Integrand size = 18, antiderivative size = 136 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {(b c-4 a d) (b c-a d)^2 x}{b^5}+\frac {3 d (b c-a d)^2 x^2}{2 b^4}+\frac {d^2 (3 b c-2 a d) x^3}{3 b^3}+\frac {d^3 x^4}{4 b^2}-\frac {a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac {a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6} \] Output:
(-4*a*d+b*c)*(-a*d+b*c)^2*x/b^5+3/2*d*(-a*d+b*c)^2*x^2/b^4+1/3*d^2*(-2*a*d +3*b*c)*x^3/b^3+1/4*d^3*x^4/b^2-a^2*(-a*d+b*c)^3/b^6/(b*x+a)-a*(-5*a*d+2*b *c)*(-a*d+b*c)^2*ln(b*x+a)/b^6
Time = 0.04 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {12 b (b c-4 a d) (b c-a d)^2 x+18 b^2 d (b c-a d)^2 x^2+4 b^3 d^2 (3 b c-2 a d) x^3+3 b^4 d^3 x^4+\frac {12 a^2 (-b c+a d)^3}{a+b x}+12 a (b c-a d)^2 (-2 b c+5 a d) \log (a+b x)}{12 b^6} \] Input:
Integrate[(x^2*(c + d*x)^3)/(a + b*x)^2,x]
Output:
(12*b*(b*c - 4*a*d)*(b*c - a*d)^2*x + 18*b^2*d*(b*c - a*d)^2*x^2 + 4*b^3*d ^2*(3*b*c - 2*a*d)*x^3 + 3*b^4*d^3*x^4 + (12*a^2*(-(b*c) + a*d)^3)/(a + b* x) + 12*a*(b*c - a*d)^2*(-2*b*c + 5*a*d)*Log[a + b*x])/(12*b^6)
Time = 0.32 (sec) , antiderivative size = 136, 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.
| \(\displaystyle \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {a^2 (a d-b c)^3}{b^5 (a+b x)^2}+\frac {a (5 a d-2 b c) (a d-b c)^2}{b^5 (a+b x)}+\frac {(b c-4 a d) (b c-a d)^2}{b^5}+\frac {3 d x (b c-a d)^2}{b^4}+\frac {d^2 x^2 (3 b c-2 a d)}{b^3}+\frac {d^3 x^3}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 (b c-a d)^3}{b^6 (a+b x)}-\frac {a (2 b c-5 a d) (b c-a d)^2 \log (a+b x)}{b^6}+\frac {x (b c-4 a d) (b c-a d)^2}{b^5}+\frac {3 d x^2 (b c-a d)^2}{2 b^4}+\frac {d^2 x^3 (3 b c-2 a d)}{3 b^3}+\frac {d^3 x^4}{4 b^2}\) |
Input:
Int[(x^2*(c + d*x)^3)/(a + b*x)^2,x]
Output:
((b*c - 4*a*d)*(b*c - a*d)^2*x)/b^5 + (3*d*(b*c - a*d)^2*x^2)/(2*b^4) + (d ^2*(3*b*c - 2*a*d)*x^3)/(3*b^3) + (d^3*x^4)/(4*b^2) - (a^2*(b*c - a*d)^3)/ (b^6*(a + b*x)) - (a*(2*b*c - 5*a*d)*(b*c - a*d)^2*Log[a + b*x])/b^6
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 = 0.18 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.60
| method | result | size |
| default | \(-\frac {-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {2}{3} x^{3} a \,b^{2} d^{3}-x^{3} b^{3} c \,d^{2}-\frac {3}{2} x^{2} a^{2} b \,d^{3}+3 x^{2} a \,b^{2} c \,d^{2}-\frac {3}{2} x^{2} b^{3} c^{2} d +4 a^{3} d^{3} x -9 a^{2} b c \,d^{2} x +6 a \,b^{2} c^{2} d x -b^{3} c^{3} x}{b^{5}}+\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{6} \left (b x +a \right )}+\frac {a \left (5 a^{3} d^{3}-12 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(218\) |
