Integrand size = 30, antiderivative size = 455 \[ \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {(b c-a d)^3 \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) (c+d x)^{1+n}}{d^7 (1+n)}-\frac {(b c-a d)^2 \left (a d \left (2 c C d-B d^2-3 c^2 D\right )-b \left (5 c^2 C d-4 B c d^2+3 A d^3-6 c^3 D\right )\right ) (c+d x)^{2+n}}{d^7 (2+n)}-\frac {(b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) (c+d x)^{3+n}}{d^7 (3+n)}+\frac {\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (4 c C d-B d^2-10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^{4+n}}{d^7 (4+n)}+\frac {b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-b^2 \left (5 c C d-B d^2-15 c^2 D\right )\right ) (c+d x)^{5+n}}{d^7 (5+n)}+\frac {b^2 (b C d-6 b c D+3 a d D) (c+d x)^{6+n}}{d^7 (6+n)}+\frac {b^3 D (c+d x)^{7+n}}{d^7 (7+n)} \]
-(-a*d+b*c)^3*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(d*x+c)^(1+n)/d^7/(1+n)-(-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*x+c)^(2+n)/d^7/(2+n)-(-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*x+c)^( 3+n)/d^7/(3+n)+(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)^(4+n)/d^7/( 4+n)+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)^(5+n)/d^7/(5+n)+b^2*(C*b*d+3*D*a*d-6*D*b*c)*(d*x+c)^(6+n)/d^7/(6+n)+b ^3*D*(d*x+c)^(7+n)/d^7/(7+n)
Time = 2.03 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.92 \[ \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {(c+d x)^{1+n} \left (\frac {(b c-a d)^3 \left (-c^2 C d+B c d^2-A d^3+c^3 D\right )}{1+n}-\frac {(b c-a d)^2 \left (-a d \left (-2 c C d+B d^2+3 c^2 D\right )+b \left (-5 c^2 C d+4 B c d^2-3 A d^3+6 c^3 D\right )\right ) (c+d x)}{2+n}+\frac {(b c-a d) \left (a^2 d^2 (-C d+3 c D)+a b d \left (8 c C d-3 B d^2-15 c^2 D\right )+b^2 \left (-10 c^2 C d+6 B c d^2-3 A d^3+15 c^3 D\right )\right ) (c+d x)^2}{3+n}+\frac {\left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)+3 a b^2 d \left (-4 c C d+B d^2+10 c^2 D\right )+b^3 \left (10 c^2 C d-4 B c d^2+A d^3-20 c^3 D\right )\right ) (c+d x)^3}{4+n}+\frac {b \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-5 c C d+B d^2+15 c^2 D\right )\right ) (c+d x)^4}{5+n}+\frac {b^2 (b C d-6 b c D+3 a d D) (c+d x)^5}{6+n}+\frac {b^3 D (c+d x)^6}{7+n}\right )}{d^7} \]
((c + d*x)^(1 + n)*(((b*c - a*d)^3*(-(c^2*C*d) + B*c*d^2 - A*d^3 + c^3*D)) /(1 + n) - ((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 + n) + ((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)^2)/(3 + n) + ((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)^3)/(4 + n) + (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)^4)/(5 + n) + (b^2*(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^5)/( 6 + n) + (b^3*D*(c + d*x)^6)/(7 + n)))/d^7
Time = 0.75 (sec) , antiderivative size = 455, 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.
| \(\displaystyle \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {(b c-a d) (c+d x)^{n+2} \left (-a^2 d^2 (C d-3 c D)+a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )-\left (b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{d^6}+\frac {b (c+d x)^{n+4} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^6}+\frac {(c+d x)^{n+3} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{d^6}+\frac {(a d-b c)^3 (c+d x)^n \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6}+\frac {(b c-a d)^2 (c+d x)^{n+1} \left (b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )-a d \left (-B d^2-3 c^2 D+2 c C d\right )\right )}{d^6}+\frac {b^2 (c+d x)^{n+5} (3 a d D-6 b c D+b C d)}{d^6}+\frac {b^3 D (c+d x)^{n+6}}{d^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(b c-a d) (c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{d^7 (n+3)}+\frac {b (c+d x)^{n+5} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)-\left (b^2 \left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{d^7 (n+5)}+\frac {(c+d x)^{n+4} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{d^7 (n+4)}-\frac {(b c-a d)^3 (c+d x)^{n+1} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 (n+1)}-\frac {(b c-a d)^2 (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7 (n+2)}+\frac {b^2 (c+d x)^{n+6} (3 a d D-6 b c D+b C d)}{d^7 (n+6)}+\frac {b^3 D (c+d x)^{n+7}}{d^7 (n+7)}\) |
-(((b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d ^7*(1 + n))) - ((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 + n))/(d^7*(2 + n)) - ( (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)^(3 + n))/(d ^7*(3 + n)) + ((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)^(4 + n))/(d^7*(4 + n)) + (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)^(5 + n))/(d^7*(5 + n)) + (b^2 *(b*C*d - 6*b*c*D + 3*a*d*D)*(c + d*x)^(6 + n))/(d^7*(6 + n)) + (b^3*D*(c + d*x)^(7 + n))/(d^7*(7 + n))
3.1.25.3.1 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. \(3931\) vs. \(2(455)=910\).
Time = 1.77 (sec) , antiderivative size = 3932, normalized size of antiderivative = 8.64
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(3932\) |
| gosper | \(\text {Expression too large to display}\) | \(5003\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(9319\) |
D*b^3/(7+n)*x^7*exp(n*ln(d*x+c))+c*(A*a^3*d^6*n^6+27*A*a^3*d^6*n^5-3*A*a^2 *b*c*d^5*n^5-B*a^3*c*d^5*n^5+295*A*a^3*d^6*n^4-75*A*a^2*b*c*d^5*n^4+6*A*a* b^2*c^2*d^4*n^4-25*B*a^3*c*d^5*n^4+6*B*a^2*b*c^2*d^4*n^4+2*C*a^3*c^2*d^4*n ^4+1665*A*a^3*d^6*n^3-735*A*a^2*b*c*d^5*n^3+132*A*a*b^2*c^2*d^4*n^3-6*A*b^ 3*c^3*d^3*n^3-245*B*a^3*c*d^5*n^3+132*B*a^2*b*c^2*d^4*n^3-18*B*a*b^2*c^3*d ^3*n^3+44*C*a^3*c^2*d^4*n^3-18*C*a^2*b*c^3*d^3*n^3-6*D*a^3*c^3*d^3*n^3+510 4*A*a^3*d^6*n^2-3525*A*a^2*b*c*d^5*n^2+1074*A*a*b^2*c^2*d^4*n^2-108*A*b^3* c^3*d^3*n^2-1175*B*a^3*c*d^5*n^2+1074*B*a^2*b*c^2*d^4*n^2-324*B*a*b^2*c^3* d^3*n^2+24*B*b^3*c^4*d^2*n^2+358*C*a^3*c^2*d^4*n^2-324*C*a^2*b*c^3*d^3*n^2 +72*C*a*b^2*c^4*d^2*n^2-108*D*a^3*c^3*d^3*n^2+72*D*a^2*b*c^4*d^2*n^2+8028* A*a^3*d^6*n-8262*A*a^2*b*c*d^5*n+3828*A*a*b^2*c^2*d^4*n-642*A*b^3*c^3*d^3* n-2754*B*a^3*c*d^5*n+3828*B*a^2*b*c^2*d^4*n-1926*B*a*b^2*c^3*d^3*n+312*B*b ^3*c^4*d^2*n+1276*C*a^3*c^2*d^4*n-1926*C*a^2*b*c^3*d^3*n+936*C*a*b^2*c^4*d ^2*n-120*C*b^3*c^5*d*n-642*D*a^3*c^3*d^3*n+936*D*a^2*b*c^4*d^2*n-360*D*a*b ^2*c^5*d*n+5040*A*a^3*d^6-7560*A*a^2*b*c*d^5+5040*A*a*b^2*c^2*d^4-1260*A*b ^3*c^3*d^3-2520*B*a^3*c*d^5+5040*B*a^2*b*c^2*d^4-3780*B*a*b^2*c^3*d^3+1008 *B*b^3*c^4*d^2+1680*C*a^3*c^2*d^4-3780*C*a^2*b*c^3*d^3+3024*C*a*b^2*c^4*d^ 2-840*C*b^3*c^5*d-1260*D*a^3*c^3*d^3+3024*D*a^2*b*c^4*d^2-2520*D*a*b^2*c^5 *d+720*D*b^3*c^6)/d^7/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+1306 8*n+5040)*exp(n*ln(d*x+c))+(A*b^3*d^3*n^3+3*B*a*b^2*d^3*n^3+B*b^3*c*d^2...
Leaf count of result is larger than twice the leaf count of optimal. 4115 vs. \(2 (455) = 910\).
Time = 0.36 (sec) , antiderivative size = 4115, normalized size of antiderivative = 9.04 \[ \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \]
(A*a^3*c*d^6*n^6 + 720*D*b^3*c^7 + 5040*A*a^3*c*d^6 + 1680*(C*a^3 + 3*B*a^ 2*b + 3*A*a*b^2)*c^3*d^4 - 2520*(B*a^3 + 3*A*a^2*b)*c^2*d^5 + (D*b^3*d^7*n ^6 + 21*D*b^3*d^7*n^5 + 175*D*b^3*d^7*n^4 + 735*D*b^3*d^7*n^3 + 1624*D*b^3 *d^7*n^2 + 1764*D*b^3*d^7*n + 720*D*b^3*d^7)*x^7 + (840*(3*D*a*b^2 + C*b^3 )*d^7 + (D*b^3*c*d^6 + (3*D*a*b^2 + C*b^3)*d^7)*n^6 + (15*D*b^3*c*d^6 + 22 *(3*D*a*b^2 + C*b^3)*d^7)*n^5 + 5*(17*D*b^3*c*d^6 + 38*(3*D*a*b^2 + C*b^3) *d^7)*n^4 + 5*(45*D*b^3*c*d^6 + 164*(3*D*a*b^2 + C*b^3)*d^7)*n^3 + (274*D* b^3*c*d^6 + 1849*(3*D*a*b^2 + C*b^3)*d^7)*n^2 + 2*(60*D*b^3*c*d^6 + 1019*( 3*D*a*b^2 + C*b^3)*d^7)*n)*x^6 + (27*A*a^3*c*d^6 - (B*a^3 + 3*A*a^2*b)*c^2 *d^5)*n^5 + (1008*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 + ((3*D*a^2*b + 3*C* a*b^2 + B*b^3)*d^7 + (3*D*a*b^2*c + C*b^3*c)*d^6)*n^6 - (6*D*b^3*c^2*d^5 - 23*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 - 17*(3*D*a*b^2*c + C*b^3*c)*d^6)* n^5 - 3*(20*D*b^3*c^2*d^5 - 69*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 - 35*(3 *D*a*b^2*c + C*b^3*c)*d^6)*n^4 - 5*(42*D*b^3*c^2*d^5 - 185*(3*D*a^2*b + 3* C*a*b^2 + B*b^3)*d^7 - 59*(3*D*a*b^2*c + C*b^3*c)*d^6)*n^3 - 2*(150*D*b^3* c^2*d^5 - 1072*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^7 - 187*(3*D*a*b^2*c + C* b^3*c)*d^6)*n^2 - 12*(12*D*b^3*c^2*d^5 - 201*(3*D*a^2*b + 3*C*a*b^2 + B*b^ 3)*d^7 - 14*(3*D*a*b^2*c + C*b^3*c)*d^6)*n)*x^5 + (295*A*a^3*c*d^6 + 2*(C* a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^3*d^4 - 25*(B*a^3 + 3*A*a^2*b)*c^2*d^5)*n^4 + (1260*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^7 + ((D*a^3 + 3*C*a^...
Leaf count of result is larger than twice the leaf count of optimal. 65321 vs. \(2 (445) = 890\).
Time = 11.92 (sec) , antiderivative size = 65321, normalized size of antiderivative = 143.56 \[ \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \]
Piecewise((c**n*(A*a**3*x + 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3*x** 4/4 + B*a**3*x**2/2 + B*a**2*b*x**3 + 3*B*a*b**2*x**4/4 + B*b**3*x**5/5 + C*a**3*x**3/3 + 3*C*a**2*b*x**4/4 + 3*C*a*b**2*x**5/5 + C*b**3*x**6/6 + D* a**3*x**4/4 + 3*D*a**2*b*x**5/5 + D*a*b**2*x**6/2 + D*b**3*x**7/7), Eq(d, 0)), (-10*A*a**3*d**6/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360*c*d**12*x**5 + 60*d**1 3*x**6) - 6*A*a**2*b*c*d**5/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c**4*d** 9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360*c*d**12*x**5 + 6 0*d**13*x**6) - 36*A*a**2*b*d**6*x/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c **4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360*c*d**12*x **5 + 60*d**13*x**6) - 3*A*a*b**2*c**2*d**4/(60*c**6*d**7 + 360*c**5*d**8* x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x**4 + 360* c*d**12*x**5 + 60*d**13*x**6) - 18*A*a*b**2*c*d**5*x/(60*c**6*d**7 + 360*c **5*d**8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c**2*d**11*x* *4 + 360*c*d**12*x**5 + 60*d**13*x**6) - 45*A*a*b**2*d**6*x**2/(60*c**6*d* *7 + 360*c**5*d**8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 900*c** 2*d**11*x**4 + 360*c*d**12*x**5 + 60*d**13*x**6) - A*b**3*c**3*d**3/(60*c* *6*d**7 + 360*c**5*d**8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**10*x**3 + 90 0*c**2*d**11*x**4 + 360*c*d**12*x**5 + 60*d**13*x**6) - 6*A*b**3*c**2*d**4 *x/(60*c**6*d**7 + 360*c**5*d**8*x + 900*c**4*d**9*x**2 + 1200*c**3*d**...
Leaf count of result is larger than twice the leaf count of optimal. 1802 vs. \(2 (455) = 910\).
Time = 0.27 (sec) , antiderivative size = 1802, normalized size of antiderivative = 3.96 \[ \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \]
(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*B*a^3/((n^2 + 3*n + 2)*d^2) + 3*(d^2*(n + 1)*x^2 + c*d*n*x - c^2)*(d*x + c)^n*A*a^2*b/((n^2 + 3*n + 2) *d^2) + (d*x + c)^(n + 1)*A*a^3/(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^3/((n^3 + 6*n^2 + 11*n + 6)*d^3) + 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*B*a^2*b/((n^3 + 6*n^2 + 11*n + 6)*d^3) + 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*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^3/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4) + 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*C*a^2*b/((n^4 + 1 0*n^3 + 35*n^2 + 50*n + 24)*d^4) + 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*B*a*b^2/((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*A*b^3/((n^4 + 10*n^3 + 35*n ^2 + 50*n + 24)*d^4) + 3*((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)*c^3*d^2*x^2 - 24*c^4*d*n*x + 24*c^5)*(d*x + c)^n*D*a^2*b/...
Leaf count of result is larger than twice the leaf count of optimal. 9032 vs. \(2 (455) = 910\).
Time = 0.35 (sec) , antiderivative size = 9032, normalized size of antiderivative = 19.85 \[ \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \]
((d*x + c)^n*D*b^3*d^7*n^6*x^7 + (d*x + c)^n*D*b^3*c*d^6*n^6*x^6 + 3*(d*x + c)^n*D*a*b^2*d^7*n^6*x^6 + (d*x + c)^n*C*b^3*d^7*n^6*x^6 + 21*(d*x + c)^ n*D*b^3*d^7*n^5*x^7 + 3*(d*x + c)^n*D*a*b^2*c*d^6*n^6*x^5 + (d*x + c)^n*C* b^3*c*d^6*n^6*x^5 + 3*(d*x + c)^n*D*a^2*b*d^7*n^6*x^5 + 3*(d*x + c)^n*C*a* b^2*d^7*n^6*x^5 + (d*x + c)^n*B*b^3*d^7*n^6*x^5 + 15*(d*x + c)^n*D*b^3*c*d ^6*n^5*x^6 + 66*(d*x + c)^n*D*a*b^2*d^7*n^5*x^6 + 22*(d*x + c)^n*C*b^3*d^7 *n^5*x^6 + 175*(d*x + c)^n*D*b^3*d^7*n^4*x^7 + 3*(d*x + c)^n*D*a^2*b*c*d^6 *n^6*x^4 + 3*(d*x + c)^n*C*a*b^2*c*d^6*n^6*x^4 + (d*x + c)^n*B*b^3*c*d^6*n ^6*x^4 + (d*x + c)^n*D*a^3*d^7*n^6*x^4 + 3*(d*x + c)^n*C*a^2*b*d^7*n^6*x^4 + 3*(d*x + c)^n*B*a*b^2*d^7*n^6*x^4 + (d*x + c)^n*A*b^3*d^7*n^6*x^4 - 6*( d*x + c)^n*D*b^3*c^2*d^5*n^5*x^5 + 51*(d*x + c)^n*D*a*b^2*c*d^6*n^5*x^5 + 17*(d*x + c)^n*C*b^3*c*d^6*n^5*x^5 + 69*(d*x + c)^n*D*a^2*b*d^7*n^5*x^5 + 69*(d*x + c)^n*C*a*b^2*d^7*n^5*x^5 + 23*(d*x + c)^n*B*b^3*d^7*n^5*x^5 + 85 *(d*x + c)^n*D*b^3*c*d^6*n^4*x^6 + 570*(d*x + c)^n*D*a*b^2*d^7*n^4*x^6 + 1 90*(d*x + c)^n*C*b^3*d^7*n^4*x^6 + 735*(d*x + c)^n*D*b^3*d^7*n^3*x^7 + (d* x + c)^n*D*a^3*c*d^6*n^6*x^3 + 3*(d*x + c)^n*C*a^2*b*c*d^6*n^6*x^3 + 3*(d* x + c)^n*B*a*b^2*c*d^6*n^6*x^3 + (d*x + c)^n*A*b^3*c*d^6*n^6*x^3 + (d*x + c)^n*C*a^3*d^7*n^6*x^3 + 3*(d*x + c)^n*B*a^2*b*d^7*n^6*x^3 + 3*(d*x + c)^n *A*a*b^2*d^7*n^6*x^3 - 15*(d*x + c)^n*D*a*b^2*c^2*d^5*n^5*x^4 - 5*(d*x + c )^n*C*b^3*c^2*d^5*n^5*x^4 + 57*(d*x + c)^n*D*a^2*b*c*d^6*n^5*x^4 + 57*(...
Timed out. \[ \int (a+b x)^3 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx=\int {\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^n\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \]