Integrand size = 30, antiderivative size = 410 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=-\frac {(b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) x}{d^6}+\frac {(b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{d^7 (c+d x)}+\frac {\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^2}{2 d^7}+\frac {b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^3}{3 d^7}+\frac {b^2 (b C d-6 b c D+3 a d D) (c+d x)^4}{4 d^7}+\frac {b^3 D (c+d x)^5}{5 d^7}-\frac {(b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) \log (c+d x)}{d^7} \] Output:
-(-a*d+b*c)*(a^2*d^2*(C*d-3*D*c)-a*b*d*(-3*B*d^2+8*C*c*d-15*D*c^2)+b^2*(3* A*d^3-6*B*c*d^2+10*C*c^2*d-15*D*c^3))*x/d^6+(-a*d+b*c)^3*(A*d^3-B*c*d^2+C* c^2*d-D*c^3)/d^7/(d*x+c)+1/2*(a^3*d^3*D+3*a^2*b*d^2*(C*d-4*D*c)-3*a*b^2*d* (-B*d^2+4*C*c*d-10*D*c^2)+b^3*(A*d^3-4*B*c*d^2+10*C*c^2*d-20*D*c^3))*(d*x+ c)^2/d^7+1/3*b*(3*a^2*d^2*D+3*a*b*d*(C*d-5*D*c)-b^2*(-B*d^2+5*C*c*d-15*D*c ^2))*(d*x+c)^3/d^7+1/4*b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^4/d^7+1/5*b^3*D *(d*x+c)^5/d^7-(-a*d+b*c)^2*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(3*A*d^3-4*B*c *d^2+5*C*c^2*d-6*D*c^3))*ln(d*x+c)/d^7
Time = 0.10 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {60 d \left (a^3 d^3 (C d-2 c D)+3 a^2 b d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+3 a b^2 d \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )+b^3 c \left (-4 c^2 C d+3 B c d^2-2 A d^3+5 c^3 D\right )\right ) x+30 d^2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-2 c D)+3 a b^2 d \left (-2 c C d+B d^2+3 c^2 D\right )+b^3 \left (3 c^2 C d-2 B c d^2+A d^3-4 c^3 D\right )\right ) x^2+20 b d^3 \left (3 a^2 d^2 D+3 a b d (C d-2 c D)+b^2 \left (-2 c C d+B d^2+3 c^2 D\right )\right ) x^3+15 b^2 d^4 (b C d-2 b c D+3 a d D) x^4+12 b^3 d^5 D x^5-\frac {60 (b c-a d)^3 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )}{c+d x}-60 (b c-a d)^2 \left (-a d \left (-2 c C d+B d^2+3 c^2 D\right )+b \left (-5 c^2 C d+4 B c d^2-3 A d^3+6 c^3 D\right )\right ) \log (c+d x)}{60 d^7} \] Input:
Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^2,x]
Output:
(60*d*(a^3*d^3*(C*d - 2*c*D) + 3*a^2*b*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + 3*a*b^2*d*(3*c^2*C*d - 2*B*c*d^2 + A*d^3 - 4*c^3*D) + b^3*c*(-4*c^2*C*d + 3*B*c*d^2 - 2*A*d^3 + 5*c^3*D))*x + 30*d^2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 2*c*D) + 3*a*b^2*d*(-2*c*C*d + B*d^2 + 3*c^2*D) + b^3*(3*c^2*C*d - 2*B*c* d^2 + A*d^3 - 4*c^3*D))*x^2 + 20*b*d^3*(3*a^2*d^2*D + 3*a*b*d*(C*d - 2*c*D ) + b^2*(-2*c*C*d + B*d^2 + 3*c^2*D))*x^3 + 15*b^2*d^4*(b*C*d - 2*b*c*D + 3*a*d*D)*x^4 + 12*b^3*d^5*D*x^5 - (60*(b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(c + d*x) - 60*(b*c - a*d)^2*(-(a*d*(-2*c*C*d + B*d^2 + 3*c^2*D)) + b*(-5*c^2*C*d + 4*B*c*d^2 - 3*A*d^3 + 6*c^3*D))*Log[c + d*x])/ (60*d^7)
Time = 0.98 (sec) , antiderivative size = 410, 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.
| \(\displaystyle \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )-\left (b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{d^6}+\frac {b (c+d x)^2 \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^6}+\frac {(c+d x) \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{d^6}+\frac {(b c-a d)^2 \left (b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^6 (c+d x)}+\frac {(a d-b c)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 (c+d x)^2}+\frac {b^2 (c+d x)^3 (3 a d D-6 b c D+b C d)}{d^6}+\frac {b^3 D (c+d x)^4}{d^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{d^6}+\frac {b (c+d x)^3 \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{3 d^7}+\frac {(c+d x)^2 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{2 d^7}+\frac {(b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 (c+d x)}-\frac {(b c-a d)^2 \log (c+d x) \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7}+\frac {b^2 (c+d x)^4 (3 a d D-6 b c D+b C d)}{4 d^7}+\frac {b^3 D (c+d x)^5}{5 d^7}\) |
Input:
Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^2,x]
Output:
-(((b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) - a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2* D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*x)/d^6) + ((b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^7*(c + d*x)) + ((a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D) - 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D) + b^3 *(10*c^2*C*d - 4*B*c*d^2 + A*d^3 - 20*c^3*D))*(c + d*x)^2)/(2*d^7) + (b*(3 *a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D) - b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^3)/(3*d^7) + (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^4)/(4*d^7) + (b^3*D*(c + d*x)^5)/(5*d^7) - ((b*c - a*d)^2*(a*d*(2*c*C*d - B*d^2 - 3*c ^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^3 - 6*c^3*D))*Log[c + d*x])/d^7
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 = 0.40 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.80
| method | result | size |
| norman | \(\frac {\frac {\left (A \,a^{3} d^{6}-3 A \,a^{2} b c \,d^{5}+6 A a \,b^{2} c^{2} d^{4}-3 A \,b^{3} c^{3} d^{3}-B \,a^{3} c \,d^{5}+6 B \,a^{2} b \,c^{2} d^{4}-9 B a \,b^{2} c^{3} d^{3}+4 B \,b^{3} c^{4} d^{2}+2 C \,a^{3} c^{2} d^{4}-9 C \,a^{2} b \,c^{3} d^{3}+12 C a \,b^{2} c^{4} d^{2}-5 C \,b^{3} c^{5} d -3 D a^{3} c^{3} d^{3}+12 D a^{2} b \,c^{4} d^{2}-15 D a \,b^{2} c^{5} d +6 D b^{3} c^{6}\right ) x}{d^{6} c}+\frac {\left (3 A \,b^{3} d^{3}+9 B a \,b^{2} d^{3}-4 B \,b^{3} c \,d^{2}+9 a^{2} b C \,d^{3}-12 C a \,b^{2} c \,d^{2}+5 C \,b^{3} c^{2} d +3 a^{3} d^{3} D-12 D a^{2} b c \,d^{2}+15 D a \,b^{2} c^{2} d -6 D b^{3} c^{3}\right ) x^{3}}{6 d^{4}}+\frac {\left (6 a \,b^{2} A \,d^{4}-3 A \,b^{3} c \,d^{3}+6 a^{2} b B \,d^{4}-9 B a \,b^{2} c \,d^{3}+4 B \,b^{3} c^{2} d^{2}+2 a^{3} C \,d^{4}-9 C \,a^{2} b c \,d^{3}+12 C a \,b^{2} c^{2} d^{2}-5 C \,b^{3} c^{3} d -3 D a^{3} c \,d^{3}+12 D a^{2} b \,c^{2} d^{2}-15 D a \,b^{2} c^{3} d +6 D b^{3} c^{4}\right ) x^{2}}{2 d^{5}}+\frac {D b^{3} x^{6}}{5 d}+\frac {b \left (4 b^{2} B \,d^{2}+12 C a b \,d^{2}-5 C \,b^{2} c d +12 a^{2} d^{2} D-15 D a b c d +6 D b^{2} c^{2}\right ) x^{4}}{12 d^{3}}+\frac {b^{2} \left (5 C b d +15 D a d -6 D b c \right ) x^{5}}{20 d^{2}}}{x d +c}+\frac {\left (3 A \,a^{2} b \,d^{5}-6 A a \,b^{2} c \,d^{4}+3 A \,b^{3} c^{2} d^{3}+B \,a^{3} d^{5}-6 B \,a^{2} b c \,d^{4}+9 B a \,b^{2} c^{2} d^{3}-4 B \,b^{3} c^{3} d^{2}-2 C \,a^{3} c \,d^{4}+9 C \,a^{2} b \,c^{2} d^{3}-12 C a \,b^{2} c^{3} d^{2}+5 C \,b^{3} c^{4} d +3 D a^{3} c^{2} d^{3}-12 D a^{2} b \,c^{3} d^{2}+15 D a \,b^{2} c^{4} d -6 D b^{3} c^{5}\right ) \ln \left (x d +c \right )}{d^{7}}\) | \(740\) |
| default | \(\frac {-2 D a \,b^{2} c \,d^{3} x^{3}-3 C a \,b^{2} c \,d^{3} x^{2}-3 D a^{2} b c \,d^{3} x^{2}+\frac {9}{2} D a \,b^{2} c^{2} d^{2} x^{2}+3 a \,b^{2} A \,d^{4} x -2 A \,b^{3} c \,d^{3} x +3 a^{2} b B \,d^{4} x +3 B \,b^{3} c^{2} d^{2} x -4 C \,b^{3} c^{3} d x -2 D a^{3} c \,d^{3} x -6 B a \,b^{2} c \,d^{3} x +\frac {1}{2} D a^{3} d^{4} x^{2}+a^{3} C \,d^{4} x +5 D b^{3} c^{4} x +\frac {1}{2} A \,b^{3} d^{4} x^{2}+9 C a \,b^{2} c^{2} d^{2} x +\frac {1}{5} b^{3} D x^{5} d^{4}+\frac {1}{4} C \,b^{3} d^{4} x^{4}+\frac {1}{3} B \,b^{3} d^{4} x^{3}-6 C \,a^{2} b c \,d^{3} x +9 D a^{2} b \,c^{2} d^{2} x -12 D a \,b^{2} c^{3} d x +D b^{3} c^{2} d^{2} x^{3}+C a \,b^{2} d^{4} x^{3}+D a^{2} b \,d^{4} x^{3}-2 D b^{3} c^{3} d \,x^{2}+\frac {3}{4} D a \,b^{2} d^{4} x^{4}-\frac {1}{2} D b^{3} c \,d^{3} x^{4}-\frac {2}{3} C \,b^{3} c \,d^{3} x^{3}+\frac {3}{2} B a \,b^{2} d^{4} x^{2}-B \,b^{3} c \,d^{3} x^{2}+\frac {3}{2} C \,a^{2} b \,d^{4} x^{2}+\frac {3}{2} C \,b^{3} c^{2} d^{2} x^{2}}{d^{6}}-\frac {A \,a^{3} d^{6}-3 A \,a^{2} b c \,d^{5}+3 A a \,b^{2} c^{2} d^{4}-A \,b^{3} c^{3} d^{3}-B \,a^{3} c \,d^{5}+3 B \,a^{2} b \,c^{2} d^{4}-3 B a \,b^{2} c^{3} d^{3}+B \,b^{3} c^{4} d^{2}+C \,a^{3} c^{2} d^{4}-3 C \,a^{2} b \,c^{3} d^{3}+3 C a \,b^{2} c^{4} d^{2}-C \,b^{3} c^{5} d -D a^{3} c^{3} d^{3}+3 D a^{2} b \,c^{4} d^{2}-3 D a \,b^{2} c^{5} d +D b^{3} c^{6}}{d^{7} \left (x d +c \right )}+\frac {\left (3 A \,a^{2} b \,d^{5}-6 A a \,b^{2} c \,d^{4}+3 A \,b^{3} c^{2} d^{3}+B \,a^{3} d^{5}-6 B \,a^{2} b c \,d^{4}+9 B a \,b^{2} c^{2} d^{3}-4 B \,b^{3} c^{3} d^{2}-2 C \,a^{3} c \,d^{4}+9 C \,a^{2} b \,c^{2} d^{3}-12 C a \,b^{2} c^{3} d^{2}+5 C \,b^{3} c^{4} d +3 D a^{3} c^{2} d^{3}-12 D a^{2} b \,c^{3} d^{2}+15 D a \,b^{2} c^{4} d -6 D b^{3} c^{5}\right ) \ln \left (x d +c \right )}{d^{7}}\) | \(794\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1198\) |
Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
((A*a^3*d^6-3*A*a^2*b*c*d^5+6*A*a*b^2*c^2*d^4-3*A*b^3*c^3*d^3-B*a^3*c*d^5+ 6*B*a^2*b*c^2*d^4-9*B*a*b^2*c^3*d^3+4*B*b^3*c^4*d^2+2*C*a^3*c^2*d^4-9*C*a^ 2*b*c^3*d^3+12*C*a*b^2*c^4*d^2-5*C*b^3*c^5*d-3*D*a^3*c^3*d^3+12*D*a^2*b*c^ 4*d^2-15*D*a*b^2*c^5*d+6*D*b^3*c^6)/d^6/c*x+1/6*(3*A*b^3*d^3+9*B*a*b^2*d^3 -4*B*b^3*c*d^2+9*C*a^2*b*d^3-12*C*a*b^2*c*d^2+5*C*b^3*c^2*d+3*D*a^3*d^3-12 *D*a^2*b*c*d^2+15*D*a*b^2*c^2*d-6*D*b^3*c^3)/d^4*x^3+1/2*(6*A*a*b^2*d^4-3* A*b^3*c*d^3+6*B*a^2*b*d^4-9*B*a*b^2*c*d^3+4*B*b^3*c^2*d^2+2*C*a^3*d^4-9*C* a^2*b*c*d^3+12*C*a*b^2*c^2*d^2-5*C*b^3*c^3*d-3*D*a^3*c*d^3+12*D*a^2*b*c^2* d^2-15*D*a*b^2*c^3*d+6*D*b^3*c^4)/d^5*x^2+1/5*D*b^3/d*x^6+1/12*b*(4*B*b^2* d^2+12*C*a*b*d^2-5*C*b^2*c*d+12*D*a^2*d^2-15*D*a*b*c*d+6*D*b^2*c^2)/d^3*x^ 4+1/20*b^2*(5*C*b*d+15*D*a*d-6*D*b*c)/d^2*x^5)/(d*x+c)+1/d^7*(3*A*a^2*b*d^ 5-6*A*a*b^2*c*d^4+3*A*b^3*c^2*d^3+B*a^3*d^5-6*B*a^2*b*c*d^4+9*B*a*b^2*c^2* d^3-4*B*b^3*c^3*d^2-2*C*a^3*c*d^4+9*C*a^2*b*c^2*d^3-12*C*a*b^2*c^3*d^2+5*C *b^3*c^4*d+3*D*a^3*c^2*d^3-12*D*a^2*b*c^3*d^2+15*D*a*b^2*c^4*d-6*D*b^3*c^5 )*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (403) = 806\).
Time = 0.08 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.18 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x, algorithm="fricas")
Output:
1/60*(12*D*b^3*d^6*x^6 - 60*D*b^3*c^6 - 60*A*a^3*d^6 + 60*(3*D*a*b^2 + C*b ^3)*c^5*d - 60*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 + 60*(D*a^3 + 3*C*a ^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 - 60*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2 *d^4 + 60*(B*a^3 + 3*A*a^2*b)*c*d^5 - 3*(6*D*b^3*c*d^5 - 5*(3*D*a*b^2 + C* b^3)*d^6)*x^5 + 5*(6*D*b^3*c^2*d^4 - 5*(3*D*a*b^2 + C*b^3)*c*d^5 + 4*(3*D* a^2*b + 3*C*a*b^2 + B*b^3)*d^6)*x^4 - 10*(6*D*b^3*c^3*d^3 - 5*(3*D*a*b^2 + C*b^3)*c^2*d^4 + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^5 - 3*(D*a^3 + 3*C *a^2*b + 3*B*a*b^2 + A*b^3)*d^6)*x^3 + 30*(6*D*b^3*c^4*d^2 - 5*(3*D*a*b^2 + C*b^3)*c^3*d^3 + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^4 - 3*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c*d^5 + 2*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d ^6)*x^2 + 60*(5*D*b^3*c^5*d - 4*(3*D*a*b^2 + C*b^3)*c^4*d^2 + 3*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^3 - 2*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)* c^2*d^4 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^5)*x - 60*(6*D*b^3*c^6 - 5*( 3*D*a*b^2 + C*b^3)*c^5*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 - 3*( D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + 2*(C*a^3 + 3*B*a^2*b + 3* A*a*b^2)*c^2*d^4 - (B*a^3 + 3*A*a^2*b)*c*d^5 + (6*D*b^3*c^5*d - 5*(3*D*a*b ^2 + C*b^3)*c^4*d^2 + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^3 - 3*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^4 + 2*(C*a^3 + 3*B*a^2*b + 3*A*a*b ^2)*c*d^5 - (B*a^3 + 3*A*a^2*b)*d^6)*x)*log(d*x + c))/(d^8*x + c*d^7)
Time = 3.52 (sec) , antiderivative size = 728, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {D b^{3} x^{5}}{5 d^{2}} + x^{4} \left (\frac {C b^{3}}{4 d^{2}} + \frac {3 D a b^{2}}{4 d^{2}} - \frac {D b^{3} c}{2 d^{3}}\right ) + x^{3} \left (\frac {B b^{3}}{3 d^{2}} + \frac {C a b^{2}}{d^{2}} - \frac {2 C b^{3} c}{3 d^{3}} + \frac {D a^{2} b}{d^{2}} - \frac {2 D a b^{2} c}{d^{3}} + \frac {D b^{3} c^{2}}{d^{4}}\right ) + x^{2} \left (\frac {A b^{3}}{2 d^{2}} + \frac {3 B a b^{2}}{2 d^{2}} - \frac {B b^{3} c}{d^{3}} + \frac {3 C a^{2} b}{2 d^{2}} - \frac {3 C a b^{2} c}{d^{3}} + \frac {3 C b^{3} c^{2}}{2 d^{4}} + \frac {D a^{3}}{2 d^{2}} - \frac {3 D a^{2} b c}{d^{3}} + \frac {9 D a b^{2} c^{2}}{2 d^{4}} - \frac {2 D b^{3} c^{3}}{d^{5}}\right ) + x \left (\frac {3 A a b^{2}}{d^{2}} - \frac {2 A b^{3} c}{d^{3}} + \frac {3 B a^{2} b}{d^{2}} - \frac {6 B a b^{2} c}{d^{3}} + \frac {3 B b^{3} c^{2}}{d^{4}} + \frac {C a^{3}}{d^{2}} - \frac {6 C a^{2} b c}{d^{3}} + \frac {9 C a b^{2} c^{2}}{d^{4}} - \frac {4 C b^{3} c^{3}}{d^{5}} - \frac {2 D a^{3} c}{d^{3}} + \frac {9 D a^{2} b c^{2}}{d^{4}} - \frac {12 D a b^{2} c^{3}}{d^{5}} + \frac {5 D b^{3} c^{4}}{d^{6}}\right ) + \frac {- A a^{3} d^{6} + 3 A a^{2} b c d^{5} - 3 A a b^{2} c^{2} d^{4} + A b^{3} c^{3} d^{3} + B a^{3} c d^{5} - 3 B a^{2} b c^{2} d^{4} + 3 B a b^{2} c^{3} d^{3} - B b^{3} c^{4} d^{2} - C a^{3} c^{2} d^{4} + 3 C a^{2} b c^{3} d^{3} - 3 C a b^{2} c^{4} d^{2} + C b^{3} c^{5} d + D a^{3} c^{3} d^{3} - 3 D a^{2} b c^{4} d^{2} + 3 D a b^{2} c^{5} d - D b^{3} c^{6}}{c d^{7} + d^{8} x} + \frac {\left (a d - b c\right )^{2} \cdot \left (3 A b d^{3} + B a d^{3} - 4 B b c d^{2} - 2 C a c d^{2} + 5 C b c^{2} d + 3 D a c^{2} d - 6 D b c^{3}\right ) \log {\left (c + d x \right )}}{d^{7}} \] Input:
integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**2,x)
Output:
D*b**3*x**5/(5*d**2) + x**4*(C*b**3/(4*d**2) + 3*D*a*b**2/(4*d**2) - D*b** 3*c/(2*d**3)) + x**3*(B*b**3/(3*d**2) + C*a*b**2/d**2 - 2*C*b**3*c/(3*d**3 ) + D*a**2*b/d**2 - 2*D*a*b**2*c/d**3 + D*b**3*c**2/d**4) + x**2*(A*b**3/( 2*d**2) + 3*B*a*b**2/(2*d**2) - B*b**3*c/d**3 + 3*C*a**2*b/(2*d**2) - 3*C* a*b**2*c/d**3 + 3*C*b**3*c**2/(2*d**4) + D*a**3/(2*d**2) - 3*D*a**2*b*c/d* *3 + 9*D*a*b**2*c**2/(2*d**4) - 2*D*b**3*c**3/d**5) + x*(3*A*a*b**2/d**2 - 2*A*b**3*c/d**3 + 3*B*a**2*b/d**2 - 6*B*a*b**2*c/d**3 + 3*B*b**3*c**2/d** 4 + C*a**3/d**2 - 6*C*a**2*b*c/d**3 + 9*C*a*b**2*c**2/d**4 - 4*C*b**3*c**3 /d**5 - 2*D*a**3*c/d**3 + 9*D*a**2*b*c**2/d**4 - 12*D*a*b**2*c**3/d**5 + 5 *D*b**3*c**4/d**6) + (-A*a**3*d**6 + 3*A*a**2*b*c*d**5 - 3*A*a*b**2*c**2*d **4 + A*b**3*c**3*d**3 + B*a**3*c*d**5 - 3*B*a**2*b*c**2*d**4 + 3*B*a*b**2 *c**3*d**3 - B*b**3*c**4*d**2 - C*a**3*c**2*d**4 + 3*C*a**2*b*c**3*d**3 - 3*C*a*b**2*c**4*d**2 + C*b**3*c**5*d + D*a**3*c**3*d**3 - 3*D*a**2*b*c**4* d**2 + 3*D*a*b**2*c**5*d - D*b**3*c**6)/(c*d**7 + d**8*x) + (a*d - b*c)**2 *(3*A*b*d**3 + B*a*d**3 - 4*B*b*c*d**2 - 2*C*a*c*d**2 + 5*C*b*c**2*d + 3*D *a*c**2*d - 6*D*b*c**3)*log(c + d*x)/d**7
Time = 0.04 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=-\frac {D b^{3} c^{6} + A a^{3} d^{6} - {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5}}{d^{8} x + c d^{7}} + \frac {12 \, D b^{3} d^{4} x^{5} - 15 \, {\left (2 \, D b^{3} c d^{3} - {\left (3 \, D a b^{2} + C b^{3}\right )} d^{4}\right )} x^{4} + 20 \, {\left (3 \, D b^{3} c^{2} d^{2} - 2 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d^{3} + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{4}\right )} x^{3} - 30 \, {\left (4 \, D b^{3} c^{3} d - 3 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d^{2} + 2 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{3} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{4}\right )} x^{2} + 60 \, {\left (5 \, D b^{3} c^{4} - 4 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 3 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 2 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )} x}{60 \, d^{6}} - \frac {{\left (6 \, D b^{3} c^{5} - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} \log \left (d x + c\right )}{d^{7}} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x, algorithm="maxima")
Output:
-(D*b^3*c^6 + A*a^3*d^6 - (3*D*a*b^2 + C*b^3)*c^5*d + (3*D*a^2*b + 3*C*a*b ^2 + B*b^3)*c^4*d^2 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + (C *a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^4 - (B*a^3 + 3*A*a^2*b)*c*d^5)/(d^8*x + c*d^7) + 1/60*(12*D*b^3*d^4*x^5 - 15*(2*D*b^3*c*d^3 - (3*D*a*b^2 + C*b^3 )*d^4)*x^4 + 20*(3*D*b^3*c^2*d^2 - 2*(3*D*a*b^2 + C*b^3)*c*d^3 + (3*D*a^2* b + 3*C*a*b^2 + B*b^3)*d^4)*x^3 - 30*(4*D*b^3*c^3*d - 3*(3*D*a*b^2 + C*b^3 )*c^2*d^2 + 2*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^3 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^4)*x^2 + 60*(5*D*b^3*c^4 - 4*(3*D*a*b^2 + C*b^3)*c^3 *d + 3*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^2 - 2*(D*a^3 + 3*C*a^2*b + 3* B*a*b^2 + A*b^3)*c*d^3 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^4)*x)/d^6 - (6* D*b^3*c^5 - 5*(3*D*a*b^2 + C*b^3)*c^4*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3 )*c^3*d^2 - 3*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^3 + 2*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^4 - (B*a^3 + 3*A*a^2*b)*d^5)*log(d*x + c)/d^7
Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (403) = 806\).
Time = 0.13 (sec) , antiderivative size = 912, normalized size of antiderivative = 2.22 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x, algorithm="giac")
Output:
1/60*(12*D*b^3 - 15*(6*D*b^3*c*d - 3*D*a*b^2*d^2 - C*b^3*d^2)/((d*x + c)*d ) + 20*(15*D*b^3*c^2*d^2 - 15*D*a*b^2*c*d^3 - 5*C*b^3*c*d^3 + 3*D*a^2*b*d^ 4 + 3*C*a*b^2*d^4 + B*b^3*d^4)/((d*x + c)^2*d^2) - 30*(20*D*b^3*c^3*d^3 - 30*D*a*b^2*c^2*d^4 - 10*C*b^3*c^2*d^4 + 12*D*a^2*b*c*d^5 + 12*C*a*b^2*c*d^ 5 + 4*B*b^3*c*d^5 - D*a^3*d^6 - 3*C*a^2*b*d^6 - 3*B*a*b^2*d^6 - A*b^3*d^6) /((d*x + c)^3*d^3) + 60*(15*D*b^3*c^4*d^4 - 30*D*a*b^2*c^3*d^5 - 10*C*b^3* c^3*d^5 + 18*D*a^2*b*c^2*d^6 + 18*C*a*b^2*c^2*d^6 + 6*B*b^3*c^2*d^6 - 3*D* a^3*c*d^7 - 9*C*a^2*b*c*d^7 - 9*B*a*b^2*c*d^7 - 3*A*b^3*c*d^7 + C*a^3*d^8 + 3*B*a^2*b*d^8 + 3*A*a*b^2*d^8)/((d*x + c)^4*d^4))*(d*x + c)^5/d^7 + (6*D *b^3*c^5 - 15*D*a*b^2*c^4*d - 5*C*b^3*c^4*d + 12*D*a^2*b*c^3*d^2 + 12*C*a* b^2*c^3*d^2 + 4*B*b^3*c^3*d^2 - 3*D*a^3*c^2*d^3 - 9*C*a^2*b*c^2*d^3 - 9*B* a*b^2*c^2*d^3 - 3*A*b^3*c^2*d^3 + 2*C*a^3*c*d^4 + 6*B*a^2*b*c*d^4 + 6*A*a* b^2*c*d^4 - B*a^3*d^5 - 3*A*a^2*b*d^5)*log(abs(d*x + c)/((d*x + c)^2*abs(d )))/d^7 - (D*b^3*c^6*d^5/(d*x + c) - 3*D*a*b^2*c^5*d^6/(d*x + c) - C*b^3*c ^5*d^6/(d*x + c) + 3*D*a^2*b*c^4*d^7/(d*x + c) + 3*C*a*b^2*c^4*d^7/(d*x + c) + B*b^3*c^4*d^7/(d*x + c) - D*a^3*c^3*d^8/(d*x + c) - 3*C*a^2*b*c^3*d^8 /(d*x + c) - 3*B*a*b^2*c^3*d^8/(d*x + c) - A*b^3*c^3*d^8/(d*x + c) + C*a^3 *c^2*d^9/(d*x + c) + 3*B*a^2*b*c^2*d^9/(d*x + c) + 3*A*a*b^2*c^2*d^9/(d*x + c) - B*a^3*c*d^10/(d*x + c) - 3*A*a^2*b*c*d^10/(d*x + c) + A*a^3*d^11/(d *x + c))/d^12
Timed out. \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\int \frac {{\left (a+b\,x\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^2,x)
Output:
int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^2, x)
Time = 0.14 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.67 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^2} \, dx=\frac {240 \,\mathrm {log}\left (d x +c \right ) a^{3} b c \,d^{5} x -720 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c^{2} d^{4} x -180 \,\mathrm {log}\left (d x +c \right ) a^{2} b \,c^{4} d^{3} x +720 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{3} d^{3} x +180 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c^{5} d^{2} x -60 \,\mathrm {log}\left (d x +c \right ) b^{3} c^{7}+60 a^{4} d^{6} x +240 \,\mathrm {log}\left (d x +c \right ) a^{3} b \,c^{2} d^{4}+60 \,\mathrm {log}\left (d x +c \right ) a^{3} c^{3} d^{4} x -720 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c^{3} d^{3}-180 \,\mathrm {log}\left (d x +c \right ) a^{2} b \,c^{5} d^{2}+720 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{4} d^{2}+180 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c^{6} d -240 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{4} d^{2} x -60 \,\mathrm {log}\left (d x +c \right ) b^{3} c^{6} d x -240 a^{3} b c \,d^{5} x +720 a^{2} b^{2} c^{2} d^{4} x +360 a^{2} b^{2} c \,d^{5} x^{2}+180 a^{2} b \,c^{4} d^{3} x +90 a^{2} b \,c^{3} d^{4} x^{2}-30 a^{2} b \,c^{2} d^{5} x^{3}+60 a^{2} b c \,d^{6} x^{4}-720 a \,b^{3} c^{3} d^{3} x -360 a \,b^{3} c^{2} d^{4} x^{2}+60 \,\mathrm {log}\left (d x +c \right ) a^{3} c^{4} d^{3}-240 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{5} d -60 a^{3} c^{3} d^{4} x -30 a^{3} c^{2} d^{5} x^{2}+30 a^{3} c \,d^{6} x^{3}+240 b^{4} c^{4} d^{2} x +120 b^{4} c^{3} d^{3} x^{2}-40 b^{4} c^{2} d^{4} x^{3}+20 b^{4} c \,d^{5} x^{4}+60 b^{3} c^{6} d x +30 b^{3} c^{5} d^{2} x^{2}-10 b^{3} c^{4} d^{3} x^{3}+5 b^{3} c^{3} d^{4} x^{4}-3 b^{3} c^{2} d^{5} x^{5}+12 b^{3} c \,d^{6} x^{6}+120 a \,b^{3} c \,d^{5} x^{3}-180 a \,b^{2} c^{5} d^{2} x -90 a \,b^{2} c^{4} d^{3} x^{2}+30 a \,b^{2} c^{3} d^{4} x^{3}-15 a \,b^{2} c^{2} d^{5} x^{4}+45 a \,b^{2} c \,d^{6} x^{5}}{60 c \,d^{6} \left (d x +c \right )} \] Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^2,x)
Output:
(240*log(c + d*x)*a**3*b*c**2*d**4 + 240*log(c + d*x)*a**3*b*c*d**5*x + 60 *log(c + d*x)*a**3*c**4*d**3 + 60*log(c + d*x)*a**3*c**3*d**4*x - 720*log( c + d*x)*a**2*b**2*c**3*d**3 - 720*log(c + d*x)*a**2*b**2*c**2*d**4*x - 18 0*log(c + d*x)*a**2*b*c**5*d**2 - 180*log(c + d*x)*a**2*b*c**4*d**3*x + 72 0*log(c + d*x)*a*b**3*c**4*d**2 + 720*log(c + d*x)*a*b**3*c**3*d**3*x + 18 0*log(c + d*x)*a*b**2*c**6*d + 180*log(c + d*x)*a*b**2*c**5*d**2*x - 240*l og(c + d*x)*b**4*c**5*d - 240*log(c + d*x)*b**4*c**4*d**2*x - 60*log(c + d *x)*b**3*c**7 - 60*log(c + d*x)*b**3*c**6*d*x + 60*a**4*d**6*x - 240*a**3* b*c*d**5*x - 60*a**3*c**3*d**4*x - 30*a**3*c**2*d**5*x**2 + 30*a**3*c*d**6 *x**3 + 720*a**2*b**2*c**2*d**4*x + 360*a**2*b**2*c*d**5*x**2 + 180*a**2*b *c**4*d**3*x + 90*a**2*b*c**3*d**4*x**2 - 30*a**2*b*c**2*d**5*x**3 + 60*a* *2*b*c*d**6*x**4 - 720*a*b**3*c**3*d**3*x - 360*a*b**3*c**2*d**4*x**2 + 12 0*a*b**3*c*d**5*x**3 - 180*a*b**2*c**5*d**2*x - 90*a*b**2*c**4*d**3*x**2 + 30*a*b**2*c**3*d**4*x**3 - 15*a*b**2*c**2*d**5*x**4 + 45*a*b**2*c*d**6*x* *5 + 240*b**4*c**4*d**2*x + 120*b**4*c**3*d**3*x**2 - 40*b**4*c**2*d**4*x* *3 + 20*b**4*c*d**5*x**4 + 60*b**3*c**6*d*x + 30*b**3*c**5*d**2*x**2 - 10* b**3*c**4*d**3*x**3 + 5*b**3*c**3*d**4*x**4 - 3*b**3*c**2*d**5*x**5 + 12*b **3*c*d**6*x**6)/(60*c*d**6*(c + d*x))