - 5*a*b^4*c*d^3 + 10*a^2*b^3*d^4)*x^3 - 30*(b^5*c^3*d - 5*a*b^4*c^2*d^2 + 10*a^2*b^3*c*d^3 - 10*a^3*b^2*d^4)*x^2 + 60*(b^5*c^4 - 5*a*b^4*c^3*d + 10* a^2*b^3*c^2*d^2 - 10*a^3*b^2*c*d^3 + 5*a^4*b*d^4)*x)/d^5 - (b^5*c^5 - 5*a* b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5* d^5)*log(d*x + c)/d^6
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (114) = 228\).
Time = 0.13 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.24 \[ \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx=\frac {12 \, b^{5} d^{4} x^{5} - 15 \, b^{5} c d^{3} x^{4} + 75 \, a b^{4} d^{4} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} - 100 \, a b^{4} c d^{3} x^{3} + 200 \, a^{2} b^{3} d^{4} x^{3} - 30 \, b^{5} c^{3} d x^{2} + 150 \, a b^{4} c^{2} d^{2} x^{2} - 300 \, a^{2} b^{3} c d^{3} x^{2} + 300 \, a^{3} b^{2} d^{4} x^{2} + 60 \, b^{5} c^{4} x - 300 \, a b^{4} c^{3} d x + 600 \, a^{2} b^{3} c^{2} d^{2} x - 600 \, a^{3} b^{2} c d^{3} x + 300 \, a^{4} b d^{4} x}{60 \, d^{5}} - \frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} \] Input:
integrate((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="giac")
Output:
1/60*(12*b^5*d^4*x^5 - 15*b^5*c*d^3*x^4 + 75*a*b^4*d^4*x^4 + 20*b^5*c^2*d^ 2*x^3 - 100*a*b^4*c*d^3*x^3 + 200*a^2*b^3*d^4*x^3 - 30*b^5*c^3*d*x^2 + 150 *a*b^4*c^2*d^2*x^2 - 300*a^2*b^3*c*d^3*x^2 + 300*a^3*b^2*d^4*x^2 + 60*b^5* c^4*x - 300*a*b^4*c^3*d*x + 600*a^2*b^3*c^2*d^2*x - 600*a^3*b^2*c*d^3*x + 300*a^4*b*d^4*x)/d^5 - (b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10* a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*log(abs(d*x + c))/d^6
Time = 5.39 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.30 \[ \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx=x\,\left (\frac {5\,a^4\,b}{d}-\frac {c\,\left (\frac {10\,a^3\,b^2}{d}+\frac {c\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{d}-\frac {10\,a^2\,b^3}{d}\right )}{d}\right )}{d}\right )+x^4\,\left (\frac {5\,a\,b^4}{4\,d}-\frac {b^5\,c}{4\,d^2}\right )+x^2\,\left (\frac {5\,a^3\,b^2}{d}+\frac {c\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{d}-\frac {10\,a^2\,b^3}{d}\right )}{2\,d}\right )-x^3\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d}-\frac {b^5\,c}{d^2}\right )}{3\,d}-\frac {10\,a^2\,b^3}{3\,d}\right )+\frac {b^5\,x^5}{5\,d}+\frac {\ln \left (c+d\,x\right )\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{d^6} \] Input:
int((a + b*x)^6/(a*c + x*(a*d + b*c) + b*d*x^2),x)
Output:
x*((5*a^4*b)/d - (c*((10*a^3*b^2)/d + (c*((c*((5*a*b^4)/d - (b^5*c)/d^2))/ d - (10*a^2*b^3)/d))/d))/d) + x^4*((5*a*b^4)/(4*d) - (b^5*c)/(4*d^2)) + x^ 2*((5*a^3*b^2)/d + (c*((c*((5*a*b^4)/d - (b^5*c)/d^2))/d - (10*a^2*b^3)/d) )/(2*d)) - x^3*((c*((5*a*b^4)/d - (b^5*c)/d^2))/(3*d) - (10*a^2*b^3)/(3*d) ) + (b^5*x^5)/(5*d) + (log(c + d*x)*(a^5*d^5 - b^5*c^5 - 10*a^2*b^3*c^3*d^ 2 + 10*a^3*b^2*c^2*d^3 + 5*a*b^4*c^4*d - 5*a^4*b*c*d^4))/d^6
Time = 0.21 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.47 \[ \int \frac {(a+b x)^6}{a c+(b c+a d) x+b d x^2} \, dx=\frac {60 \,\mathrm {log}\left (d x +c \right ) a^{5} d^{5}-300 \,\mathrm {log}\left (d x +c \right ) a^{4} b c \,d^{4}+600 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c^{2} d^{3}-600 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{3} d^{2}+300 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c^{4} d -60 \,\mathrm {log}\left (d x +c \right ) b^{5} c^{5}+300 a^{4} b \,d^{5} x -600 a^{3} b^{2} c \,d^{4} x +300 a^{3} b^{2} d^{5} x^{2}+600 a^{2} b^{3} c^{2} d^{3} x -300 a^{2} b^{3} c \,d^{4} x^{2}+200 a^{2} b^{3} d^{5} x^{3}-300 a \,b^{4} c^{3} d^{2} x +150 a \,b^{4} c^{2} d^{3} x^{2}-100 a \,b^{4} c \,d^{4} x^{3}+75 a \,b^{4} d^{5} x^{4}+60 b^{5} c^{4} d x -30 b^{5} c^{3} d^{2} x^{2}+20 b^{5} c^{2} d^{3} x^{3}-15 b^{5} c \,d^{4} x^{4}+12 b^{5} d^{5} x^{5}}{60 d^{6}} \] Input:
int((b*x+a)^6/(a*c+(a*d+b*c)*x+b*d*x^2),x)
Output:
(60*log(c + d*x)*a**5*d**5 - 300*log(c + d*x)*a**4*b*c*d**4 + 600*log(c + d*x)*a**3*b**2*c**2*d**3 - 600*log(c + d*x)*a**2*b**3*c**3*d**2 + 300*log( c + d*x)*a*b**4*c**4*d - 60*log(c + d*x)*b**5*c**5 + 300*a**4*b*d**5*x - 6 00*a**3*b**2*c*d**4*x + 300*a**3*b**2*d**5*x**2 + 600*a**2*b**3*c**2*d**3* x - 300*a**2*b**3*c*d**4*x**2 + 200*a**2*b**3*d**5*x**3 - 300*a*b**4*c**3* d**2*x + 150*a*b**4*c**2*d**3*x**2 - 100*a*b**4*c*d**4*x**3 + 75*a*b**4*d* *5*x**4 + 60*b**5*c**4*d*x - 30*b**5*c**3*d**2*x**2 + 20*b**5*c**2*d**3*x* *3 - 15*b**5*c*d**4*x**4 + 12*b**5*d**5*x**5)/(60*d**6)