Integrand size = 30, antiderivative size = 338 \[ \int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {(b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{1+n}}{d^6 (1+n)}+\frac {(b c-a d) \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (4 c^2 C d-3 B c d^2+2 A d^3-5 c^3 D\right )\right ) (c+d x)^{2+n}}{d^6 (2+n)}+\frac {\left (a^2 d^2 (C d-3 c D)-2 a b d \left (3 c C d-B d^2-6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^{3+n}}{d^6 (3+n)}+\frac {\left (a^2 d^2 D+2 a b d (C d-4 c D)-b^2 \left (4 c C d-B d^2-10 c^2 D\right )\right ) (c+d x)^{4+n}}{d^6 (4+n)}+\frac {b (b C d-5 b c D+2 a d D) (c+d x)^{5+n}}{d^6 (5+n)}+\frac {b^2 D (c+d x)^{6+n}}{d^6 (6+n)} \]
(-a*d+b*c)^2*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1+n)/d^6/(1+n)+(-a*d+b *c)*(a*d*(-B*d^2+2*C*c*d-3*D*c^2)-b*(2*A*d^3-3*B*c*d^2+4*C*c^2*d-5*D*c^3)) *(d*x+c)^(2+n)/d^6/(2+n)+(a^2*d^2*(C*d-3*D*c)-2*a*b*d*(-B*d^2+3*C*c*d-6*D* c^2)+b^2*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*(d*x+c)^(3+n)/d^6/(3+n)+(a^ 2*d^2*D+2*a*b*d*(C*d-4*D*c)-b^2*(-B*d^2+4*C*c*d-10*D*c^2))*(d*x+c)^(4+n)/d ^6/(4+n)+b*(C*b*d+2*D*a*d-5*D*b*c)*(d*x+c)^(5+n)/d^6/(5+n)+b^2*D*(d*x+c)^( 6+n)/d^6/(6+n)
Time = 0.77 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.91 \[ \int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {(c+d x)^{1+n} \left (\frac {(b c-a d)^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )}{1+n}+\frac {(b c-a d) \left (-a d \left (-2 c C d+B d^2+3 c^2 D\right )+b \left (-4 c^2 C d+3 B c d^2-2 A d^3+5 c^3 D\right )\right ) (c+d x)}{2+n}+\frac {\left (a^2 d^2 (C d-3 c D)+2 a b d \left (-3 c C d+B d^2+6 c^2 D\right )+b^2 \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) (c+d x)^2}{3+n}+\frac {\left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-4 c C d+B d^2+10 c^2 D\right )\right ) (c+d x)^3}{4+n}+\frac {b (b C d-5 b c D+2 a d D) (c+d x)^4}{5+n}+\frac {b^2 D (c+d x)^5}{6+n}\right )}{d^6} \]
((c + d*x)^(1 + n)*(((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(1 + n) + ((b*c - a*d)*(-(a*d*(-2*c*C*d + B*d^2 + 3*c^2*D)) + b*(-4*c^2*C*d + 3*B*c*d^2 - 2*A*d^3 + 5*c^3*D))*(c + d*x))/(2 + n) + ((a^2*d^2*(C*d - 3* c*D) + 2*a*b*d*(-3*c*C*d + B*d^2 + 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^2)/(3 + n) + ((a^2*d^2*D + 2*a*b*d*(C*d - 4* c*D) + b^2*(-4*c*C*d + B*d^2 + 10*c^2*D))*(c + d*x)^3)/(4 + n) + (b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^4)/(5 + n) + (b^2*D*(c + d*x)^5)/(6 + n))) /d^6
Time = 0.58 (sec) , antiderivative size = 338, 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.