| norman | \(\frac {-\frac {\left (5 a^{3} d^{3}-12 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{2}}{2 b^{4}}+\frac {d^{3} x^{5}}{4 b}+\frac {d \left (5 a^{2} d^{2}-12 a b c d +9 b^{2} c^{2}\right ) x^{3}}{6 b^{3}}-\frac {d^{2} \left (5 a d -12 b c \right ) x^{4}}{12 b^{2}}-\frac {\left (5 a^{5} d^{3}-12 a^{4} b c \,d^{2}+9 a^{3} b^{2} c^{2} d -2 a^{2} b^{3} c^{3}\right ) x}{b^{5} a}}{b x +a}+\frac {a \left (5 a^{3} d^{3}-12 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(218\) |
| risch | \(\frac {d^{3} x^{4}}{4 b^{2}}-\frac {2 x^{3} a \,d^{3}}{3 b^{3}}+\frac {x^{3} c \,d^{2}}{b^{2}}+\frac {3 x^{2} a^{2} d^{3}}{2 b^{4}}-\frac {3 x^{2} a c \,d^{2}}{b^{3}}+\frac {3 x^{2} c^{2} d}{2 b^{2}}-\frac {4 a^{3} d^{3} x}{b^{5}}+\frac {9 a^{2} c \,d^{2} x}{b^{4}}-\frac {6 a \,c^{2} d x}{b^{3}}+\frac {c^{3} x}{b^{2}}+\frac {a^{5} d^{3}}{b^{6} \left (b x +a \right )}-\frac {3 a^{4} c \,d^{2}}{b^{5} \left (b x +a \right )}+\frac {3 a^{3} c^{2} d}{b^{4} \left (b x +a \right )}-\frac {a^{2} c^{3}}{b^{3} \left (b x +a \right )}+\frac {5 a^{4} \ln \left (b x +a \right ) d^{3}}{b^{6}}-\frac {12 a^{3} \ln \left (b x +a \right ) c \,d^{2}}{b^{5}}+\frac {9 a^{2} \ln \left (b x +a \right ) c^{2} d}{b^{4}}-\frac {2 a \ln \left (b x +a \right ) c^{3}}{b^{3}}\) | \(260\) |
| parallelrisch | \(\frac {3 d^{3} x^{5} b^{5}-5 x^{4} a \,b^{4} d^{3}+12 x^{4} b^{5} c \,d^{2}+10 x^{3} a^{2} b^{3} d^{3}-24 x^{3} a \,b^{4} c \,d^{2}+18 x^{3} b^{5} c^{2} d +60 \ln \left (b x +a \right ) x \,a^{4} b \,d^{3}-144 \ln \left (b x +a \right ) x \,a^{3} b^{2} c \,d^{2}+108 \ln \left (b x +a \right ) x \,a^{2} b^{3} c^{2} d -24 \ln \left (b x +a \right ) x a \,b^{4} c^{3}-30 x^{2} a^{3} b^{2} d^{3}+72 x^{2} a^{2} b^{3} c \,d^{2}-54 x^{2} a \,b^{4} c^{2} d +12 x^{2} b^{5} c^{3}+60 \ln \left (b x +a \right ) a^{5} d^{3}-144 \ln \left (b x +a \right ) a^{4} b c \,d^{2}+108 \ln \left (b x +a \right ) a^{3} b^{2} c^{2} d -24 \ln \left (b x +a \right ) a^{2} b^{3} c^{3}+60 a^{5} d^{3}-144 a^{4} b c \,d^{2}+108 a^{3} b^{2} c^{2} d -24 a^{2} b^{3} c^{3}}{12 b^{6} \left (b x +a \right )}\) | \(317\) |
Input:
int(x^2*(d*x+c)^3/(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/b^5*(-1/4*d^3*x^4*b^3+2/3*x^3*a*b^2*d^3-x^3*b^3*c*d^2-3/2*x^2*a^2*b*d^3 +3*x^2*a*b^2*c*d^2-3/2*x^2*b^3*c^2*d+4*a^3*d^3*x-9*a^2*b*c*d^2*x+6*a*b^2*c ^2*d*x-b^3*c^3*x)+a^2*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^6/(b *x+a)+a/b^6*(5*a^3*d^3-12*a^2*b*c*d^2+9*a*b^2*c^2*d-2*b^3*c^3)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (130) = 260\).
