Integrand size = 30, antiderivative size = 397 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {\left (a^3 d^3 D+3 a^2 b d^2 (C d-3 c D)-3 a b^2 d \left (3 c C d-B d^2-6 c^2 D\right )+b^3 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) x}{d^6}+\frac {b \left (3 a^2 d^2 D+3 a b d (C d-3 c D)-b^2 \left (3 c C d-B d^2-6 c^2 D\right )\right ) x^2}{2 d^5}+\frac {b^2 (b C d-3 b c D+3 a d D) x^3}{3 d^4}+\frac {b^3 D x^4}{4 d^3}+\frac {(b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{2 d^7 (c+d x)^2}+\frac {(b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right )}{d^7 (c+d x)}-\frac {(b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) \log (c+d x)}{d^7} \] Output:
(a^3*d^3*D+3*a^2*b*d^2*(C*d-3*D*c)-3*a*b^2*d*(-B*d^2+3*C*c*d-6*D*c^2)+b^3* (A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*x/d^6+1/2*b*(3*a^2*d^2*D+3*a*b*d*(C* d-3*D*c)-b^2*(-B*d^2+3*C*c*d-6*D*c^2))*x^2/d^5+1/3*b^2*(C*b*d+3*D*a*d-3*D* b*c)*x^3/d^4+1/4*b^3*D*x^4/d^3+1/2*(-a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c ^3)/d^7/(d*x+c)^2+(-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))/d^7/(d*x+c)-(-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 ))*ln(d*x+c)/d^7
Time = 0.11 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {12 d \left (a^3 d^3 D+3 a^2 b d^2 (C d-3 c D)+3 a b^2 d \left (-3 c C d+B d^2+6 c^2 D\right )+b^3 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) x+6 b d^2 \left (3 a^2 d^2 D+3 a b d (C d-3 c D)+b^2 \left (-3 c C d+B d^2+6 c^2 D\right )\right ) x^2+4 b^2 d^3 (b C d-3 b c D+3 a d D) x^3+3 b^3 d^4 D x^4-\frac {6 (b c-a d)^3 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )}{(c+d x)^2}+\frac {12 (b c-a d)^2 \left (-a d \left (-2 c C d+B d^2+3 c^2 D\right )+b \left (-5 c^2 C d+4 B c d^2-3 A d^3+6 c^3 D\right )\right )}{c+d x}+12 (b c-a d) \left (a^2 d^2 (-C d+3 c D)+a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (-10 c^2 C d+6 B c d^2-3 A d^3+15 c^3 D\right )\right ) \log (c+d x)}{12 d^7} \] Input:
Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^3,x]
Output:
(12*d*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 3*c*D) + 3*a*b^2*d*(-3*c*C*d + B*d^2 + 6*c^2*D) + b^3*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*x + 6*b*d^2* (3*a^2*d^2*D + 3*a*b*d*(C*d - 3*c*D) + b^2*(-3*c*C*d + B*d^2 + 6*c^2*D))*x ^2 + 4*b^2*d^3*(b*C*d - 3*b*c*D + 3*a*d*D)*x^3 + 3*b^3*d^4*D*x^4 - (6*(b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(c + d*x)^2 + (12*(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)))/(c + d*x) + 12*(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))*Log[c + d*x])/(12*d^7)
Time = 0.98 (sec) , antiderivative size = 397, 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.
| \(\displaystyle \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )-\left (b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{d^6 (c+d x)}+\frac {b x \left (3 a^2 d^2 D+3 a b d (C d-3 c D)-\left (b^2 \left (-B d^2-6 c^2 D+3 c C d\right )\right )\right )}{d^5}+\frac {a^3 d^3 D+3 a^2 b d^2 (C d-3 c D)-3 a b^2 d \left (-B d^2-6 c^2 D+3 c C d\right )+b^3 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )}{d^6}+\frac {(a d-b c)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 (c+d x)^3}+\frac {(b c-a d)^2 \left (b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^6 (c+d x)^2}+\frac {b^2 x^2 (3 a d D-3 b c D+b C d)}{d^4}+\frac {b^3 D x^3}{d^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(b c-a d) \log (c+d x) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{d^7}+\frac {b x^2 \left (3 a^2 d^2 D+3 a b d (C d-3 c D)-\left (b^2 \left (-B d^2-6 c^2 D+3 c C d\right )\right )\right )}{2 d^5}+\frac {x \left (a^3 d^3 D+3 a^2 b d^2 (C d-3 c D)-3 a b^2 d \left (-B d^2-6 c^2 D+3 c C d\right )+b^3 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6}+\frac {(b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7 (c+d x)}+\frac {(b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^7 (c+d x)^2}+\frac {b^2 x^3 (3 a d D-3 b c D+b C d)}{3 d^4}+\frac {b^3 D x^4}{4 d^3}\) |
Input:
Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^3,x]
Output:
((a^3*d^3*D + 3*a^2*b*d^2*(C*d - 3*c*D) - 3*a*b^2*d*(3*c*C*d - B*d^2 - 6*c ^2*D) + b^3*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*x)/d^6 + (b*(3*a^2 *d^2*D + 3*a*b*d*(C*d - 3*c*D) - b^2*(3*c*C*d - B*d^2 - 6*c^2*D))*x^2)/(2* d^5) + (b^2*(b*C*d - 3*b*c*D + 3*a*d*D)*x^3)/(3*d^4) + (b^3*D*x^4)/(4*d^3) + ((b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(2*d^7*(c + d*x)^2) + ((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)))/(d^7*(c + d*x)) - ((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))*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 = 0.37 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.85
| method | result | size |
| norman | \(\frac {-\frac {A \,a^{3} d^{6}+3 A \,a^{2} b c \,d^{5}-9 A a \,b^{2} c^{2} d^{4}+9 A \,b^{3} c^{3} d^{3}+B \,a^{3} c \,d^{5}-9 B \,a^{2} b \,c^{2} d^{4}+27 B a \,b^{2} c^{3} d^{3}-18 B \,b^{3} c^{4} d^{2}-3 C \,a^{3} c^{2} d^{4}+27 C \,a^{2} b \,c^{3} d^{3}-54 C a \,b^{2} c^{4} d^{2}+30 C \,b^{3} c^{5} d +9 D a^{3} c^{3} d^{3}-54 D a^{2} b \,c^{4} d^{2}+90 D a \,b^{2} c^{5} d -45 D b^{3} c^{6}}{2 d^{7}}+\frac {\left (3 A \,b^{3} d^{3}+9 B a \,b^{2} d^{3}-6 B \,b^{3} c \,d^{2}+9 a^{2} b C \,d^{3}-18 C a \,b^{2} c \,d^{2}+10 C \,b^{3} c^{2} d +3 a^{3} d^{3} D-18 D a^{2} b c \,d^{2}+30 D a \,b^{2} c^{2} d -15 D b^{3} c^{3}\right ) x^{3}}{3 d^{4}}-\frac {\left (3 A \,a^{2} b \,d^{5}-6 A a \,b^{2} c \,d^{4}+6 A \,b^{3} c^{2} d^{3}+B \,a^{3} d^{5}-6 B \,a^{2} b c \,d^{4}+18 B a \,b^{2} c^{2} d^{3}-12 B \,b^{3} c^{3} d^{2}-2 C \,a^{3} c \,d^{4}+18 C \,a^{2} b \,c^{2} d^{3}-36 C a \,b^{2} c^{3} d^{2}+20 C \,b^{3} c^{4} d +6 D a^{3} c^{2} d^{3}-36 D a^{2} b \,c^{3} d^{2}+60 D a \,b^{2} c^{4} d -30 D b^{3} c^{5}\right ) x}{d^{6}}+\frac {D b^{3} x^{6}}{4 d}+\frac {b \left (6 b^{2} B \,d^{2}+18 C a b \,d^{2}-10 C \,b^{2} c d +18 a^{2} d^{2} D-30 D a b c d +15 D b^{2} c^{2}\right ) x^{4}}{12 d^{3}}+\frac {b^{2} \left (2 C b d +6 D a d -3 D b c \right ) x^{5}}{6 d^{2}}}{\left (x d +c \right )^{2}}+\frac {\left (3 a \,b^{2} A \,d^{4}-3 A \,b^{3} c \,d^{3}+3 a^{2} b B \,d^{4}-9 B a \,b^{2} c \,d^{3}+6 B \,b^{3} c^{2} d^{2}+a^{3} C \,d^{4}-9 C \,a^{2} b c \,d^{3}+18 C a \,b^{2} c^{2} d^{2}-10 C \,b^{3} c^{3} d -3 D a^{3} c \,d^{3}+18 D a^{2} b \,c^{2} d^{2}-30 D a \,b^{2} c^{3} d +15 D b^{3} c^{4}\right ) \ln \left (x d +c \right )}{d^{7}}\) | \(733\) |
| default | \(\frac {\frac {1}{4} D x^{4} b^{3} d^{3}+\frac {1}{3} C \,b^{3} d^{3} x^{3}+D x^{3} a \,b^{2} d^{3}-D x^{3} b^{3} c \,d^{2}+\frac {1}{2} b^{3} B \,d^{3} x^{2}+\frac {3}{2} C \,x^{2} a \,b^{2} d^{3}-\frac {3}{2} C \,x^{2} b^{3} c \,d^{2}+\frac {3}{2} D x^{2} a^{2} b \,d^{3}-\frac {9}{2} D a \,b^{2} c \,d^{2} x^{2}+3 D x^{2} b^{3} c^{2} d +A \,b^{3} d^{3} x +3 B a \,b^{2} d^{3} x -3 B \,b^{3} c \,d^{2} x +3 a^{2} b C \,d^{3} x -9 C a \,b^{2} c \,d^{2} x +6 C \,b^{3} c^{2} d x +a^{3} d^{3} D x -9 D a^{2} b c \,d^{2} x +18 D a \,b^{2} c^{2} d x -10 D b^{3} c^{3} x}{d^{6}}-\frac {3 A \,a^{2} b \,d^{5}-6 A a \,b^{2} c \,d^{4}+3 A \,b^{3} c^{2} d^{3}+B \,a^{3} d^{5}-6 B \,a^{2} b c \,d^{4}+9 B a \,b^{2} c^{2} d^{3}-4 B \,b^{3} c^{3} d^{2}-2 C \,a^{3} c \,d^{4}+9 C \,a^{2} b \,c^{2} d^{3}-12 C a \,b^{2} c^{3} d^{2}+5 C \,b^{3} c^{4} d +3 D a^{3} c^{2} d^{3}-12 D a^{2} b \,c^{3} d^{2}+15 D a \,b^{2} c^{4} d -6 D b^{3} c^{5}}{d^{7} \left (x d +c \right )}+\frac {\left (3 a \,b^{2} A \,d^{4}-3 A \,b^{3} c \,d^{3}+3 a^{2} b B \,d^{4}-9 B a \,b^{2} c \,d^{3}+6 B \,b^{3} c^{2} d^{2}+a^{3} C \,d^{4}-9 C \,a^{2} b c \,d^{3}+18 C a \,b^{2} c^{2} d^{2}-10 C \,b^{3} c^{3} d -3 D a^{3} c \,d^{3}+18 D a^{2} b \,c^{2} d^{2}-30 D a \,b^{2} c^{3} d +15 D b^{3} c^{4}\right ) \ln \left (x d +c \right )}{d^{7}}-\frac {A \,a^{3} d^{6}-3 A \,a^{2} b c \,d^{5}+3 A a \,b^{2} c^{2} d^{4}-A \,b^{3} c^{3} d^{3}-B \,a^{3} c \,d^{5}+3 B \,a^{2} b \,c^{2} d^{4}-3 B a \,b^{2} c^{3} d^{3}+B \,b^{3} c^{4} d^{2}+C \,a^{3} c^{2} d^{4}-3 C \,a^{2} b \,c^{3} d^{3}+3 C a \,b^{2} c^{4} d^{2}-C \,b^{3} c^{5} d -D a^{3} c^{3} d^{3}+3 D a^{2} b \,c^{4} d^{2}-3 D a \,b^{2} c^{5} d +D b^{3} c^{6}}{2 d^{7} \left (x d +c \right )^{2}}\) | \(767\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1399\) |
Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
(-1/2*(A*a^3*d^6+3*A*a^2*b*c*d^5-9*A*a*b^2*c^2*d^4+9*A*b^3*c^3*d^3+B*a^3*c *d^5-9*B*a^2*b*c^2*d^4+27*B*a*b^2*c^3*d^3-18*B*b^3*c^4*d^2-3*C*a^3*c^2*d^4 +27*C*a^2*b*c^3*d^3-54*C*a*b^2*c^4*d^2+30*C*b^3*c^5*d+9*D*a^3*c^3*d^3-54*D *a^2*b*c^4*d^2+90*D*a*b^2*c^5*d-45*D*b^3*c^6)/d^7+1/3*(3*A*b^3*d^3+9*B*a*b ^2*d^3-6*B*b^3*c*d^2+9*C*a^2*b*d^3-18*C*a*b^2*c*d^2+10*C*b^3*c^2*d+3*D*a^3 *d^3-18*D*a^2*b*c*d^2+30*D*a*b^2*c^2*d-15*D*b^3*c^3)/d^4*x^3-(3*A*a^2*b*d^ 5-6*A*a*b^2*c*d^4+6*A*b^3*c^2*d^3+B*a^3*d^5-6*B*a^2*b*c*d^4+18*B*a*b^2*c^2 *d^3-12*B*b^3*c^3*d^2-2*C*a^3*c*d^4+18*C*a^2*b*c^2*d^3-36*C*a*b^2*c^3*d^2+ 20*C*b^3*c^4*d+6*D*a^3*c^2*d^3-36*D*a^2*b*c^3*d^2+60*D*a*b^2*c^4*d-30*D*b^ 3*c^5)/d^6*x+1/4*D*b^3/d*x^6+1/12*b*(6*B*b^2*d^2+18*C*a*b*d^2-10*C*b^2*c*d +18*D*a^2*d^2-30*D*a*b*c*d+15*D*b^2*c^2)/d^3*x^4+1/6*b^2*(2*C*b*d+6*D*a*d- 3*D*b*c)/d^2*x^5)/(d*x+c)^2+1/d^7*(3*A*a*b^2*d^4-3*A*b^3*c*d^3+3*B*a^2*b*d ^4-9*B*a*b^2*c*d^3+6*B*b^3*c^2*d^2+C*a^3*d^4-9*C*a^2*b*c*d^3+18*C*a*b^2*c^ 2*d^2-10*C*b^3*c^3*d-3*D*a^3*c*d^3+18*D*a^2*b*c^2*d^2-30*D*a*b^2*c^3*d+15* D*b^3*c^4)*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 978 vs. \(2 (391) = 782\).
Time = 0.09 (sec) , antiderivative size = 978, normalized size of antiderivative = 2.46 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="fricas")
Output:
1/12*(3*D*b^3*d^6*x^6 + 66*D*b^3*c^6 - 6*A*a^3*d^6 - 54*(3*D*a*b^2 + C*b^3 )*c^5*d + 42*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 - 30*(D*a^3 + 3*C*a^2 *b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + 18*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d ^4 - 6*(B*a^3 + 3*A*a^2*b)*c*d^5 - 2*(3*D*b^3*c*d^5 - 2*(3*D*a*b^2 + C*b^3 )*d^6)*x^5 + (15*D*b^3*c^2*d^4 - 10*(3*D*a*b^2 + C*b^3)*c*d^5 + 6*(3*D*a^2 *b + 3*C*a*b^2 + B*b^3)*d^6)*x^4 - 4*(15*D*b^3*c^3*d^3 - 10*(3*D*a*b^2 + C *b^3)*c^2*d^4 + 6*(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 - 6*(34*D*b^3*c^4*d^2 - 21*(3*D*a*b^2 + C*b^3)*c^3*d^3 + 11*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^4 - 4*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c*d^5)*x^2 - 12*(4*D*b^3*c^5*d - (3*D*a*b^2 + C*b^3)*c^4*d^2 - (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 - 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 + 12*(15*D*b^3*c^6 - 10*(3*D*a*b^2 + C* b^3)*c^5*d + 6*(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 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^4 + (15*D*b^3*c^4*d^2 - 10*(3*D*a*b^2 + C*b^3)*c^3*d^3 + 6*(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 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6)*x^2 + 2*(15*D*b^3*c^5*d - 10*(3*D*a *b^2 + C*b^3)*c^4*d^2 + 6*(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 + (C*a^3 + 3*B*a^2*b + 3*A*...
Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (400) = 800\).
Time = 30.36 (sec) , antiderivative size = 814, normalized size of antiderivative = 2.05 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {D b^{3} x^{4}}{4 d^{3}} + x^{3} \left (\frac {C b^{3}}{3 d^{3}} + \frac {D a b^{2}}{d^{3}} - \frac {D b^{3} c}{d^{4}}\right ) + x^{2} \left (\frac {B b^{3}}{2 d^{3}} + \frac {3 C a b^{2}}{2 d^{3}} - \frac {3 C b^{3} c}{2 d^{4}} + \frac {3 D a^{2} b}{2 d^{3}} - \frac {9 D a b^{2} c}{2 d^{4}} + \frac {3 D b^{3} c^{2}}{d^{5}}\right ) + x \left (\frac {A b^{3}}{d^{3}} + \frac {3 B a b^{2}}{d^{3}} - \frac {3 B b^{3} c}{d^{4}} + \frac {3 C a^{2} b}{d^{3}} - \frac {9 C a b^{2} c}{d^{4}} + \frac {6 C b^{3} c^{2}}{d^{5}} + \frac {D a^{3}}{d^{3}} - \frac {9 D a^{2} b c}{d^{4}} + \frac {18 D a b^{2} c^{2}}{d^{5}} - \frac {10 D b^{3} c^{3}}{d^{6}}\right ) + \frac {- A a^{3} d^{6} - 3 A a^{2} b c d^{5} + 9 A a b^{2} c^{2} d^{4} - 5 A b^{3} c^{3} d^{3} - B a^{3} c d^{5} + 9 B a^{2} b c^{2} d^{4} - 15 B a b^{2} c^{3} d^{3} + 7 B b^{3} c^{4} d^{2} + 3 C a^{3} c^{2} d^{4} - 15 C a^{2} b c^{3} d^{3} + 21 C a b^{2} c^{4} d^{2} - 9 C b^{3} c^{5} d - 5 D a^{3} c^{3} d^{3} + 21 D a^{2} b c^{4} d^{2} - 27 D a b^{2} c^{5} d + 11 D b^{3} c^{6} + x \left (- 6 A a^{2} b d^{6} + 12 A a b^{2} c d^{5} - 6 A b^{3} c^{2} d^{4} - 2 B a^{3} d^{6} + 12 B a^{2} b c d^{5} - 18 B a b^{2} c^{2} d^{4} + 8 B b^{3} c^{3} d^{3} + 4 C a^{3} c d^{5} - 18 C a^{2} b c^{2} d^{4} + 24 C a b^{2} c^{3} d^{3} - 10 C b^{3} c^{4} d^{2} - 6 D a^{3} c^{2} d^{4} + 24 D a^{2} b c^{3} d^{3} - 30 D a b^{2} c^{4} d^{2} + 12 D b^{3} c^{5} d\right )}{2 c^{2} d^{7} + 4 c d^{8} x + 2 d^{9} x^{2}} - \frac {\left (a d - b c\right ) \left (- 3 A b^{2} d^{3} - 3 B a b d^{3} + 6 B b^{2} c d^{2} - C a^{2} d^{3} + 8 C a b c d^{2} - 10 C b^{2} c^{2} d + 3 D a^{2} c d^{2} - 15 D a b c^{2} d + 15 D b^{2} c^{3}\right ) \log {\left (c + d x \right )}}{d^{7}} \] Input:
integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**3,x)
Output:
D*b**3*x**4/(4*d**3) + x**3*(C*b**3/(3*d**3) + D*a*b**2/d**3 - D*b**3*c/d* *4) + x**2*(B*b**3/(2*d**3) + 3*C*a*b**2/(2*d**3) - 3*C*b**3*c/(2*d**4) + 3*D*a**2*b/(2*d**3) - 9*D*a*b**2*c/(2*d**4) + 3*D*b**3*c**2/d**5) + x*(A*b **3/d**3 + 3*B*a*b**2/d**3 - 3*B*b**3*c/d**4 + 3*C*a**2*b/d**3 - 9*C*a*b** 2*c/d**4 + 6*C*b**3*c**2/d**5 + D*a**3/d**3 - 9*D*a**2*b*c/d**4 + 18*D*a*b **2*c**2/d**5 - 10*D*b**3*c**3/d**6) + (-A*a**3*d**6 - 3*A*a**2*b*c*d**5 + 9*A*a*b**2*c**2*d**4 - 5*A*b**3*c**3*d**3 - B*a**3*c*d**5 + 9*B*a**2*b*c* *2*d**4 - 15*B*a*b**2*c**3*d**3 + 7*B*b**3*c**4*d**2 + 3*C*a**3*c**2*d**4 - 15*C*a**2*b*c**3*d**3 + 21*C*a*b**2*c**4*d**2 - 9*C*b**3*c**5*d - 5*D*a* *3*c**3*d**3 + 21*D*a**2*b*c**4*d**2 - 27*D*a*b**2*c**5*d + 11*D*b**3*c**6 + x*(-6*A*a**2*b*d**6 + 12*A*a*b**2*c*d**5 - 6*A*b**3*c**2*d**4 - 2*B*a** 3*d**6 + 12*B*a**2*b*c*d**5 - 18*B*a*b**2*c**2*d**4 + 8*B*b**3*c**3*d**3 + 4*C*a**3*c*d**5 - 18*C*a**2*b*c**2*d**4 + 24*C*a*b**2*c**3*d**3 - 10*C*b* *3*c**4*d**2 - 6*D*a**3*c**2*d**4 + 24*D*a**2*b*c**3*d**3 - 30*D*a*b**2*c* *4*d**2 + 12*D*b**3*c**5*d))/(2*c**2*d**7 + 4*c*d**8*x + 2*d**9*x**2) - (a *d - b*c)*(-3*A*b**2*d**3 - 3*B*a*b*d**3 + 6*B*b**2*c*d**2 - C*a**2*d**3 + 8*C*a*b*c*d**2 - 10*C*b**2*c**2*d + 3*D*a**2*c*d**2 - 15*D*a*b*c**2*d + 1 5*D*b**2*c**3)*log(c + d*x)/d**7
Time = 0.05 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {11 \, D b^{3} c^{6} - A a^{3} d^{6} - 9 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + 7 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - 5 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + 3 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} + 2 \, {\left (6 \, D b^{3} c^{5} d - 5 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d^{2} + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{3} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{4} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6}\right )} x}{2 \, {\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} + \frac {3 \, D b^{3} d^{3} x^{4} - 4 \, {\left (3 \, D b^{3} c d^{2} - {\left (3 \, D a b^{2} + C b^{3}\right )} d^{3}\right )} x^{3} + 6 \, {\left (6 \, D b^{3} c^{2} d - 3 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d^{2} + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{3}\right )} x^{2} - 12 \, {\left (10 \, D b^{3} c^{3} - 6 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 3 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} x}{12 \, d^{6}} + \frac {{\left (15 \, D b^{3} c^{4} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )} \log \left (d x + c\right )}{d^{7}} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="maxima")
Output:
1/2*(11*D*b^3*c^6 - A*a^3*d^6 - 9*(3*D*a*b^2 + C*b^3)*c^5*d + 7*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 - 5*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)* c^3*d^3 + 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 + 2*(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)/(d^9*x^2 + 2*c*d^8*x + c^2*d^7) + 1/12*(3*D*b^3*d^3*x^4 - 4*(3*D*b^3*c* d^2 - (3*D*a*b^2 + C*b^3)*d^3)*x^3 + 6*(6*D*b^3*c^2*d - 3*(3*D*a*b^2 + C*b ^3)*c*d^2 + (3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^3)*x^2 - 12*(10*D*b^3*c^3 - 6*(3*D*a*b^2 + C*b^3)*c^2*d + 3*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^2 - (D *a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^3)*x)/d^6 + (15*D*b^3*c^4 - 10*(3* D*a*b^2 + C*b^3)*c^3*d + 6*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^2 - 3*(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)*log(d*x + c)/d^7
Time = 0.13 (sec) , antiderivative size = 781, normalized size of antiderivative = 1.97 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {{\left (15 \, D b^{3} c^{4} - 30 \, D a b^{2} c^{3} d - 10 \, C b^{3} c^{3} d + 18 \, D a^{2} b c^{2} d^{2} + 18 \, C a b^{2} c^{2} d^{2} + 6 \, B b^{3} c^{2} d^{2} - 3 \, D a^{3} c d^{3} - 9 \, C a^{2} b c d^{3} - 9 \, B a b^{2} c d^{3} - 3 \, A b^{3} c d^{3} + C a^{3} d^{4} + 3 \, B a^{2} b d^{4} + 3 \, A a b^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{7}} + \frac {11 \, D b^{3} c^{6} - 27 \, D a b^{2} c^{5} d - 9 \, C b^{3} c^{5} d + 21 \, D a^{2} b c^{4} d^{2} + 21 \, C a b^{2} c^{4} d^{2} + 7 \, B b^{3} c^{4} d^{2} - 5 \, D a^{3} c^{3} d^{3} - 15 \, C a^{2} b c^{3} d^{3} - 15 \, B a b^{2} c^{3} d^{3} - 5 \, A b^{3} c^{3} d^{3} + 3 \, C a^{3} c^{2} d^{4} + 9 \, B a^{2} b c^{2} d^{4} + 9 \, A a b^{2} c^{2} d^{4} - B a^{3} c d^{5} - 3 \, A a^{2} b c d^{5} - A a^{3} d^{6} + 2 \, {\left (6 \, D b^{3} c^{5} d - 15 \, D a b^{2} c^{4} d^{2} - 5 \, C b^{3} c^{4} d^{2} + 12 \, D a^{2} b c^{3} d^{3} + 12 \, C a b^{2} c^{3} d^{3} + 4 \, B b^{3} c^{3} d^{3} - 3 \, D a^{3} c^{2} d^{4} - 9 \, C a^{2} b c^{2} d^{4} - 9 \, B a b^{2} c^{2} d^{4} - 3 \, A b^{3} c^{2} d^{4} + 2 \, C a^{3} c d^{5} + 6 \, B a^{2} b c d^{5} + 6 \, A a b^{2} c d^{5} - B a^{3} d^{6} - 3 \, A a^{2} b d^{6}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{7}} + \frac {3 \, D b^{3} d^{9} x^{4} - 12 \, D b^{3} c d^{8} x^{3} + 12 \, D a b^{2} d^{9} x^{3} + 4 \, C b^{3} d^{9} x^{3} + 36 \, D b^{3} c^{2} d^{7} x^{2} - 54 \, D a b^{2} c d^{8} x^{2} - 18 \, C b^{3} c d^{8} x^{2} + 18 \, D a^{2} b d^{9} x^{2} + 18 \, C a b^{2} d^{9} x^{2} + 6 \, B b^{3} d^{9} x^{2} - 120 \, D b^{3} c^{3} d^{6} x + 216 \, D a b^{2} c^{2} d^{7} x + 72 \, C b^{3} c^{2} d^{7} x - 108 \, D a^{2} b c d^{8} x - 108 \, C a b^{2} c d^{8} x - 36 \, B b^{3} c d^{8} x + 12 \, D a^{3} d^{9} x + 36 \, C a^{2} b d^{9} x + 36 \, B a b^{2} d^{9} x + 12 \, A b^{3} d^{9} x}{12 \, d^{12}} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="giac")
Output:
(15*D*b^3*c^4 - 30*D*a*b^2*c^3*d - 10*C*b^3*c^3*d + 18*D*a^2*b*c^2*d^2 + 1 8*C*a*b^2*c^2*d^2 + 6*B*b^3*c^2*d^2 - 3*D*a^3*c*d^3 - 9*C*a^2*b*c*d^3 - 9* B*a*b^2*c*d^3 - 3*A*b^3*c*d^3 + C*a^3*d^4 + 3*B*a^2*b*d^4 + 3*A*a*b^2*d^4) *log(abs(d*x + c))/d^7 + 1/2*(11*D*b^3*c^6 - 27*D*a*b^2*c^5*d - 9*C*b^3*c^ 5*d + 21*D*a^2*b*c^4*d^2 + 21*C*a*b^2*c^4*d^2 + 7*B*b^3*c^4*d^2 - 5*D*a^3* c^3*d^3 - 15*C*a^2*b*c^3*d^3 - 15*B*a*b^2*c^3*d^3 - 5*A*b^3*c^3*d^3 + 3*C* a^3*c^2*d^4 + 9*B*a^2*b*c^2*d^4 + 9*A*a*b^2*c^2*d^4 - B*a^3*c*d^5 - 3*A*a^ 2*b*c*d^5 - A*a^3*d^6 + 2*(6*D*b^3*c^5*d - 15*D*a*b^2*c^4*d^2 - 5*C*b^3*c^ 4*d^2 + 12*D*a^2*b*c^3*d^3 + 12*C*a*b^2*c^3*d^3 + 4*B*b^3*c^3*d^3 - 3*D*a^ 3*c^2*d^4 - 9*C*a^2*b*c^2*d^4 - 9*B*a*b^2*c^2*d^4 - 3*A*b^3*c^2*d^4 + 2*C* a^3*c*d^5 + 6*B*a^2*b*c*d^5 + 6*A*a*b^2*c*d^5 - B*a^3*d^6 - 3*A*a^2*b*d^6) *x)/((d*x + c)^2*d^7) + 1/12*(3*D*b^3*d^9*x^4 - 12*D*b^3*c*d^8*x^3 + 12*D* a*b^2*d^9*x^3 + 4*C*b^3*d^9*x^3 + 36*D*b^3*c^2*d^7*x^2 - 54*D*a*b^2*c*d^8* x^2 - 18*C*b^3*c*d^8*x^2 + 18*D*a^2*b*d^9*x^2 + 18*C*a*b^2*d^9*x^2 + 6*B*b ^3*d^9*x^2 - 120*D*b^3*c^3*d^6*x + 216*D*a*b^2*c^2*d^7*x + 72*C*b^3*c^2*d^ 7*x - 108*D*a^2*b*c*d^8*x - 108*C*a*b^2*c*d^8*x - 36*B*b^3*c*d^8*x + 12*D* a^3*d^9*x + 36*C*a^2*b*d^9*x + 36*B*a*b^2*d^9*x + 12*A*b^3*d^9*x)/d^12
Timed out. \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^3,x)
Output:
int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^3, x)
Time = 0.15 (sec) , antiderivative size = 794, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x)
Output:
( - 24*log(c + d*x)*a**3*c**4*d**3 - 48*log(c + d*x)*a**3*c**3*d**4*x - 24 *log(c + d*x)*a**3*c**2*d**5*x**2 + 72*log(c + d*x)*a**2*b**2*c**3*d**3 + 144*log(c + d*x)*a**2*b**2*c**2*d**4*x + 72*log(c + d*x)*a**2*b**2*c*d**5* x**2 + 108*log(c + d*x)*a**2*b*c**5*d**2 + 216*log(c + d*x)*a**2*b*c**4*d* *3*x + 108*log(c + d*x)*a**2*b*c**3*d**4*x**2 - 144*log(c + d*x)*a*b**3*c* *4*d**2 - 288*log(c + d*x)*a*b**3*c**3*d**3*x - 144*log(c + d*x)*a*b**3*c* *2*d**4*x**2 - 144*log(c + d*x)*a*b**2*c**6*d - 288*log(c + d*x)*a*b**2*c* *5*d**2*x - 144*log(c + d*x)*a*b**2*c**4*d**3*x**2 + 72*log(c + d*x)*b**4* c**5*d + 144*log(c + d*x)*b**4*c**4*d**2*x + 72*log(c + d*x)*b**4*c**3*d** 3*x**2 + 60*log(c + d*x)*b**3*c**7 + 120*log(c + d*x)*b**3*c**6*d*x + 60*l og(c + d*x)*b**3*c**5*d**2*x**2 - 6*a**4*c*d**5 + 24*a**3*b*d**6*x**2 - 12 *a**3*c**4*d**3 + 24*a**3*c**2*d**5*x**2 + 12*a**3*c*d**6*x**3 + 36*a**2*b **2*c**3*d**3 - 72*a**2*b**2*c*d**5*x**2 + 54*a**2*b*c**5*d**2 - 108*a**2* b*c**3*d**4*x**2 - 36*a**2*b*c**2*d**5*x**3 + 18*a**2*b*c*d**6*x**4 - 72*a *b**3*c**4*d**2 + 144*a*b**3*c**2*d**4*x**2 + 48*a*b**3*c*d**5*x**3 - 72*a *b**2*c**6*d + 144*a*b**2*c**4*d**3*x**2 + 48*a*b**2*c**3*d**4*x**3 - 12*a *b**2*c**2*d**5*x**4 + 12*a*b**2*c*d**6*x**5 + 36*b**4*c**5*d - 72*b**4*c* *3*d**3*x**2 - 24*b**4*c**2*d**4*x**3 + 6*b**4*c*d**5*x**4 + 30*b**3*c**7 - 60*b**3*c**5*d**2*x**2 - 20*b**3*c**4*d**3*x**3 + 5*b**3*c**3*d**4*x**4 - 2*b**3*c**2*d**5*x**5 + 3*b**3*c*d**6*x**6)/(12*c*d**6*(c**2 + 2*c*d*...