\(\displaystyle \int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \int \left (\frac {(c+d x)^{n+2} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^5}+\frac {(c+d x)^{n+3} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^5}+\frac {(a d-b c)^2 (c+d x)^n \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^5}+\frac {(b c-a d) (c+d x)^{n+1} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^5}+\frac {b (c+d x)^{n+4} (2 a d D-5 b c D+b C d)}{d^5}+\frac {b^2 D (c+d x)^{n+5}}{d^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6 (n+3)}+\frac {(c+d x)^{n+4} \left (a^2 d^2 D+2 a b d (C d-4 c D)-\left (b^2 \left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^6 (n+4)}+\frac {(b c-a d)^2 (c+d x)^{n+1} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 (n+1)}+\frac {(b c-a d) (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6 (n+2)}+\frac {b (c+d x)^{n+5} (2 a d D-5 b c D+b C d)}{d^6 (n+5)}+\frac {b^2 D (c+d x)^{n+6}}{d^6 (n+6)}\) |
((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^6 *(1 + n)) + ((b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 + 2*A*d^3 - 5*c^3*D))*(c + d*x)^(2 + n))/(d^6*(2 + n)) + ((a^2* d^2*(C*d - 3*c*D) - 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*c^3*D))*(c + d*x)^(3 + n))/(d^6*(3 + n)) + ((a^2*d ^2*D + 2*a*b*d*(C*d - 4*c*D) - b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x) ^(4 + n))/(d^6*(4 + n)) + (b*(b*C*d - 5*b*c*D + 2*a*d*D)*(c + d*x)^(5 + n) )/(d^6*(5 + n)) + (b^2*D*(c + d*x)^(6 + n))/(d^6*(6 + n))
3.1.26.3.1 Defintions of rubi rules used
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])
Leaf count of result is larger than twice the leaf count of optimal. \(2174\) vs. \(2(338)=676\).
Time = 1.69 (sec) , antiderivative size = 2175, normalized size of antiderivative = 6.43
method | result | size |
norman | \(\text {Expression too large to display}\) | \(2175\) |
gosper | \(\text {Expression too large to display}\) | \(2588\) |
parallelrisch | \(\text {Expression too large to display}\) | \(5150\) |
D*b^2/(6+n)*x^6*exp(n*ln(d*x+c))+c*(A*a^2*d^5*n^5+20*A*a^2*d^5*n^4-2*A*a*b *c*d^4*n^4-B*a^2*c*d^4*n^4+155*A*a^2*d^5*n^3-36*A*a*b*c*d^4*n^3+2*A*b^2*c^ 2*d^3*n^3-18*B*a^2*c*d^4*n^3+4*B*a*b*c^2*d^3*n^3+2*C*a^2*c^2*d^3*n^3+580*A *a^2*d^5*n^2-238*A*a*b*c*d^4*n^2+30*A*b^2*c^2*d^3*n^2-119*B*a^2*c*d^4*n^2+ 60*B*a*b*c^2*d^3*n^2-6*B*b^2*c^3*d^2*n^2+30*C*a^2*c^2*d^3*n^2-12*C*a*b*c^3 *d^2*n^2-6*D*a^2*c^3*d^2*n^2+1044*A*a^2*d^5*n-684*A*a*b*c*d^4*n+148*A*b^2* c^2*d^3*n-342*B*a^2*c*d^4*n+296*B*a*b*c^2*d^3*n-66*B*b^2*c^3*d^2*n+148*C*a ^2*c^2*d^3*n-132*C*a*b*c^3*d^2*n+24*C*b^2*c^4*d*n-66*D*a^2*c^3*d^2*n+48*D* a*b*c^4*d*n+720*A*a^2*d^5-720*A*a*b*c*d^4+240*A*b^2*c^2*d^3-360*B*a^2*c*d^ 