Time = 0.09 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.31 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {3 \, b^{5} d^{3} x^{5} - 12 \, a^{2} b^{3} c^{3} + 36 \, a^{3} b^{2} c^{2} d - 36 \, a^{4} b c d^{2} + 12 \, a^{5} d^{3} + {\left (12 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{4} + 2 \, {\left (9 \, b^{5} c^{2} d - 12 \, a b^{4} c d^{2} + 5 \, a^{2} b^{3} d^{3}\right )} x^{3} + 6 \, {\left (2 \, b^{5} c^{3} - 9 \, a b^{4} c^{2} d + 12 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{2} + 12 \, {\left (a b^{4} c^{3} - 6 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 4 \, a^{4} b d^{3}\right )} x - 12 \, {\left (2 \, a^{2} b^{3} c^{3} - 9 \, a^{3} b^{2} c^{2} d + 12 \, a^{4} b c d^{2} - 5 \, a^{5} d^{3} + {\left (2 \, a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 12 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \] Input:
integrate(x^2*(d*x+c)^3/(b*x+a)^2,x, algorithm="fricas")
Output:
1/12*(3*b^5*d^3*x^5 - 12*a^2*b^3*c^3 + 36*a^3*b^2*c^2*d - 36*a^4*b*c*d^2 + 12*a^5*d^3 + (12*b^5*c*d^2 - 5*a*b^4*d^3)*x^4 + 2*(9*b^5*c^2*d - 12*a*b^4 *c*d^2 + 5*a^2*b^3*d^3)*x^3 + 6*(2*b^5*c^3 - 9*a*b^4*c^2*d + 12*a^2*b^3*c* d^2 - 5*a^3*b^2*d^3)*x^2 + 12*(a*b^4*c^3 - 6*a^2*b^3*c^2*d + 9*a^3*b^2*c*d ^2 - 4*a^4*b*d^3)*x - 12*(2*a^2*b^3*c^3 - 9*a^3*b^2*c^2*d + 12*a^4*b*c*d^2 - 5*a^5*d^3 + (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)*log(b*x + a))/(b^7*x + a*b^6)
Time = 0.44 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.50 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {a \left (a d - b c\right )^{2} \cdot \left (5 a d - 2 b c\right ) \log {\left (a + b x \right )}}{b^{6}} + x^{3} \left (- \frac {2 a d^{3}}{3 b^{3}} + \frac {c d^{2}}{b^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{3}}{2 b^{4}} - \frac {3 a c d^{2}}{b^{3}} + \frac {3 c^{2} d}{2 b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{3}}{b^{5}} + \frac {9 a^{2} c d^{2}}{b^{4}} - \frac {6 a c^{2} d}{b^{3}} + \frac {c^{3}}{b^{2}}\right ) + \frac {a^{5} d^{3} - 3 a^{4} b c d^{2} + 3 a^{3} b^{2} c^{2} d - a^{2} b^{3} c^{3}}{a b^{6} + b^{7} x} + \frac {d^{3} x^{4}}{4 b^{2}} \] Input:
integrate(x**2*(d*x+c)**3/(b*x+a)**2,x)
Output:
a*(a*d - b*c)**2*(5*a*d - 2*b*c)*log(a + b*x)/b**6 + x**3*(-2*a*d**3/(3*b* *3) + c*d**2/b**2) + x**2*(3*a**2*d**3/(2*b**4) - 3*a*c*d**2/b**3 + 3*c**2 *d/(2*b**2)) + x*(-4*a**3*d**3/b**5 + 9*a**2*c*d**2/b**4 - 6*a*c**2*d/b**3 + c**3/b**2) + (a**5*d**3 - 3*a**4*b*c*d**2 + 3*a**3*b**2*c**2*d - a**2*b **3*c**3)/(a*b**6 + b**7*x) + d**3*x**4/(4*b**2)
Time = 0.03 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.62 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=-\frac {a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} d^{3} x^{4} + 4 \, {\left (3 \, b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{3} + 18 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 12 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} x}{12 \, b^{5}} - \frac {{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \] Input:
integrate(x^2*(d*x+c)^3/(b*x+a)^2,x, algorithm="maxima")
Output:
-(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)/(b^7*x + a*b^6) + 1/12*(3*b^3*d^3*x^4 + 4*(3*b^3*c*d^2 - 2*a*b^2*d^3)*x^3 + 18*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x^2 + 12*(b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b* c*d^2 - 4*a^3*d^3)*x)/b^5 - (2*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 12*a^3*b*c*d^ 2 - 5*a^4*d^3)*log(b*x + a)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (130) = 260\).
Time = 0.13 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.10 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {{\left (3 \, d^{3} + \frac {4 \, {\left (3 \, b^{2} c d^{2} - 5 \, a b d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {6 \, {\left (3 \, b^{4} c^{2} d - 12 \, a b^{3} c d^{2} + 10 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac {12 \, {\left (b^{6} c^{3} - 9 \, a b^{5} c^{2} d + 18 \, a^{2} b^{4} c d^{2} - 10 \, a^{3} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{4}}{12 \, b^{6}} + \frac {{\left (2 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{6}} - \frac {\frac {a^{2} b^{7} c^{3}}{b x + a} - \frac {3 \, a^{3} b^{6} c^{2} d}{b x + a} + \frac {3 \, a^{4} b^{5} c d^{2}}{b x + a} - \frac {a^{5} b^{4} d^{3}}{b x + a}}{b^{10}} \] Input:
integrate(x^2*(d*x+c)^3/(b*x+a)^2,x, algorithm="giac")
Output:
1/12*(3*d^3 + 4*(3*b^2*c*d^2 - 5*a*b*d^3)/((b*x + a)*b) + 6*(3*b^4*c^2*d - 12*a*b^3*c*d^2 + 10*a^2*b^2*d^3)/((b*x + a)^2*b^2) + 12*(b^6*c^3 - 9*a*b^ 5*c^2*d + 18*a^2*b^4*c*d^2 - 10*a^3*b^3*d^3)/((b*x + a)^3*b^3))*(b*x + a)^ 4/b^6 + (2*a*b^3*c^3 - 9*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 - 5*a^4*d^3)*log(a bs(b*x + a)/((b*x + a)^2*abs(b)))/b^6 - (a^2*b^7*c^3/(b*x + a) - 3*a^3*b^6 *c^2*d/(b*x + a) + 3*a^4*b^5*c*d^2/(b*x + a) - a^5*b^4*d^3/(b*x + a))/b^10
Time = 0.05 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.07 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=x\,\left (\frac {c^3}{b^2}-\frac {2\,a\,\left (\frac {3\,c^2\,d}{b^2}+\frac {2\,a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{b^4}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b^2}\right )-x^3\,\left (\frac {2\,a\,d^3}{3\,b^3}-\frac {c\,d^2}{b^2}\right )+x^2\,\left (\frac {3\,c^2\,d}{2\,b^2}+\frac {a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{2\,b^4}\right )+\frac {a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {d^3\,x^4}{4\,b^2}+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,d^3-12\,a^3\,b\,c\,d^2+9\,a^2\,b^2\,c^2\,d-2\,a\,b^3\,c^3\right )}{b^6} \] Input:
int((x^2*(c + d*x)^3)/(a + b*x)^2,x)
Output:
x*(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) - x^3*(( 2*a*d^3)/(3*b^3) - (c*d^2)/b^2) + x^2*((3*c^2*d)/(2*b^2) + (a*((2*a*d^3)/b ^3 - (3*c*d^2)/b^2))/b - (a^2*d^3)/(2*b^4)) + (a^5*d^3 - a^2*b^3*c^3 + 3*a ^3*b^2*c^2*d - 3*a^4*b*c*d^2)/(b*(a*b^5 + b^6*x)) + (d^3*x^4)/(4*b^2) + (l og(a + b*x)*(5*a^4*d^3 - 2*a*b^3*c^3 + 9*a^2*b^2*c^2*d - 12*a^3*b*c*d^2))/ b^6
Time = 0.18 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.36 \[ \int \frac {x^2 (c+d x)^3}{(a+b x)^2} \, dx=\frac {60 \,\mathrm {log}\left (b x +a \right ) a^{5} d^{3}-144 \,\mathrm {log}\left (b x +a \right ) a^{4} b c \,d^{2}+60 \,\mathrm {log}\left (b x +a \right ) a^{4} b \,d^{3} x +108 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} c^{2} d -144 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} c \,d^{2} x -24 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c^{3}+108 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c^{2} d x -24 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} c^{3} x -60 a^{4} b \,d^{3} x +144 a^{3} b^{2} c \,d^{2} x -30 a^{3} b^{2} d^{3} x^{2}-108 a^{2} b^{3} c^{2} d x +72 a^{2} b^{3} c \,d^{2} x^{2}+10 a^{2} b^{3} d^{3} x^{3}+24 a \,b^{4} c^{3} x -54 a \,b^{4} c^{2} d \,x^{2}-24 a \,b^{4} c \,d^{2} x^{3}-5 a \,b^{4} d^{3} x^{4}+12 b^{5} c^{3} x^{2}+18 b^{5} c^{2} d \,x^{3}+12 b^{5} c \,d^{2} x^{4}+3 b^{5} d^{3} x^{5}}{12 b^{6} \left (b x +a \right )} \] Input:
int(x^2*(d*x+c)^3/(b*x+a)^2,x)
Output:
(60*log(a + b*x)*a**5*d**3 - 144*log(a + b*x)*a**4*b*c*d**2 + 60*log(a + b *x)*a**4*b*d**3*x + 108*log(a + b*x)*a**3*b**2*c**2*d - 144*log(a + b*x)*a **3*b**2*c*d**2*x - 24*log(a + b*x)*a**2*b**3*c**3 + 108*log(a + b*x)*a**2 *b**3*c**2*d*x - 24*log(a + b*x)*a*b**4*c**3*x - 60*a**4*b*d**3*x + 144*a* *3*b**2*c*d**2*x - 30*a**3*b**2*d**3*x**2 - 108*a**2*b**3*c**2*d*x + 72*a* *2*b**3*c*d**2*x**2 + 10*a**2*b**3*d**3*x**3 + 24*a*b**4*c**3*x - 54*a*b** 4*c**2*d*x**2 - 24*a*b**4*c*d**2*x**3 - 5*a*b**4*d**3*x**4 + 12*b**5*c**3* x**2 + 18*b**5*c**2*d*x**3 + 12*b**5*c*d**2*x**4 + 3*b**5*d**3*x**5)/(12*b **6*(a + b*x))