Integrand size = 30, antiderivative size = 400 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\frac {b \left (3 a^2 d^2 D+3 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) x}{d^6}+\frac {b^2 (b C d-4 b c D+3 a d D) x^2}{2 d^5}+\frac {b^3 D x^3}{3 d^4}+\frac {(b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{3 d^7 (c+d x)^3}+\frac {(b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right )}{2 d^7 (c+d x)^2}+\frac {(b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right )}{d^7 (c+d x)}+\frac {\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) \log (c+d x)}{d^7} \] Output:
b*(3*a^2*d^2*D+3*a*b*d*(C*d-4*D*c)-b^2*(-B*d^2+4*C*c*d-10*D*c^2))*x/d^6+1/ 2*b^2*(C*b*d+3*D*a*d-4*D*b*c)*x^2/d^5+1/3*b^3*D*x^3/d^4+1/3*(-a*d+b*c)^3*( A*d^3-B*c*d^2+C*c^2*d-D*c^3)/d^7/(d*x+c)^3+1/2*(-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)^2+(-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))/d^7/(d*x+c)+(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))*ln(d*x+c)/d^7
Time = 0.12 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\frac {6 b d \left (3 a^2 d^2 D+3 a b d (C d-4 c D)+b^2 \left (-4 c C d+B d^2+10 c^2 D\right )\right ) x+3 b^2 d^2 (b C d-4 b c D+3 a d D) x^2+2 b^3 d^3 D x^3-\frac {2 (b c-a d)^3 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )}{(c+d x)^3}+\frac {3 (b c-a d)^2 \left (-a d \left (-2 c C d+B d^2+3 c^2 D\right )+b \left (-5 c^2 C d+4 B c d^2-3 A d^3+6 c^3 D\right )\right )}{(c+d x)^2}-\frac {6 (b c-a d) \left (a^2 d^2 (-C d+3 c D)+a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (-10 c^2 C d+6 B c d^2-3 A d^3+15 c^3 D\right )\right )}{c+d x}+6 \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)+3 a b^2 d \left (-4 c C d+B d^2+10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) \log (c+d x)}{6 d^7} \] Input:
Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^4,x]
Output:
(6*b*d*(3*a^2*d^2*D + 3*a*b*d*(C*d - 4*c*D) + b^2*(-4*c*C*d + B*d^2 + 10*c ^2*D))*x + 3*b^2*d^2*(b*C*d - 4*b*c*D + 3*a*d*D)*x^2 + 2*b^3*d^3*D*x^3 - ( 2*(b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D))/(c + d*x)^3 + (3*( 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)^2 - (6*(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)))/(c + d*x) + 6*(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))*Log[c + d*x])/(6*d^7)
Time = 0.99 (sec) , antiderivative size = 400, 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.
| \(\displaystyle \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(b c-a d) \left (-a^2 d^2 (C d-3 c D)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )-\left (b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{d^6 (c+d x)^2}+\frac {b \left (3 a^2 d^2 D+3 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^6}+\frac {a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )}{d^6 (c+d x)}+\frac {(b c-a d)^2 \left (b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^6 (c+d x)^3}+\frac {(a d-b c)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 (c+d x)^4}+\frac {b^2 x (3 a d D-4 b c D+b C d)}{d^5}+\frac {b^3 D x^2}{d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{d^7 (c+d x)}+\frac {b x \left (3 a^2 d^2 D+3 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^6}+\frac {\log (c+d x) \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{d^7}+\frac {(b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^7 (c+d x)^3}+\frac {(b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{2 d^7 (c+d x)^2}+\frac {b^2 x^2 (3 a d D-4 b c D+b C d)}{2 d^5}+\frac {b^3 D x^3}{3 d^4}\) |
Input:
Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^4,x]
Output:
(b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D) )*x)/d^6 + (b^2*(b*C*d - 4*b*c*D + 3*a*d*D)*x^2)/(2*d^5) + (b^3*D*x^3)/(3* d^4) + ((b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^7*(c + d*x )^3) + ((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)))/(2*d^7*(c + d*x)^2) + ((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)))/(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))*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 = 731, normalized size of antiderivative = 1.83
| method | result | size |
| norman | \(\frac {-\frac {2 A \,a^{3} d^{6}+3 A \,a^{2} b c \,d^{5}+6 A a \,b^{2} c^{2} d^{4}-11 A \,b^{3} c^{3} d^{3}+B \,a^{3} c \,d^{5}+6 B \,a^{2} b \,c^{2} d^{4}-33 B a \,b^{2} c^{3} d^{3}+44 B \,b^{3} c^{4} d^{2}+2 C \,a^{3} c^{2} d^{4}-33 C \,a^{2} b \,c^{3} d^{3}+132 C a \,b^{2} c^{4} d^{2}-110 C \,b^{3} c^{5} d -11 D a^{3} c^{3} d^{3}+132 D a^{2} b \,c^{4} d^{2}-330 D a \,b^{2} c^{5} d +220 D b^{3} c^{6}}{6 d^{7}}-\frac {\left (3 a \,b^{2} A \,d^{4}-3 A \,b^{3} c \,d^{3}+3 B \,a^{2} b \,d^{4}-9 B a \,b^{2} c \,d^{3}+12 B \,b^{3} c^{2} d^{2}+a^{3} C \,d^{4}-9 C \,a^{2} b c \,d^{3}+36 C a \,b^{2} c^{2} d^{2}-30 C \,b^{3} c^{3} d -3 D a^{3} c \,d^{3}+36 D a^{2} b \,c^{2} d^{2}-90 D a \,b^{2} c^{3} d +60 D b^{3} c^{4}\right ) x^{2}}{d^{5}}-\frac {\left (3 A \,a^{2} b \,d^{5}+6 A a \,b^{2} c \,d^{4}-9 A \,b^{3} c^{2} d^{3}+B \,a^{3} d^{5}+6 B \,a^{2} b c \,d^{4}-27 B a \,b^{2} c^{2} d^{3}+36 B \,b^{3} c^{3} d^{2}+2 C \,a^{3} c \,d^{4}-27 C \,a^{2} b \,c^{2} d^{3}+108 C a \,b^{2} c^{3} d^{2}-90 C \,b^{3} c^{4} d -9 D a^{3} c^{2} d^{3}+108 D a^{2} b \,c^{3} d^{2}-270 D a \,b^{2} c^{4} d +180 D b^{3} c^{5}\right ) x}{2 d^{6}}+\frac {D b^{3} x^{6}}{3 d}+\frac {b \left (2 b^{2} B \,d^{2}+6 C a b \,d^{2}-5 C \,b^{2} c d +6 a^{2} d^{2} D-15 D a b c d +10 D b^{2} c^{2}\right ) x^{4}}{2 d^{3}}+\frac {b^{2} \left (C b d +3 D a d -2 D b c \right ) x^{5}}{2 d^{2}}}{\left (x d +c \right )^{3}}+\frac {\left (A \,b^{3} d^{3}+3 B a \,b^{2} d^{3}-4 B \,b^{3} c \,d^{2}+3 a^{2} b C \,d^{3}-12 C a \,b^{2} c \,d^{2}+10 C \,b^{3} c^{2} d +a^{3} d^{3} D-12 D a^{2} b c \,d^{2}+30 D a \,b^{2} c^{2} d -20 D b^{3} c^{3}\right ) \ln \left (x d +c \right )}{d^{7}}\) | \(731\) |
| default | \(\frac {b \left (\frac {1}{3} d^{2} D x^{3} b^{2}+\frac {1}{2} C \,b^{2} d^{2} x^{2}+\frac {3}{2} D a b \,d^{2} x^{2}-2 D b^{2} c d \,x^{2}+b^{2} B \,d^{2} x +3 C a b \,d^{2} x -4 C \,b^{2} c d x +3 a^{2} d^{2} D x -12 D a b c d x +10 D b^{2} c^{2} x \right )}{d^{6}}-\frac {3 a \,b^{2} A \,d^{4}-3 A \,b^{3} c \,d^{3}+3 B \,a^{2} b \,d^{4}-9 B a \,b^{2} c \,d^{3}+6 B \,b^{3} c^{2} d^{2}+a^{3} C \,d^{4}-9 C \,a^{2} b c \,d^{3}+18 C a \,b^{2} c^{2} d^{2}-10 C \,b^{3} c^{3} d -3 D a^{3} c \,d^{3}+18 D a^{2} b \,c^{2} d^{2}-30 D a \,b^{2} c^{3} d +15 D b^{3} c^{4}}{d^{7} \left (x d +c \right )}+\frac {\left (A \,b^{3} d^{3}+3 B a \,b^{2} d^{3}-4 B \,b^{3} c \,d^{2}+3 a^{2} b C \,d^{3}-12 C a \,b^{2} c \,d^{2}+10 C \,b^{3} c^{2} d +a^{3} d^{3} D-12 D a^{2} b c \,d^{2}+30 D a \,b^{2} c^{2} d -20 D b^{3} c^{3}\right ) \ln \left (x d +c \right )}{d^{7}}-\frac {3 A \,a^{2} b \,d^{5}-6 A a \,b^{2} c \,d^{4}+3 A \,b^{3} c^{2} d^{3}+B \,a^{3} d^{5}-6 B \,a^{2} b c \,d^{4}+9 B a \,b^{2} c^{2} d^{3}-4 B \,b^{3} c^{3} d^{2}-2 C \,a^{3} c \,d^{4}+9 C \,a^{2} b \,c^{2} d^{3}-12 C a \,b^{2} c^{3} d^{2}+5 C \,b^{3} c^{4} d +3 D a^{3} c^{2} d^{3}-12 D a^{2} b \,c^{3} d^{2}+15 D a \,b^{2} c^{4} d -6 D b^{3} c^{5}}{2 d^{7} \left (x d +c \right )^{2}}-\frac {A \,a^{3} d^{6}-3 A \,a^{2} b c \,d^{5}+3 A a \,b^{2} c^{2} d^{4}-A \,b^{3} c^{3} d^{3}-B \,a^{3} c \,d^{5}+3 B \,a^{2} b \,c^{2} d^{4}-3 B a \,b^{2} c^{3} d^{3}+B \,b^{3} c^{4} d^{2}+C \,a^{3} c^{2} d^{4}-3 C \,a^{2} b \,c^{3} d^{3}+3 C a \,b^{2} c^{4} d^{2}-C \,b^{3} c^{5} d -D a^{3} c^{3} d^{3}+3 D a^{2} b \,c^{4} d^{2}-3 D a \,b^{2} c^{5} d +D b^{3} c^{6}}{3 d^{7} \left (x d +c \right )^{3}}\) | \(745\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1479\) |
Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x,method=_RETURNVERBOSE)
Output:
(-1/6*(2*A*a^3*d^6+3*A*a^2*b*c*d^5+6*A*a*b^2*c^2*d^4-11*A*b^3*c^3*d^3+B*a^ 3*c*d^5+6*B*a^2*b*c^2*d^4-33*B*a*b^2*c^3*d^3+44*B*b^3*c^4*d^2+2*C*a^3*c^2* d^4-33*C*a^2*b*c^3*d^3+132*C*a*b^2*c^4*d^2-110*C*b^3*c^5*d-11*D*a^3*c^3*d^ 3+132*D*a^2*b*c^4*d^2-330*D*a*b^2*c^5*d+220*D*b^3*c^6)/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+12*B*b^3*c^2*d^2+C*a^3*d^4-9*C *a^2*b*c*d^3+36*C*a*b^2*c^2*d^2-30*C*b^3*c^3*d-3*D*a^3*c*d^3+36*D*a^2*b*c^ 2*d^2-90*D*a*b^2*c^3*d+60*D*b^3*c^4)/d^5*x^2-1/2*(3*A*a^2*b*d^5+6*A*a*b^2* c*d^4-9*A*b^3*c^2*d^3+B*a^3*d^5+6*B*a^2*b*c*d^4-27*B*a*b^2*c^2*d^3+36*B*b^ 3*c^3*d^2+2*C*a^3*c*d^4-27*C*a^2*b*c^2*d^3+108*C*a*b^2*c^3*d^2-90*C*b^3*c^ 4*d-9*D*a^3*c^2*d^3+108*D*a^2*b*c^3*d^2-270*D*a*b^2*c^4*d+180*D*b^3*c^5)/d ^6*x+1/3*D*b^3/d*x^6+1/2*b*(2*B*b^2*d^2+6*C*a*b*d^2-5*C*b^2*c*d+6*D*a^2*d^ 2-15*D*a*b*c*d+10*D*b^2*c^2)/d^3*x^4+1/2*b^2*(C*b*d+3*D*a*d-2*D*b*c)/d^2*x ^5)/(d*x+c)^3+1/d^7*(A*b^3*d^3+3*B*a*b^2*d^3-4*B*b^3*c*d^2+3*C*a^2*b*d^3-1 2*C*a*b^2*c*d^2+10*C*b^3*c^2*d+D*a^3*d^3-12*D*a^2*b*c*d^2+30*D*a*b^2*c^2*d -20*D*b^3*c^3)*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (395) = 790\).
Time = 0.09 (sec) , antiderivative size = 1003, normalized size of antiderivative = 2.51 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x, algorithm="fricas")
Output:
1/6*(2*D*b^3*d^6*x^6 - 74*D*b^3*c^6 - 2*A*a^3*d^6 + 47*(3*D*a*b^2 + C*b^3) *c^5*d - 26*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 + 11*(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 - 3*(2*D*b^3*c*d^5 - (3*D*a*b^2 + C*b^3)*d^6) *x^5 + 3*(10*D*b^3*c^2*d^4 - 5*(3*D*a*b^2 + C*b^3)*c*d^5 + 2*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^6)*x^4 + (146*D*b^3*c^3*d^3 - 63*(3*D*a*b^2 + C*b^3)* c^2*d^4 + 18*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^5)*x^3 + 3*(26*D*b^3*c^4* d^2 - 3*(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 + 6*(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 - 3*(34*D*b^3*c^5*d - 27*(3*D*a*b^2 + C*b^3)*c^ 4*d^2 + 18*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^3 - 9*(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 - 6*(20*D*b^3*c^6 - 10*(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 - (D*a^3 + 3*C*a^2*b + 3*B*a* b^2 + A*b^3)*c^3*d^3 + (20*D*b^3*c^3*d^3 - 10*(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 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^6)*x^3 + 3*(20*D*b^3*c^4*d^2 - 10*(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 - (D*a^3 + 3*C*a^2*b + 3*B*a*b ^2 + A*b^3)*c*d^5)*x^2 + 3*(20*D*b^3*c^5*d - 10*(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 - (D*a^3 + 3*C*a^2*b + 3*...
Timed out. \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**4,x)
Output:
Timed out
Time = 0.06 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=-\frac {74 \, D b^{3} c^{6} + 2 \, A a^{3} d^{6} - 47 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + 26 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - 11 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} + {\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} + 6 \, {\left (15 \, D b^{3} c^{4} d^{2} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d^{3} + 6 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{4} - 3 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{5} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6}\right )} x^{2} + 3 \, {\left (54 \, D b^{3} c^{5} d - 35 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d^{2} + 20 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{3} - 9 \, {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{4} + 2 \, {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6}\right )} x}{6 \, {\left (d^{10} x^{3} + 3 \, c d^{9} x^{2} + 3 \, c^{2} d^{8} x + c^{3} d^{7}\right )}} + \frac {2 \, D b^{3} d^{2} x^{3} - 3 \, {\left (4 \, D b^{3} c d - {\left (3 \, D a b^{2} + C b^{3}\right )} d^{2}\right )} x^{2} + 6 \, {\left (10 \, D b^{3} c^{2} - 4 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c d + {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )} x}{6 \, d^{6}} - \frac {{\left (20 \, D b^{3} c^{3} - 10 \, {\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \, {\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} - {\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )} \log \left (d x + c\right )}{d^{7}} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x, algorithm="maxima")
Output:
-1/6*(74*D*b^3*c^6 + 2*A*a^3*d^6 - 47*(3*D*a*b^2 + C*b^3)*c^5*d + 26*(3*D* a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 - 11*(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*(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 + 3*(54*D*b^3*c^5* d - 35*(3*D*a*b^2 + C*b^3)*c^4*d^2 + 20*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^ 3*d^3 - 9*(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^10*x^3 + 3*c*d^ 9*x^2 + 3*c^2*d^8*x + c^3*d^7) + 1/6*(2*D*b^3*d^2*x^3 - 3*(4*D*b^3*c*d - ( 3*D*a*b^2 + C*b^3)*d^2)*x^2 + 6*(10*D*b^3*c^2 - 4*(3*D*a*b^2 + C*b^3)*c*d + (3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^2)*x)/d^6 - (20*D*b^3*c^3 - 10*(3*D*a* b^2 + C*b^3)*c^2*d + 4*(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)*log(d*x + c)/d^7
Time = 0.13 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=-\frac {{\left (20 \, D b^{3} c^{3} - 30 \, D a b^{2} c^{2} d - 10 \, C b^{3} c^{2} d + 12 \, D a^{2} b c d^{2} + 12 \, C a b^{2} c d^{2} + 4 \, B b^{3} c d^{2} - D a^{3} d^{3} - 3 \, C a^{2} b d^{3} - 3 \, B a b^{2} d^{3} - A b^{3} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{7}} - \frac {74 \, D b^{3} c^{6} - 141 \, D a b^{2} c^{5} d - 47 \, C b^{3} c^{5} d + 78 \, D a^{2} b c^{4} d^{2} + 78 \, C a b^{2} c^{4} d^{2} + 26 \, B b^{3} c^{4} d^{2} - 11 \, D a^{3} c^{3} d^{3} - 33 \, C a^{2} b c^{3} d^{3} - 33 \, B a b^{2} c^{3} d^{3} - 11 \, A b^{3} c^{3} d^{3} + 2 \, C a^{3} c^{2} d^{4} + 6 \, B a^{2} b c^{2} d^{4} + 6 \, A a b^{2} c^{2} d^{4} + B a^{3} c d^{5} + 3 \, A a^{2} b c d^{5} + 2 \, A a^{3} d^{6} + 6 \, {\left (15 \, D b^{3} c^{4} d^{2} - 30 \, D a b^{2} c^{3} d^{3} - 10 \, C b^{3} c^{3} d^{3} + 18 \, D a^{2} b c^{2} d^{4} + 18 \, C a b^{2} c^{2} d^{4} + 6 \, B b^{3} c^{2} d^{4} - 3 \, D a^{3} c d^{5} - 9 \, C a^{2} b c d^{5} - 9 \, B a b^{2} c d^{5} - 3 \, A b^{3} c d^{5} + C a^{3} d^{6} + 3 \, B a^{2} b d^{6} + 3 \, A a b^{2} d^{6}\right )} x^{2} + 3 \, {\left (54 \, D b^{3} c^{5} d - 105 \, D a b^{2} c^{4} d^{2} - 35 \, C b^{3} c^{4} d^{2} + 60 \, D a^{2} b c^{3} d^{3} + 60 \, C a b^{2} c^{3} d^{3} + 20 \, B b^{3} c^{3} d^{3} - 9 \, D a^{3} c^{2} d^{4} - 27 \, C a^{2} b c^{2} d^{4} - 27 \, B a b^{2} c^{2} d^{4} - 9 \, A b^{3} c^{2} d^{4} + 2 \, C a^{3} c d^{5} + 6 \, B a^{2} b c d^{5} + 6 \, A a b^{2} c d^{5} + B a^{3} d^{6} + 3 \, A a^{2} b d^{6}\right )} x}{6 \, {\left (d x + c\right )}^{3} d^{7}} + \frac {2 \, D b^{3} d^{8} x^{3} - 12 \, D b^{3} c d^{7} x^{2} + 9 \, D a b^{2} d^{8} x^{2} + 3 \, C b^{3} d^{8} x^{2} + 60 \, D b^{3} c^{2} d^{6} x - 72 \, D a b^{2} c d^{7} x - 24 \, C b^{3} c d^{7} x + 18 \, D a^{2} b d^{8} x + 18 \, C a b^{2} d^{8} x + 6 \, B b^{3} d^{8} x}{6 \, d^{12}} \] Input:
integrate((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x, algorithm="giac")
Output:
-(20*D*b^3*c^3 - 30*D*a*b^2*c^2*d - 10*C*b^3*c^2*d + 12*D*a^2*b*c*d^2 + 12 *C*a*b^2*c*d^2 + 4*B*b^3*c*d^2 - D*a^3*d^3 - 3*C*a^2*b*d^3 - 3*B*a*b^2*d^3 - A*b^3*d^3)*log(abs(d*x + c))/d^7 - 1/6*(74*D*b^3*c^6 - 141*D*a*b^2*c^5* d - 47*C*b^3*c^5*d + 78*D*a^2*b*c^4*d^2 + 78*C*a*b^2*c^4*d^2 + 26*B*b^3*c^ 4*d^2 - 11*D*a^3*c^3*d^3 - 33*C*a^2*b*c^3*d^3 - 33*B*a*b^2*c^3*d^3 - 11*A* b^3*c^3*d^3 + 2*C*a^3*c^2*d^4 + 6*B*a^2*b*c^2*d^4 + 6*A*a*b^2*c^2*d^4 + B* a^3*c*d^5 + 3*A*a^2*b*c*d^5 + 2*A*a^3*d^6 + 6*(15*D*b^3*c^4*d^2 - 30*D*a*b ^2*c^3*d^3 - 10*C*b^3*c^3*d^3 + 18*D*a^2*b*c^2*d^4 + 18*C*a*b^2*c^2*d^4 + 6*B*b^3*c^2*d^4 - 3*D*a^3*c*d^5 - 9*C*a^2*b*c*d^5 - 9*B*a*b^2*c*d^5 - 3*A* b^3*c*d^5 + C*a^3*d^6 + 3*B*a^2*b*d^6 + 3*A*a*b^2*d^6)*x^2 + 3*(54*D*b^3*c ^5*d - 105*D*a*b^2*c^4*d^2 - 35*C*b^3*c^4*d^2 + 60*D*a^2*b*c^3*d^3 + 60*C* a*b^2*c^3*d^3 + 20*B*b^3*c^3*d^3 - 9*D*a^3*c^2*d^4 - 27*C*a^2*b*c^2*d^4 - 27*B*a*b^2*c^2*d^4 - 9*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)^3*d^7) + 1/6*(2* D*b^3*d^8*x^3 - 12*D*b^3*c*d^7*x^2 + 9*D*a*b^2*d^8*x^2 + 3*C*b^3*d^8*x^2 + 60*D*b^3*c^2*d^6*x - 72*D*a*b^2*c*d^7*x - 24*C*b^3*c*d^7*x + 18*D*a^2*b*d ^8*x + 18*C*a*b^2*d^8*x + 6*B*b^3*d^8*x)/d^12
Timed out. \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\int \frac {{\left (a+b\,x\right )}^3\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^4} \,d x \] Input:
int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^4,x)
Output:
int(((a + b*x)^3*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^4, x)
Time = 0.15 (sec) , antiderivative size = 844, normalized size of antiderivative = 2.11 \[ \int \frac {(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x)
Output:
(6*log(c + d*x)*a**3*c**4*d**3 + 18*log(c + d*x)*a**3*c**3*d**4*x + 18*log (c + d*x)*a**3*c**2*d**5*x**2 + 6*log(c + d*x)*a**3*c*d**6*x**3 - 54*log(c + d*x)*a**2*b*c**5*d**2 - 162*log(c + d*x)*a**2*b*c**4*d**3*x - 162*log(c + d*x)*a**2*b*c**3*d**4*x**2 - 54*log(c + d*x)*a**2*b*c**2*d**5*x**3 + 24 *log(c + d*x)*a*b**3*c**4*d**2 + 72*log(c + d*x)*a*b**3*c**3*d**3*x + 72*l og(c + d*x)*a*b**3*c**2*d**4*x**2 + 24*log(c + d*x)*a*b**3*c*d**5*x**3 + 1 08*log(c + d*x)*a*b**2*c**6*d + 324*log(c + d*x)*a*b**2*c**5*d**2*x + 324* log(c + d*x)*a*b**2*c**4*d**3*x**2 + 108*log(c + d*x)*a*b**2*c**3*d**4*x** 3 - 24*log(c + d*x)*b**4*c**5*d - 72*log(c + d*x)*b**4*c**4*d**2*x - 72*lo g(c + d*x)*b**4*c**3*d**3*x**2 - 24*log(c + d*x)*b**4*c**2*d**4*x**3 - 60* log(c + d*x)*b**3*c**7 - 180*log(c + d*x)*b**3*c**6*d*x - 180*log(c + d*x) *b**3*c**5*d**2*x**2 - 60*log(c + d*x)*b**3*c**4*d**3*x**3 - 2*a**4*c*d**5 - 4*a**3*b*c**2*d**4 - 12*a**3*b*c*d**5*x + 5*a**3*c**4*d**3 + 9*a**3*c** 3*d**4*x - 4*a**3*c*d**6*x**3 + 12*a**2*b**2*d**6*x**3 - 45*a**2*b*c**5*d* *2 - 81*a**2*b*c**4*d**3*x + 54*a**2*b*c**2*d**5*x**3 + 18*a**2*b*c*d**6*x **4 + 20*a*b**3*c**4*d**2 + 36*a*b**3*c**3*d**3*x - 24*a*b**3*c*d**5*x**3 + 90*a*b**2*c**6*d + 162*a*b**2*c**5*d**2*x - 108*a*b**2*c**3*d**4*x**3 - 27*a*b**2*c**2*d**5*x**4 + 9*a*b**2*c*d**6*x**5 - 20*b**4*c**5*d - 36*b**4 *c**4*d**2*x + 24*b**4*c**2*d**4*x**3 + 6*b**4*c*d**5*x**4 - 50*b**3*c**7 - 90*b**3*c**6*d*x + 60*b**3*c**4*d**3*x**3 + 15*b**3*c**3*d**4*x**4 - ...