Integrand size = 30, antiderivative size = 284 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\frac {\left (6 a^2 d^2 D-3 a b d (C d+2 c D)+b^2 \left (2 c C d+B d^2+c^2 D\right )\right ) x}{b^5}+\frac {d (b C d+2 b c D-3 a d D) x^2}{2 b^4}+\frac {d^2 D x^3}{3 b^3}-\frac {(b c-a d)^2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right )}{2 b^6 (a+b x)^2}-\frac {(b c-a d) \left (b^3 (B c+2 A d)-a b^2 (2 c C+3 B d)-5 a^3 d D+a^2 b (4 C d+3 c D)\right )}{b^6 (a+b x)}+\frac {\left (b^3 \left (c^2 C+2 B c d+A d^2\right )-10 a^3 d^2 D+6 a^2 b d (C d+2 c D)-3 a b^2 \left (2 c C d+B d^2+c^2 D\right )\right ) \log (a+b x)}{b^6} \] Output:
(6*a^2*d^2*D-3*a*b*d*(C*d+2*D*c)+b^2*(B*d^2+2*C*c*d+D*c^2))*x/b^5+1/2*d*(C *b*d-3*D*a*d+2*D*b*c)*x^2/b^4+1/3*d^2*D*x^3/b^3-1/2*(-a*d+b*c)^2*(A*b^3-a* (B*b^2-C*a*b+D*a^2))/b^6/(b*x+a)^2-(-a*d+b*c)*(b^3*(2*A*d+B*c)-a*b^2*(3*B* d+2*C*c)-5*a^3*d*D+a^2*b*(4*C*d+3*D*c))/b^6/(b*x+a)+(b^3*(A*d^2+2*B*c*d+C* c^2)-10*a^3*d^2*D+6*a^2*b*d*(C*d+2*D*c)-3*a*b^2*(B*d^2+2*C*c*d+D*c^2))*ln( b*x+a)/b^6
Time = 0.07 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\frac {6 b \left (6 a^2 d^2 D-3 a b d (C d+2 c D)+b^2 \left (2 c C d+B d^2+c^2 D\right )\right ) x+3 b^2 d (b C d+2 b c D-3 a d D) x^2+2 b^3 d^2 D x^3-\frac {3 (b c-a d)^2 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right )}{(a+b x)^2}-\frac {6 (b c-a d) \left (b^3 (B c+2 A d)-a b^2 (2 c C+3 B d)-5 a^3 d D+a^2 b (4 C d+3 c D)\right )}{a+b x}+6 \left (b^3 \left (c^2 C+2 B c d+A d^2\right )-10 a^3 d^2 D+6 a^2 b d (C d+2 c D)-3 a b^2 \left (2 c C d+B d^2+c^2 D\right )\right ) \log (a+b x)}{6 b^6} \] Input:
Integrate[((c + d*x)^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^3,x]
Output:
(6*b*(6*a^2*d^2*D - 3*a*b*d*(C*d + 2*c*D) + b^2*(2*c*C*d + B*d^2 + c^2*D)) *x + 3*b^2*d*(b*C*d + 2*b*c*D - 3*a*d*D)*x^2 + 2*b^3*d^2*D*x^3 - (3*(b*c - a*d)^2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D)))/(a + b*x)^2 - (6*(b*c - a*d)* (b^3*(B*c + 2*A*d) - a*b^2*(2*c*C + 3*B*d) - 5*a^3*d*D + a^2*b*(4*C*d + 3* c*D)))/(a + b*x) + 6*(b^3*(c^2*C + 2*B*c*d + A*d^2) - 10*a^3*d^2*D + 6*a^2 *b*d*(C*d + 2*c*D) - 3*a*b^2*(2*c*C*d + B*d^2 + c^2*D))*Log[a + b*x])/(6*b ^6)
Time = 0.66 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2123, 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.
| \(\displaystyle \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(b c-a d)^2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{b^5 (a+b x)^3}+\frac {6 a^2 d^2 D-3 a b d (2 c D+C d)+b^2 \left (B d^2+c^2 D+2 c C d\right )}{b^5}+\frac {-10 a^3 d^2 D+6 a^2 b d (2 c D+C d)-3 a b^2 \left (B d^2+c^2 D+2 c C d\right )+b^3 \left (A d^2+2 B c d+c^2 C\right )}{b^5 (a+b x)}+\frac {(b c-a d) \left (-5 a^3 d D+a^2 b (3 c D+4 C d)-a b^2 (3 B d+2 c C)+b^3 (2 A d+B c)\right )}{b^5 (a+b x)^2}+\frac {d x (-3 a d D+2 b c D+b C d)}{b^4}+\frac {d^2 D x^2}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(b c-a d)^2 \left (A b^3-a \left (a^2 D-a b C+b^2 B\right )\right )}{2 b^6 (a+b x)^2}+\frac {x \left (6 a^2 d^2 D-3 a b d (2 c D+C d)+b^2 \left (B d^2+c^2 D+2 c C d\right )\right )}{b^5}+\frac {\log (a+b x) \left (-10 a^3 d^2 D+6 a^2 b d (2 c D+C d)-3 a b^2 \left (B d^2+c^2 D+2 c C d\right )+b^3 \left (A d^2+2 B c d+c^2 C\right )\right )}{b^6}-\frac {(b c-a d) \left (-5 a^3 d D+a^2 b (3 c D+4 C d)-a b^2 (3 B d+2 c C)+b^3 (2 A d+B c)\right )}{b^6 (a+b x)}+\frac {d x^2 (-3 a d D+2 b c D+b C d)}{2 b^4}+\frac {d^2 D x^3}{3 b^3}\) |
Input:
Int[((c + d*x)^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x)^3,x]
Output:
((6*a^2*d^2*D - 3*a*b*d*(C*d + 2*c*D) + b^2*(2*c*C*d + B*d^2 + c^2*D))*x)/ b^5 + (d*(b*C*d + 2*b*c*D - 3*a*d*D)*x^2)/(2*b^4) + (d^2*D*x^3)/(3*b^3) - ((b*c - a*d)^2*(A*b^3 - a*(b^2*B - a*b*C + a^2*D)))/(2*b^6*(a + b*x)^2) - ((b*c - a*d)*(b^3*(B*c + 2*A*d) - a*b^2*(2*c*C + 3*B*d) - 5*a^3*d*D + a^2* b*(4*C*d + 3*c*D)))/(b^6*(a + b*x)) + ((b^3*(c^2*C + 2*B*c*d + A*d^2) - 10 *a^3*d^2*D + 6*a^2*b*d*(C*d + 2*c*D) - 3*a*b^2*(2*c*C*d + B*d^2 + c^2*D))* Log[a + b*x])/b^6
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 0.37 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.57
| method | result | size |
| norman | \(\frac {\frac {\left (2 A a \,b^{3} d^{2}-2 A \,b^{4} c d -6 B \,a^{2} b^{2} d^{2}+4 B a \,b^{3} c d -B \,b^{4} c^{2}+12 C \,a^{3} b \,d^{2}-12 C \,a^{2} b^{2} c d +2 C a \,b^{3} c^{2}-20 D a^{4} d^{2}+24 D a^{3} b c d -6 D a^{2} b^{2} c^{2}\right ) x}{b^{5}}+\frac {3 A \,a^{2} b^{3} d^{2}-2 A a \,b^{4} c d -A \,b^{5} c^{2}-9 B \,a^{3} b^{2} d^{2}+6 B \,a^{2} b^{3} c d -B a \,b^{4} c^{2}+18 C \,a^{4} b \,d^{2}-18 C \,a^{3} b^{2} c d +3 C \,a^{2} b^{3} c^{2}-30 D a^{5} d^{2}+36 D a^{4} b c d -9 D a^{3} b^{2} c^{2}}{2 b^{6}}+\frac {\left (3 b^{2} B \,d^{2}-6 C a b \,d^{2}+6 C \,b^{2} c d +10 a^{2} d^{2} D-12 D a b c d +3 D b^{2} c^{2}\right ) x^{3}}{3 b^{3}}+\frac {D d^{2} x^{5}}{3 b}+\frac {d \left (3 C b d -5 D a d +6 D b c \right ) x^{4}}{6 b^{2}}}{\left (b x +a \right )^{2}}+\frac {\left (b^{3} d^{2} A -3 B a \,b^{2} d^{2}+2 B \,b^{3} c d +6 C \,a^{2} b \,d^{2}-6 C a \,b^{2} c d +C \,b^{3} c^{2}-10 a^{3} d^{2} D+12 a^{2} b c d D-3 a \,b^{2} c^{2} D\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(446\) |