4+480*B*a*b*c^2*d^3-180*B*b^2*c^3*d^2+240*C*a^2*c^2*d^3-360*C*a*b*c^3*d^2+ 144*C*b^2*c^4*d-180*D*a^2*c^3*d^2+288*D*a*b*c^4*d-120*D*b^2*c^5)/d^6/(n^6+ 21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)*exp(n*ln(d*x+c))+(B*b^2*d^2*n^ 2+2*C*a*b*d^2*n^2+C*b^2*c*d*n^2+D*a^2*d^2*n^2+2*D*a*b*c*d*n^2+11*B*b^2*d^2 *n+22*C*a*b*d^2*n+6*C*b^2*c*d*n+11*D*a^2*d^2*n+12*D*a*b*c*d*n-5*D*b^2*c^2* n+30*B*b^2*d^2+60*C*a*b*d^2+30*D*a^2*d^2)/d^2/(n^3+15*n^2+74*n+120)*x^4*ex p(n*ln(d*x+c))+(A*b^2*d^3*n^3+2*B*a*b*d^3*n^3+B*b^2*c*d^2*n^3+C*a^2*d^3*n^ 3+2*C*a*b*c*d^2*n^3+D*a^2*c*d^2*n^3+15*A*b^2*d^3*n^2+30*B*a*b*d^3*n^2+11*B *b^2*c*d^2*n^2+15*C*a^2*d^3*n^2+22*C*a*b*c*d^2*n^2-4*C*b^2*c^2*d*n^2+11*D* a^2*c*d^2*n^2-8*D*a*b*c^2*d*n^2+74*A*b^2*d^3*n+148*B*a*b*d^3*n+30*B*b^2*c* d^2*n+74*C*a^2*d^3*n+60*C*a*b*c*d^2*n-24*C*b^2*c^2*d*n+30*D*a^2*c*d^2*n...
Leaf count of result is larger than twice the leaf count of optimal. 2258 vs. \(2 (342) = 684\).
Time = 0.32 (sec) , antiderivative size = 2258, normalized size of antiderivative = 6.68 \[ \int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \]
(A*a^2*c*d^5*n^5 - 120*D*b^2*c^6 + 720*A*a^2*c*d^5 + 240*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 360*(B*a^2 + 2*A*a*b)*c^2*d^4 + (D*b^2*d^6*n^5 + 15*D*b ^2*d^6*n^4 + 85*D*b^2*d^6*n^3 + 225*D*b^2*d^6*n^2 + 274*D*b^2*d^6*n + 120* D*b^2*d^6)*x^6 + (144*(2*D*a*b + C*b^2)*d^6 + (D*b^2*c*d^5 + (2*D*a*b + C* b^2)*d^6)*n^5 + 2*(5*D*b^2*c*d^5 + 8*(2*D*a*b + C*b^2)*d^6)*n^4 + 5*(7*D*b ^2*c*d^5 + 19*(2*D*a*b + C*b^2)*d^6)*n^3 + 10*(5*D*b^2*c*d^5 + 26*(2*D*a*b + C*b^2)*d^6)*n^2 + 12*(2*D*b^2*c*d^5 + 27*(2*D*a*b + C*b^2)*d^6)*n)*x^5 + (20*A*a^2*c*d^5 - (B*a^2 + 2*A*a*b)*c^2*d^4)*n^4 + (180*(D*a^2 + 2*C*a*b + B*b^2)*d^6 + ((D*a^2 + 2*C*a*b + B*b^2)*d^6 + (2*D*a*b*c + C*b^2*c)*d^5 )*n^5 - (5*D*b^2*c^2*d^4 - 17*(D*a^2 + 2*C*a*b + B*b^2)*d^6 - 12*(2*D*a*b* c + C*b^2*c)*d^5)*n^4 - (30*D*b^2*c^2*d^4 - 107*(D*a^2 + 2*C*a*b + B*b^2)* d^6 - 47*(2*D*a*b*c + C*b^2*c)*d^5)*n^3 - (55*D*b^2*c^2*d^4 - 307*(D*a^2 + 2*C*a*b + B*b^2)*d^6 - 72*(2*D*a*b*c + C*b^2*c)*d^5)*n^2 - 6*(5*D*b^2*c^2 *d^4 - 66*(D*a^2 + 2*C*a*b + B*b^2)*d^6 - 6*(2*D*a*b*c + C*b^2*c)*d^5)*n)* x^4 + (155*A*a^2*c*d^5 + 2*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 18*(B*a^2 + 2*A*a*b)*c^2*d^4)*n^3 + (240*(C*a^2 + 2*B*a*b + A*b^2)*d^6 + ((C*a^2 + 2* B*a*b + A*b^2)*d^6 + (D*a^2*c + (2*C*a*b + B*b^2)*c)*d^5)*n^5 + 2*(9*(C*a^ 2 + 2*B*a*b + A*b^2)*d^6 + 7*(D*a^2*c + (2*C*a*b + B*b^2)*c)*d^5 - 2*(2*D* a*b*c^2 + C*b^2*c^2)*d^4)*n^4 + (20*D*b^2*c^3*d^3 + 121*(C*a^2 + 2*B*a*b + A*b^2)*d^6 + 65*(D*a^2*c + (2*C*a*b + B*b^2)*c)*d^5 - 36*(2*D*a*b*c^2 ...
Leaf count of result is larger than twice the leaf count of optimal. 32849 vs. \(2 (328) = 656\).
Time = 6.73 (sec) , antiderivative size = 32849, normalized size of antiderivative = 97.19 \[ \int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \]
Piecewise((c**n*(A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2*x**2/2 + 2 *B*a*b*x**3/3 + B*b**2*x**4/4 + C*a**2*x**3/3 + C*a*b*x**4/2 + C*b**2*x**5 /5 + D*a**2*x**4/4 + 2*D*a*b*x**5/5 + D*b**2*x**6/6), Eq(d, 0)), (-12*A*a* *2*d**5/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d* *9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 6*A*a*b*c*d**4/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10 *x**4 + 60*d**11*x**5) - 30*A*a*b*d**5*x/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x** 5) - 2*A*b**2*c**2*d**3/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x* *2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 10*A*b**2*c* d**4*x/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d** 9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 20*A*b**2*d**5*x**2/(60*c**5* d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d **10*x**4 + 60*d**11*x**5) - 3*B*a**2*c*d**4/(60*c**5*d**6 + 300*c**4*d**7 *x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11 *x**5) - 15*B*a**2*d**5*x/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8* x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 4*B*a*b*c* *2*d**3/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d* *9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 20*B*a*b*c*d**4*x/(60*c**5*d **6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c...
Leaf count of result is larger than twice the leaf count of optimal. 1118 vs. \(2 (342) = 684\).
Time = 0.24 (sec) , antiderivative size = 1118, normalized size of antiderivative = 3.31 \[ \int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \]
(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*B*a^2/((n^2 + 3*n + 2)*d^2) + 2*(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*A*a*b/((n^2 + 3*n + 2)*d ^2) + (d*x + c)^(n + 1)*A*a^2/(d*(n + 1)) + ((n^2 + 3*n + 2)*d^3*x^3 + (n^ 2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n*C*a^2/((n^3 + 6*n^2 + 11*n + 6)*d^3) + 2*((n^2 + 3*n + 2)*d^3*x^3 + (n^2 + n)*c*d^2*x^2 - 2*c^2* d*n*x + 2*c^3)*(d*x + c)^n*B*a*b/((n^3 + 6*n^2 + 11*n + 6)*d^3) + ((n^2 + 3*n + 2)*d^3*x^3 + (n^2 + n)*c*d^2*x^2 - 2*c^2*d*n*x + 2*c^3)*(d*x + c)^n* A*b^2/((n^3 + 6*n^2 + 11*n + 6)*d^3) + ((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x - 6 *c^4)*(d*x + c)^n*D*a^2/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4) + 2*((n^ 3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n )*c^2*d^2*x^2 + 6*c^3*d*n*x - 6*c^4)*(d*x + c)^n*C*a*b/((n^4 + 10*n^3 + 35 *n^2 + 50*n + 24)*d^4) + ((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x - 6*c^4)*(d*x + c )^n*B*b^2/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4) + 2*((n^4 + 10*n^3 + 3 5*n^2 + 50*n + 24)*d^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*c*d^4*x^4 - 4*(n ^3 + 3*n^2 + 2*n)*c^2*d^3*x^3 + 12*(n^2 + n)*c^3*d^2*x^2 - 24*c^4*d*n*x + 24*c^5)*(d*x + c)^n*D*a*b/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120) *d^5) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^5*x^5 + (n^4 + 6*n^3 + 11*n ^2 + 6*n)*c*d^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*c^2*d^3*x^3 + 12*(n^2 + n)*...