| default | \(\frac {\frac {1}{3} d^{2} D x^{3} b^{2}+\frac {1}{2} C \,b^{2} d^{2} x^{2}-\frac {3}{2} D a b \,d^{2} x^{2}+D b^{2} c d \,x^{2}+b^{2} B \,d^{2} x -3 C a b \,d^{2} x +2 C \,b^{2} c d x +6 a^{2} d^{2} D x -6 D a b c d x +D b^{2} c^{2} x}{b^{5}}-\frac {A \,a^{2} b^{3} d^{2}-2 A a \,b^{4} c d +A \,b^{5} c^{2}-B \,a^{3} b^{2} d^{2}+2 B \,a^{2} b^{3} c d -B a \,b^{4} c^{2}+C \,a^{4} b \,d^{2}-2 C \,a^{3} b^{2} c d +C \,a^{2} b^{3} c^{2}-D a^{5} d^{2}+2 D a^{4} b c d -D a^{3} b^{2} c^{2}}{2 b^{6} \left (b x +a \right )^{2}}-\frac {-2 A a \,b^{3} d^{2}+2 A \,b^{4} c d +3 B \,a^{2} b^{2} d^{2}-4 B a \,b^{3} c d +B \,b^{4} c^{2}-4 C \,a^{3} b \,d^{2}+6 C \,a^{2} b^{2} c d -2 C a \,b^{3} c^{2}+5 D a^{4} d^{2}-8 D a^{3} b c d +3 D a^{2} b^{2} c^{2}}{b^{6} \left (b x +a \right )}+\frac {\left (b^{3} d^{2} A -3 B a \,b^{2} d^{2}+2 B \,b^{3} c d +6 C \,a^{2} b \,d^{2}-6 C a \,b^{2} c d +C \,b^{3} c^{2}-10 a^{3} d^{2} D+12 a^{2} b c d D-3 a \,b^{2} c^{2} D\right ) \ln \left (b x +a \right )}{b^{6}}\) | \(454\) |
| parallelrisch | \(\frac {12 C \,x^{3} b^{5} c d +20 D x^{3} a^{2} b^{3} d^{2}+24 B x a \,b^{4} c d +12 B \ln \left (b x +a \right ) a^{2} b^{3} c d -36 C \ln \left (b x +a \right ) a^{3} b^{2} c d -24 D x^{3} a \,b^{4} c d +12 A \ln \left (b x +a \right ) x a \,b^{4} d^{2}-36 B \ln \left (b x +a \right ) x \,a^{2} b^{3} d^{2}+72 C \ln \left (b x +a \right ) x \,a^{3} b^{2} d^{2}+12 C \ln \left (b x +a \right ) x a \,b^{4} c^{2}-120 D \ln \left (b x +a \right ) x \,a^{4} b \,d^{2}-36 D \ln \left (b x +a \right ) x \,a^{2} b^{3} c^{2}-18 B \ln \left (b x +a \right ) x^{2} a \,b^{4} d^{2}+144 D x \,a^{3} b^{2} c d +72 D \ln \left (b x +a \right ) a^{4} b c d +9 A \,a^{2} b^{3} d^{2}-54 C \,a^{3} b^{2} c d +9 C \,a^{2} b^{3} c^{2}-27 D a^{3} b^{2} c^{2}+18 B \,a^{2} b^{3} c d -72 C \ln \left (b x +a \right ) x \,a^{2} b^{3} c d +144 D \ln \left (b x +a \right ) x \,a^{3} b^{2} c d +24 B \ln \left (b x +a \right ) x a \,b^{4} c d +2 D x^{5} d^{2} b^{5}+6 B \,x^{3} b^{5} d^{2}+6 D x^{3} b^{5} c^{2}-6 B x \,b^{5} c^{2}-60 D \ln \left (b x +a \right ) a^{5} d^{2}+3 C \,x^{4} b^{5} d^{2}-72 C x \,a^{2} b^{3} c d -27 B \,a^{3} b^{2} d^{2}-3 B a \,b^{4} c^{2}+54 C \,a^{4} b \,d^{2}+12 A x a \,b^{4} d^{2}-12 A x \,b^{5} c d -36 B x \,a^{2} b^{3} d^{2}+72 C x \,a^{3} b^{2} d^{2}+12 C x a \,b^{4} c^{2}+6 A \ln \left (b x +a \right ) a^{2} b^{3} d^{2}+72 D \ln \left (b x +a \right ) x^{2} a^{2} b^{3} c d +108 D a^{4} b c d -36 D x \,a^{2} b^{3} c^{2}-5 D x^{4} a \,b^{4} d^{2}+6 D x^{4} b^{5} c d -12 C \,x^{3} a \,b^{4} d^{2}+6 A \ln \left (b x +a \right ) x^{2} b^{5} d^{2}+6 C \ln \left (b x +a \right ) x^{2} b^{5} c^{2}-36 C \ln \left (b x +a \right ) x^{2} a \,b^{4} c d -18 B \ln \left (b x +a \right ) a^{3} b^{2} d^{2}+36 C \ln \left (b x +a \right ) a^{4} b \,d^{2}+6 C \ln \left (b x +a \right ) a^{2} b^{3} c^{2}-18 D \ln \left (b x +a \right ) a^{3} b^{2} c^{2}-120 D x \,a^{4} b \,d^{2}+12 B \ln \left (b x +a \right ) x^{2} b^{5} c d +36 C \ln \left (b x +a \right ) x^{2} a^{2} b^{3} d^{2}-60 D \ln \left (b x +a \right ) x^{2} a^{3} b^{2} d^{2}-18 D \ln \left (b x +a \right ) x^{2} a \,b^{4} c^{2}-6 A a \,b^{4} c d -90 D a^{5} d^{2}-3 A \,b^{5} c^{2}}{6 b^{6} \left (b x +a \right )^{2}}\) | \(871\) |
Input:
int((d*x+c)^2*(D*x^3+C*x^2+B*x+A)/(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
((2*A*a*b^3*d^2-2*A*b^4*c*d-6*B*a^2*b^2*d^2+4*B*a*b^3*c*d-B*b^4*c^2+12*C*a ^3*b*d^2-12*C*a^2*b^2*c*d+2*C*a*b^3*c^2-20*D*a^4*d^2+24*D*a^3*b*c*d-6*D*a^ 2*b^2*c^2)/b^5*x+1/2*(3*A*a^2*b^3*d^2-2*A*a*b^4*c*d-A*b^5*c^2-9*B*a^3*b^2* d^2+6*B*a^2*b^3*c*d-B*a*b^4*c^2+18*C*a^4*b*d^2-18*C*a^3*b^2*c*d+3*C*a^2*b^ 3*c^2-30*D*a^5*d^2+36*D*a^4*b*c*d-9*D*a^3*b^2*c^2)/b^6+1/3*(3*B*b^2*d^2-6* C*a*b*d^2+6*C*b^2*c*d+10*D*a^2*d^2-12*D*a*b*c*d+3*D*b^2*c^2)/b^3*x^3+1/3*D /b*d^2*x^5+1/6*d*(3*C*b*d-5*D*a*d+6*D*b*c)/b^2*x^4)/(b*x+a)^2+1/b^6*(A*b^3 *d^2-3*B*a*b^2*d^2+2*B*b^3*c*d+6*C*a^2*b*d^2-6*C*a*b^2*c*d+C*b^3*c^2-10*D* a^3*d^2+12*D*a^2*b*c*d-3*D*a*b^2*c^2)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 681 vs. \(2 (280) = 560\).
Time = 0.08 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.40 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\frac {2 \, D b^{5} d^{2} x^{5} + {\left (6 \, D b^{5} c d - {\left (5 \, D a b^{4} - 3 \, C b^{5}\right )} d^{2}\right )} x^{4} + 2 \, {\left (3 \, D b^{5} c^{2} - 6 \, {\left (2 \, D a b^{4} - C b^{5}\right )} c d + {\left (10 \, D a^{2} b^{3} - 6 \, C a b^{4} + 3 \, B b^{5}\right )} d^{2}\right )} x^{3} - 3 \, {\left (5 \, D a^{3} b^{2} - 3 \, C a^{2} b^{3} + B a b^{4} + A b^{5}\right )} c^{2} + 6 \, {\left (7 \, D a^{4} b - 5 \, C a^{3} b^{2} + 3 \, B a^{2} b^{3} - A a b^{4}\right )} c d - 3 \, {\left (9 \, D a^{5} - 7 \, C a^{4} b + 5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} + 3 \, {\left (4 \, D a b^{4} c^{2} - 2 \, {\left (11 \, D a^{2} b^{3} - 4 \, C a b^{4}\right )} c d + {\left (21 \, D a^{3} b^{2} - 11 \, C a^{2} b^{3} + 4 \, B a b^{4}\right )} d^{2}\right )} x^{2} - 6 \, {\left ({\left (2 \, D a^{2} b^{3} - 2 \, C a b^{4} + B b^{5}\right )} c^{2} - 2 \, {\left (D a^{3} b^{2} - 2 \, C a^{2} b^{3} + 2 \, B a b^{4} - A b^{5}\right )} c d - {\left (D a^{4} b + C a^{3} b^{2} - 2 \, B a^{2} b^{3} + 2 \, A a b^{4}\right )} d^{2}\right )} x - 6 \, {\left ({\left (3 \, D a^{3} b^{2} - C a^{2} b^{3}\right )} c^{2} - 2 \, {\left (6 \, D a^{4} b - 3 \, C a^{3} b^{2} + B a^{2} b^{3}\right )} c d + {\left (10 \, D a^{5} - 6 \, C a^{4} b + 3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} + {\left ({\left (3 \, D a b^{4} - C b^{5}\right )} c^{2} - 2 \, {\left (6 \, D a^{2} b^{3} - 3 \, C a b^{4} + B b^{5}\right )} c d + {\left (10 \, D a^{3} b^{2} - 6 \, C a^{2} b^{3} + 3 \, B a b^{4} - A b^{5}\right )} d^{2}\right )} x^{2} + 2 \, {\left ({\left (3 \, D a^{2} b^{3} - C a b^{4}\right )} c^{2} - 2 \, {\left (6 \, D a^{3} b^{2} - 3 \, C a^{2} b^{3} + B a b^{4}\right )} c d + {\left (10 \, D a^{4} b - 6 \, C a^{3} b^{2} + 3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \] Input:
integrate((d*x+c)^2*(D*x^3+C*x^2+B*x+A)/(b*x+a)^3,x, algorithm="fricas")
Output:
1/6*(2*D*b^5*d^2*x^5 + (6*D*b^5*c*d - (5*D*a*b^4 - 3*C*b^5)*d^2)*x^4 + 2*( 3*D*b^5*c^2 - 6*(2*D*a*b^4 - C*b^5)*c*d + (10*D*a^2*b^3 - 6*C*a*b^4 + 3*B* b^5)*d^2)*x^3 - 3*(5*D*a^3*b^2 - 3*C*a^2*b^3 + B*a*b^4 + A*b^5)*c^2 + 6*(7 *D*a^4*b - 5*C*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*c*d - 3*(9*D*a^5 - 7*C*a^4 *b + 5*B*a^3*b^2 - 3*A*a^2*b^3)*d^2 + 3*(4*D*a*b^4*c^2 - 2*(11*D*a^2*b^3 - 4*C*a*b^4)*c*d + (21*D*a^3*b^2 - 11*C*a^2*b^3 + 4*B*a*b^4)*d^2)*x^2 - 6*( (2*D*a^2*b^3 - 2*C*a*b^4 + B*b^5)*c^2 - 2*(D*a^3*b^2 - 2*C*a^2*b^3 + 2*B*a *b^4 - A*b^5)*c*d - (D*a^4*b + C*a^3*b^2 - 2*B*a^2*b^3 + 2*A*a*b^4)*d^2)*x - 6*((3*D*a^3*b^2 - C*a^2*b^3)*c^2 - 2*(6*D*a^4*b - 3*C*a^3*b^2 + B*a^2*b ^3)*c*d + (10*D*a^5 - 6*C*a^4*b + 3*B*a^3*b^2 - A*a^2*b^3)*d^2 + ((3*D*a*b ^4 - C*b^5)*c^2 - 2*(6*D*a^2*b^3 - 3*C*a*b^4 + B*b^5)*c*d + (10*D*a^3*b^2 - 6*C*a^2*b^3 + 3*B*a*b^4 - A*b^5)*d^2)*x^2 + 2*((3*D*a^2*b^3 - C*a*b^4)*c ^2 - 2*(6*D*a^3*b^2 - 3*C*a^2*b^3 + B*a*b^4)*c*d + (10*D*a^4*b - 6*C*a^3*b ^2 + 3*B*a^2*b^3 - A*a*b^4)*d^2)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a ^2*b^6)
Time = 20.49 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.87 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\frac {D d^{2} x^{3}}{3 b^{3}} + x^{2} \left (\frac {C d^{2}}{2 b^{3}} - \frac {3 D a d^{2}}{2 b^{4}} + \frac {D c d}{b^{3}}\right ) + x \left (\frac {B d^{2}}{b^{3}} - \frac {3 C a d^{2}}{b^{4}} + \frac {2 C c d}{b^{3}} + \frac {6 D a^{2} d^{2}}{b^{5}} - \frac {6 D a c d}{b^{4}} + \frac {D c^{2}}{b^{3}}\right ) + \frac {3 A a^{2} b^{3} d^{2} - 2 A a b^{4} c d - A b^{5} c^{2} - 5 B a^{3} b^{2} d^{2} + 6 B a^{2} b^{3} c d - B a b^{4} c^{2} + 7 C a^{4} b d^{2} - 10 C a^{3} b^{2} c d + 3 C a^{2} b^{3} c^{2} - 9 D a^{5} d^{2} + 14 D a^{4} b c d - 5 D a^{3} b^{2} c^{2} + x \left (4 A a b^{4} d^{2} - 4 A b^{5} c d - 6 B a^{2} b^{3} d^{2} + 8 B a b^{4} c d - 2 B b^{5} c^{2} + 8 C a^{3} b^{2} d^{2} - 12 C a^{2} b^{3} c d + 4 C a b^{4} c^{2} - 10 D a^{4} b d^{2} + 16 D a^{3} b^{2} c d - 6 D a^{2} b^{3} c^{2}\right )}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} - \frac {\left (- A b^{3} d^{2} + 3 B a b^{2} d^{2} - 2 B b^{3} c d - 6 C a^{2} b d^{2} + 6 C a b^{2} c d - C b^{3} c^{2} + 10 D a^{3} d^{2} - 12 D a^{2} b c d + 3 D a b^{2} c^{2}\right ) \log {\left (a + b x \right )}}{b^{6}} \] Input:
integrate((d*x+c)**2*(D*x**3+C*x**2+B*x+A)/(b*x+a)**3,x)
Output:
D*d**2*x**3/(3*b**3) + x**2*(C*d**2/(2*b**3) - 3*D*a*d**2/(2*b**4) + D*c*d /b**3) + x*(B*d**2/b**3 - 3*C*a*d**2/b**4 + 2*C*c*d/b**3 + 6*D*a**2*d**2/b **5 - 6*D*a*c*d/b**4 + D*c**2/b**3) + (3*A*a**2*b**3*d**2 - 2*A*a*b**4*c*d - A*b**5*c**2 - 5*B*a**3*b**2*d**2 + 6*B*a**2*b**3*c*d - B*a*b**4*c**2 + 7*C*a**4*b*d**2 - 10*C*a**3*b**2*c*d + 3*C*a**2*b**3*c**2 - 9*D*a**5*d**2 + 14*D*a**4*b*c*d - 5*D*a**3*b**2*c**2 + x*(4*A*a*b**4*d**2 - 4*A*b**5*c*d - 6*B*a**2*b**3*d**2 + 8*B*a*b**4*c*d - 2*B*b**5*c**2 + 8*C*a**3*b**2*d** 2 - 12*C*a**2*b**3*c*d + 4*C*a*b**4*c**2 - 10*D*a**4*b*d**2 + 16*D*a**3*b* *2*c*d - 6*D*a**2*b**3*c**2))/(2*a**2*b**6 + 4*a*b**7*x + 2*b**8*x**2) - ( -A*b**3*d**2 + 3*B*a*b**2*d**2 - 2*B*b**3*c*d - 6*C*a**2*b*d**2 + 6*C*a*b* *2*c*d - C*b**3*c**2 + 10*D*a**3*d**2 - 12*D*a**2*b*c*d + 3*D*a*b**2*c**2) *log(a + b*x)/b**6
Time = 0.05 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.48 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=-\frac {{\left (5 \, D a^{3} b^{2} - 3 \, C a^{2} b^{3} + B a b^{4} + A b^{5}\right )} c^{2} - 2 \, {\left (7 \, D a^{4} b - 5 \, C a^{3} b^{2} + 3 \, B a^{2} b^{3} - A a b^{4}\right )} c d + {\left (9 \, D a^{5} - 7 \, C a^{4} b + 5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} + 2 \, {\left ({\left (3 \, D a^{2} b^{3} - 2 \, C a b^{4} + B b^{5}\right )} c^{2} - 2 \, {\left (4 \, D a^{3} b^{2} - 3 \, C a^{2} b^{3} + 2 \, B a b^{4} - A b^{5}\right )} c d + {\left (5 \, D a^{4} b - 4 \, C a^{3} b^{2} + 3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2}\right )} x}{2 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} + \frac {2 \, D b^{2} d^{2} x^{3} + 3 \, {\left (2 \, D b^{2} c d - {\left (3 \, D a b - C b^{2}\right )} d^{2}\right )} x^{2} + 6 \, {\left (D b^{2} c^{2} - 2 \, {\left (3 \, D a b - C b^{2}\right )} c d + {\left (6 \, D a^{2} - 3 \, C a b + B b^{2}\right )} d^{2}\right )} x}{6 \, b^{5}} - \frac {{\left ({\left (3 \, D a b^{2} - C b^{3}\right )} c^{2} - 2 \, {\left (6 \, D a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} c d + {\left (10 \, D a^{3} - 6 \, C a^{2} b + 3 \, B a b^{2} - A b^{3}\right )} d^{2}\right )} \log \left (b x + a\right )}{b^{6}} \] Input:
integrate((d*x+c)^2*(D*x^3+C*x^2+B*x+A)/(b*x+a)^3,x, algorithm="maxima")
Output:
-1/2*((5*D*a^3*b^2 - 3*C*a^2*b^3 + B*a*b^4 + A*b^5)*c^2 - 2*(7*D*a^4*b - 5 *C*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*c*d + (9*D*a^5 - 7*C*a^4*b + 5*B*a^3*b ^2 - 3*A*a^2*b^3)*d^2 + 2*((3*D*a^2*b^3 - 2*C*a*b^4 + B*b^5)*c^2 - 2*(4*D* a^3*b^2 - 3*C*a^2*b^3 + 2*B*a*b^4 - A*b^5)*c*d + (5*D*a^4*b - 4*C*a^3*b^2 + 3*B*a^2*b^3 - 2*A*a*b^4)*d^2)*x)/(b^8*x^2 + 2*a*b^7*x + a^2*b^6) + 1/6*( 2*D*b^2*d^2*x^3 + 3*(2*D*b^2*c*d - (3*D*a*b - C*b^2)*d^2)*x^2 + 6*(D*b^2*c ^2 - 2*(3*D*a*b - C*b^2)*c*d + (6*D*a^2 - 3*C*a*b + B*b^2)*d^2)*x)/b^5 - ( (3*D*a*b^2 - C*b^3)*c^2 - 2*(6*D*a^2*b - 3*C*a*b^2 + B*b^3)*c*d + (10*D*a^ 3 - 6*C*a^2*b + 3*B*a*b^2 - A*b^3)*d^2)*log(b*x + a)/b^6
Time = 0.12 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.65 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=-\frac {{\left (3 \, D a b^{2} c^{2} - C b^{3} c^{2} - 12 \, D a^{2} b c d + 6 \, C a b^{2} c d - 2 \, B b^{3} c d + 10 \, D a^{3} d^{2} - 6 \, C a^{2} b d^{2} + 3 \, B a b^{2} d^{2} - A b^{3} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {5 \, D a^{3} b^{2} c^{2} - 3 \, C a^{2} b^{3} c^{2} + B a b^{4} c^{2} + A b^{5} c^{2} - 14 \, D a^{4} b c d + 10 \, C a^{3} b^{2} c d - 6 \, B a^{2} b^{3} c d + 2 \, A a b^{4} c d + 9 \, D a^{5} d^{2} - 7 \, C a^{4} b d^{2} + 5 \, B a^{3} b^{2} d^{2} - 3 \, A a^{2} b^{3} d^{2} + 2 \, {\left (3 \, D a^{2} b^{3} c^{2} - 2 \, C a b^{4} c^{2} + B b^{5} c^{2} - 8 \, D a^{3} b^{2} c d + 6 \, C a^{2} b^{3} c d - 4 \, B a b^{4} c d + 2 \, A b^{5} c d + 5 \, D a^{4} b d^{2} - 4 \, C a^{3} b^{2} d^{2} + 3 \, B a^{2} b^{3} d^{2} - 2 \, A a b^{4} d^{2}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{6}} + \frac {2 \, D b^{6} d^{2} x^{3} + 6 \, D b^{6} c d x^{2} - 9 \, D a b^{5} d^{2} x^{2} + 3 \, C b^{6} d^{2} x^{2} + 6 \, D b^{6} c^{2} x - 36 \, D a b^{5} c d x + 12 \, C b^{6} c d x + 36 \, D a^{2} b^{4} d^{2} x - 18 \, C a b^{5} d^{2} x + 6 \, B b^{6} d^{2} x}{6 \, b^{9}} \] Input:
integrate((d*x+c)^2*(D*x^3+C*x^2+B*x+A)/(b*x+a)^3,x, algorithm="giac")
Output:
-(3*D*a*b^2*c^2 - C*b^3*c^2 - 12*D*a^2*b*c*d + 6*C*a*b^2*c*d - 2*B*b^3*c*d + 10*D*a^3*d^2 - 6*C*a^2*b*d^2 + 3*B*a*b^2*d^2 - A*b^3*d^2)*log(abs(b*x + a))/b^6 - 1/2*(5*D*a^3*b^2*c^2 - 3*C*a^2*b^3*c^2 + B*a*b^4*c^2 + A*b^5*c^ 2 - 14*D*a^4*b*c*d + 10*C*a^3*b^2*c*d - 6*B*a^2*b^3*c*d + 2*A*a*b^4*c*d + 9*D*a^5*d^2 - 7*C*a^4*b*d^2 + 5*B*a^3*b^2*d^2 - 3*A*a^2*b^3*d^2 + 2*(3*D*a ^2*b^3*c^2 - 2*C*a*b^4*c^2 + B*b^5*c^2 - 8*D*a^3*b^2*c*d + 6*C*a^2*b^3*c*d - 4*B*a*b^4*c*d + 2*A*b^5*c*d + 5*D*a^4*b*d^2 - 4*C*a^3*b^2*d^2 + 3*B*a^2 *b^3*d^2 - 2*A*a*b^4*d^2)*x)/((b*x + a)^2*b^6) + 1/6*(2*D*b^6*d^2*x^3 + 6* D*b^6*c*d*x^2 - 9*D*a*b^5*d^2*x^2 + 3*C*b^6*d^2*x^2 + 6*D*b^6*c^2*x - 36*D *a*b^5*c*d*x + 12*C*b^6*c*d*x + 36*D*a^2*b^4*d^2*x - 18*C*a*b^5*d^2*x + 6* B*b^6*d^2*x)/b^9
Timed out. \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\int \frac {{\left (c+d\,x\right )}^2\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a+b\,x\right )}^3} \,d x \] Input:
int(((c + d*x)^2*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^3,x)
Output:
int(((c + d*x)^2*(A + B*x + C*x^2 + x^3*D))/(a + b*x)^3, x)
Time = 0.14 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.14 \[ \int \frac {(c+d x)^2 \left (A+B x+C x^2+D x^3\right )}{(a+b x)^3} \, dx=\frac {216 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} c \,d^{2} x -108 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} c^{2} d x +108 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} c \,d^{2} x^{2}+24 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{5} c d x -54 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{4} c^{2} d \,x^{2}+12 \,\mathrm {log}\left (b x +a \right ) a \,b^{6} c d \,x^{2}+108 \,\mathrm {log}\left (b x +a \right ) a^{5} b c \,d^{2}-120 \,\mathrm {log}\left (b x +a \right ) a^{5} b \,d^{3} x -54 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} c^{2} d -60 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} d^{3} x^{2}+12 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{4} c d -24 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{4} d^{2} x -12 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{5} d^{2} x^{2}+12 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{4} c^{3} x +6 \,\mathrm {log}\left (b x +a \right ) a \,b^{5} c^{3} x^{2}-108 a^{3} b^{3} c \,d^{2} x^{2}+54 a^{2} b^{4} c^{2} d \,x^{2}-36 a^{2} b^{4} c \,d^{2} x^{3}-6 a \,b^{6} c d \,x^{2}+18 a \,b^{5} c^{2} d \,x^{3}+9 a \,b^{5} c \,d^{2} x^{4}-30 a^{6} d^{3}-60 \,\mathrm {log}\left (b x +a \right ) a^{6} d^{3}-6 a^{4} b^{3} d^{2}+3 a^{3} b^{3} c^{3}-3 a^{2} b^{5} c^{2}+3 b^{7} c^{2} x^{2}-12 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{3} d^{2}+6 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} c^{3}+54 a^{5} b c \,d^{2}-27 a^{4} b^{2} c^{2} d +60 a^{4} b^{2} d^{3} x^{2}+6 a^{3} b^{4} c d +20 a^{3} b^{3} d^{3} x^{3}+12 a^{2} b^{5} d^{2} x^{2}-5 a^{2} b^{4} d^{3} x^{4}+6 a \,b^{6} d^{2} x^{3}-6 a \,b^{5} c^{3} x^{2}+2 a \,b^{5} d^{3} x^{5}}{6 a \,b^{6} \left (b^{2} x^{2}+2 a b x +a^{2}\right )} \] Input:
int((d*x+c)^2*(D*x^3+C*x^2+B*x+A)/(b*x+a)^3,x)
Output:
( - 60*log(a + b*x)*a**6*d**3 + 108*log(a + b*x)*a**5*b*c*d**2 - 120*log(a + b*x)*a**5*b*d**3*x - 12*log(a + b*x)*a**4*b**3*d**2 - 54*log(a + b*x)*a **4*b**2*c**2*d + 216*log(a + b*x)*a**4*b**2*c*d**2*x - 60*log(a + b*x)*a* *4*b**2*d**3*x**2 + 12*log(a + b*x)*a**3*b**4*c*d - 24*log(a + b*x)*a**3*b **4*d**2*x + 6*log(a + b*x)*a**3*b**3*c**3 - 108*log(a + b*x)*a**3*b**3*c* *2*d*x + 108*log(a + b*x)*a**3*b**3*c*d**2*x**2 + 24*log(a + b*x)*a**2*b** 5*c*d*x - 12*log(a + b*x)*a**2*b**5*d**2*x**2 + 12*log(a + b*x)*a**2*b**4* c**3*x - 54*log(a + b*x)*a**2*b**4*c**2*d*x**2 + 12*log(a + b*x)*a*b**6*c* d*x**2 + 6*log(a + b*x)*a*b**5*c**3*x**2 - 30*a**6*d**3 + 54*a**5*b*c*d**2 - 6*a**4*b**3*d**2 - 27*a**4*b**2*c**2*d + 60*a**4*b**2*d**3*x**2 + 6*a** 3*b**4*c*d + 3*a**3*b**3*c**3 - 108*a**3*b**3*c*d**2*x**2 + 20*a**3*b**3*d **3*x**3 - 3*a**2*b**5*c**2 + 12*a**2*b**5*d**2*x**2 + 54*a**2*b**4*c**2*d *x**2 - 36*a**2*b**4*c*d**2*x**3 - 5*a**2*b**4*d**3*x**4 - 6*a*b**6*c*d*x* *2 + 6*a*b**6*d**2*x**3 - 6*a*b**5*c**3*x**2 + 18*a*b**5*c**2*d*x**3 + 9*a *b**5*c*d**2*x**4 + 2*a*b**5*d**3*x**5 + 3*b**7*c**2*x**2)/(6*a*b**6*(a**2 + 2*a*b*x + b**2*x**2))