Leaf count of result is larger than twice the leaf count of optimal. 4972 vs. \(2 (342) = 684\).
Time = 0.34 (sec) , antiderivative size = 4972, normalized size of antiderivative = 14.71 \[ \int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \]
((d*x + c)^n*D*b^2*d^6*n^5*x^6 + (d*x + c)^n*D*b^2*c*d^5*n^5*x^5 + 2*(d*x + c)^n*D*a*b*d^6*n^5*x^5 + (d*x + c)^n*C*b^2*d^6*n^5*x^5 + 15*(d*x + c)^n* D*b^2*d^6*n^4*x^6 + 2*(d*x + c)^n*D*a*b*c*d^5*n^5*x^4 + (d*x + c)^n*C*b^2* c*d^5*n^5*x^4 + (d*x + c)^n*D*a^2*d^6*n^5*x^4 + 2*(d*x + c)^n*C*a*b*d^6*n^ 5*x^4 + (d*x + c)^n*B*b^2*d^6*n^5*x^4 + 10*(d*x + c)^n*D*b^2*c*d^5*n^4*x^5 + 32*(d*x + c)^n*D*a*b*d^6*n^4*x^5 + 16*(d*x + c)^n*C*b^2*d^6*n^4*x^5 + 8 5*(d*x + c)^n*D*b^2*d^6*n^3*x^6 + (d*x + c)^n*D*a^2*c*d^5*n^5*x^3 + 2*(d*x + c)^n*C*a*b*c*d^5*n^5*x^3 + (d*x + c)^n*B*b^2*c*d^5*n^5*x^3 + (d*x + c)^ n*C*a^2*d^6*n^5*x^3 + 2*(d*x + c)^n*B*a*b*d^6*n^5*x^3 + (d*x + c)^n*A*b^2* d^6*n^5*x^3 - 5*(d*x + c)^n*D*b^2*c^2*d^4*n^4*x^4 + 24*(d*x + c)^n*D*a*b*c *d^5*n^4*x^4 + 12*(d*x + c)^n*C*b^2*c*d^5*n^4*x^4 + 17*(d*x + c)^n*D*a^2*d ^6*n^4*x^4 + 34*(d*x + c)^n*C*a*b*d^6*n^4*x^4 + 17*(d*x + c)^n*B*b^2*d^6*n ^4*x^4 + 35*(d*x + c)^n*D*b^2*c*d^5*n^3*x^5 + 190*(d*x + c)^n*D*a*b*d^6*n^ 3*x^5 + 95*(d*x + c)^n*C*b^2*d^6*n^3*x^5 + 225*(d*x + c)^n*D*b^2*d^6*n^2*x ^6 + (d*x + c)^n*C*a^2*c*d^5*n^5*x^2 + 2*(d*x + c)^n*B*a*b*c*d^5*n^5*x^2 + (d*x + c)^n*A*b^2*c*d^5*n^5*x^2 + (d*x + c)^n*B*a^2*d^6*n^5*x^2 + 2*(d*x + c)^n*A*a*b*d^6*n^5*x^2 - 8*(d*x + c)^n*D*a*b*c^2*d^4*n^4*x^3 - 4*(d*x + c)^n*C*b^2*c^2*d^4*n^4*x^3 + 14*(d*x + c)^n*D*a^2*c*d^5*n^4*x^3 + 28*(d*x + c)^n*C*a*b*c*d^5*n^4*x^3 + 14*(d*x + c)^n*B*b^2*c*d^5*n^4*x^3 + 18*(d*x + c)^n*C*a^2*d^6*n^4*x^3 + 36*(d*x + c)^n*B*a*b*d^6*n^4*x^3 + 18*(d*x +...
Timed out. \[ \int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\int {\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \]