Integrand size = 20, antiderivative size = 294 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {a^2 \left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^9 (1+n)}-\frac {2 a \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^9 (2+n)}+\frac {\left (b^6 c^2-20 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^{3+n}}{b^9 (3+n)}+\frac {4 a^2 d \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{4+n}}{b^9 (4+n)}-\frac {10 a d \left (b^3 c-7 a^3 d\right ) (a+b x)^{5+n}}{b^9 (5+n)}+\frac {2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{6+n}}{b^9 (6+n)}+\frac {28 a^2 d^2 (a+b x)^{7+n}}{b^9 (7+n)}-\frac {8 a d^2 (a+b x)^{8+n}}{b^9 (8+n)}+\frac {d^2 (a+b x)^{9+n}}{b^9 (9+n)} \] Output:
a^2*(-a^3*d+b^3*c)^2*(b*x+a)^(1+n)/b^9/(1+n)-2*a*(-4*a^3*d+b^3*c)*(-a^3*d+ b^3*c)*(b*x+a)^(2+n)/b^9/(2+n)+(28*a^6*d^2-20*a^3*b^3*c*d+b^6*c^2)*(b*x+a) ^(3+n)/b^9/(3+n)+4*a^2*d*(-14*a^3*d+5*b^3*c)*(b*x+a)^(4+n)/b^9/(4+n)-10*a* d*(-7*a^3*d+b^3*c)*(b*x+a)^(5+n)/b^9/(5+n)+2*d*(-28*a^3*d+b^3*c)*(b*x+a)^( 6+n)/b^9/(6+n)+28*a^2*d^2*(b*x+a)^(7+n)/b^9/(7+n)-8*a*d^2*(b*x+a)^(8+n)/b^ 9/(8+n)+d^2*(b*x+a)^(9+n)/b^9/(9+n)
Time = 0.35 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.86 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {(a+b x)^{1+n} \left (\frac {\left (a b^3 c-a^4 d\right )^2}{1+n}-\frac {2 a \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)}{2+n}+\frac {\left (b^6 c^2-20 a^3 b^3 c d+28 a^6 d^2\right ) (a+b x)^2}{3+n}+\frac {4 a^2 d \left (5 b^3 c-14 a^3 d\right ) (a+b x)^3}{4+n}+\frac {10 a d \left (-b^3 c+7 a^3 d\right ) (a+b x)^4}{5+n}+\frac {2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^5}{6+n}+\frac {28 a^2 d^2 (a+b x)^6}{7+n}-\frac {8 a d^2 (a+b x)^7}{8+n}+\frac {d^2 (a+b x)^8}{9+n}\right )}{b^9} \] Input:
Integrate[x^2*(a + b*x)^n*(c + d*x^3)^2,x]
Output:
((a + b*x)^(1 + n)*((a*b^3*c - a^4*d)^2/(1 + n) - (2*a*(b^3*c - 4*a^3*d)*( b^3*c - a^3*d)*(a + b*x))/(2 + n) + ((b^6*c^2 - 20*a^3*b^3*c*d + 28*a^6*d^ 2)*(a + b*x)^2)/(3 + n) + (4*a^2*d*(5*b^3*c - 14*a^3*d)*(a + b*x)^3)/(4 + n) + (10*a*d*(-(b^3*c) + 7*a^3*d)*(a + b*x)^4)/(5 + n) + (2*d*(b^3*c - 28* a^3*d)*(a + b*x)^5)/(6 + n) + (28*a^2*d^2*(a + b*x)^6)/(7 + n) - (8*a*d^2* (a + b*x)^7)/(8 + n) + (d^2*(a + b*x)^8)/(9 + n)))/b^9
Time = 0.89 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int x^2 \left (c+d x^3\right )^2 (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {\left (a b^3 c-a^4 d\right )^2 (a+b x)^n}{b^8}+\frac {10 a d \left (7 a^3 d-b^3 c\right ) (a+b x)^{n+4}}{b^8}+\frac {2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+5}}{b^8}+\frac {28 a^2 d^2 (a+b x)^{n+6}}{b^8}-\frac {2 \left (4 a^7 d^2-5 a^4 b^3 c d+a b^6 c^2\right ) (a+b x)^{n+1}}{b^8}+\frac {\left (28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+2}}{b^8}-\frac {4 a^2 d \left (14 a^3 d-5 b^3 c\right ) (a+b x)^{n+3}}{b^8}-\frac {8 a d^2 (a+b x)^{n+7}}{b^8}+\frac {d^2 (a+b x)^{n+8}}{b^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a \left (b^3 c-4 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^9 (n+2)}-\frac {10 a d \left (b^3 c-7 a^3 d\right ) (a+b x)^{n+5}}{b^9 (n+5)}+\frac {2 d \left (b^3 c-28 a^3 d\right ) (a+b x)^{n+6}}{b^9 (n+6)}+\frac {28 a^2 d^2 (a+b x)^{n+7}}{b^9 (n+7)}+\frac {\left (28 a^6 d^2-20 a^3 b^3 c d+b^6 c^2\right ) (a+b x)^{n+3}}{b^9 (n+3)}+\frac {a^2 \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^9 (n+1)}+\frac {4 a^2 d \left (5 b^3 c-14 a^3 d\right ) (a+b x)^{n+4}}{b^9 (n+4)}-\frac {8 a d^2 (a+b x)^{n+8}}{b^9 (n+8)}+\frac {d^2 (a+b x)^{n+9}}{b^9 (n+9)}\) |
Input:
Int[x^2*(a + b*x)^n*(c + d*x^3)^2,x]
Output:
(a^2*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^9*(1 + n)) - (2*a*(b^3*c - 4* a^3*d)*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^9*(2 + n)) + ((b^6*c^2 - 20*a ^3*b^3*c*d + 28*a^6*d^2)*(a + b*x)^(3 + n))/(b^9*(3 + n)) + (4*a^2*d*(5*b^ 3*c - 14*a^3*d)*(a + b*x)^(4 + n))/(b^9*(4 + n)) - (10*a*d*(b^3*c - 7*a^3* d)*(a + b*x)^(5 + n))/(b^9*(5 + n)) + (2*d*(b^3*c - 28*a^3*d)*(a + b*x)^(6 + n))/(b^9*(6 + n)) + (28*a^2*d^2*(a + b*x)^(7 + n))/(b^9*(7 + n)) - (8*a *d^2*(a + b*x)^(8 + n))/(b^9*(8 + n)) + (d^2*(a + b*x)^(9 + n))/(b^9*(9 + n))
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. \(1564\) vs. \(2(294)=588\).
Time = 0.29 (sec) , antiderivative size = 1565, normalized size of antiderivative = 5.32
| method | result | size |
| gosper | \(\text {Expression too large to display}\) | \(1565\) |
| orering | \(\text {Expression too large to display}\) | \(1568\) |
| risch | \(\text {Expression too large to display}\) | \(1799\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2710\) |
Input:
int(x^2*(b*x+a)^n*(d*x^3+c)^2,x,method=_RETURNVERBOSE)
Output:
1/b^9*(b*x+a)^(1+n)/(n^9+45*n^8+870*n^7+9450*n^6+63273*n^5+269325*n^4+7236 80*n^3+1172700*n^2+1026576*n+362880)*(b^8*d^2*n^8*x^8+36*b^8*d^2*n^7*x^8-8 *a*b^7*d^2*n^7*x^7+546*b^8*d^2*n^6*x^8-224*a*b^7*d^2*n^6*x^7+2*b^8*c*d*n^8 *x^5+4536*b^8*d^2*n^5*x^8+56*a^2*b^6*d^2*n^6*x^6-2576*a*b^7*d^2*n^5*x^7+78 *b^8*c*d*n^7*x^5+22449*b^8*d^2*n^4*x^8+1176*a^2*b^6*d^2*n^5*x^6-10*a*b^7*c *d*n^7*x^4-15680*a*b^7*d^2*n^4*x^7+1272*b^8*c*d*n^6*x^5+67284*b^8*d^2*n^3* x^8-336*a^3*b^5*d^2*n^5*x^5+9800*a^2*b^6*d^2*n^4*x^6-340*a*b^7*c*d*n^6*x^4 -54152*a*b^7*d^2*n^3*x^7+b^8*c^2*n^8*x^2+11268*b^8*c*d*n^5*x^5+118124*b^8* d^2*n^2*x^8-5040*a^3*b^5*d^2*n^4*x^5+40*a^2*b^6*c*d*n^6*x^3+41160*a^2*b^6* d^2*n^3*x^6-4660*a*b^7*c*d*n^5*x^4-105056*a*b^7*d^2*n^2*x^7+42*b^8*c^2*n^7 *x^2+58938*b^8*c*d*n^4*x^5+109584*b^8*d^2*n*x^8+1680*a^4*b^4*d^2*n^4*x^4-2 8560*a^3*b^5*d^2*n^3*x^5+1200*a^2*b^6*c*d*n^5*x^3+90944*a^2*b^6*d^2*n^2*x^ 6-2*a*b^7*c^2*n^7*x-33040*a*b^7*c*d*n^4*x^4-104544*a*b^7*d^2*n*x^7+744*b^8 *c^2*n^6*x^2+185022*b^8*c*d*n^3*x^5+40320*b^8*d^2*x^8+16800*a^4*b^4*d^2*n^ 3*x^4-120*a^3*b^5*c*d*n^5*x^2-75600*a^3*b^5*d^2*n^2*x^5+13840*a^2*b^6*c*d* n^4*x^3+98784*a^2*b^6*d^2*n*x^6-80*a*b^7*c^2*n^6*x-129490*a*b^7*c*d*n^3*x^ 4-40320*a*b^7*d^2*x^7+7218*b^8*c^2*n^5*x^2+337228*b^8*c*d*n^2*x^5-6720*a^5 *b^3*d^2*n^3*x^3+58800*a^4*b^4*d^2*n^2*x^4-3240*a^3*b^5*c*d*n^4*x^2-92064* a^3*b^5*d^2*n*x^5+2*a^2*b^6*c^2*n^6+76800*a^2*b^6*c*d*n^3*x^3+40320*a^2*b^ 6*d^2*x^6-1328*a*b^7*c^2*n^5*x-277660*a*b^7*c*d*n^2*x^4+41619*b^8*c^2*n...
Leaf count of result is larger than twice the leaf count of optimal. 1565 vs. \(2 (294) = 588\).
Time = 0.10 (sec) , antiderivative size = 1565, normalized size of antiderivative = 5.32 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x^2*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="fricas")
Output:
(2*a^3*b^6*c^2*n^6 + 78*a^3*b^6*c^2*n^5 + 1250*a^3*b^6*c^2*n^4 + 120960*a^ 3*b^6*c^2 - 120960*a^6*b^3*c*d + 40320*a^9*d^2 + (b^9*d^2*n^8 + 36*b^9*d^2 *n^7 + 546*b^9*d^2*n^6 + 4536*b^9*d^2*n^5 + 22449*b^9*d^2*n^4 + 67284*b^9* d^2*n^3 + 118124*b^9*d^2*n^2 + 109584*b^9*d^2*n + 40320*b^9*d^2)*x^9 + (a* b^8*d^2*n^8 + 28*a*b^8*d^2*n^7 + 322*a*b^8*d^2*n^6 + 1960*a*b^8*d^2*n^5 + 6769*a*b^8*d^2*n^4 + 13132*a*b^8*d^2*n^3 + 13068*a*b^8*d^2*n^2 + 5040*a*b^ 8*d^2*n)*x^8 - 8*(a^2*b^7*d^2*n^7 + 21*a^2*b^7*d^2*n^6 + 175*a^2*b^7*d^2*n ^5 + 735*a^2*b^7*d^2*n^4 + 1624*a^2*b^7*d^2*n^3 + 1764*a^2*b^7*d^2*n^2 + 7 20*a^2*b^7*d^2*n)*x^7 + 2*(b^9*c*d*n^8 + 39*b^9*c*d*n^7 + 60480*b^9*c*d + 4*(159*b^9*c*d + 7*a^3*b^6*d^2)*n^6 + 6*(939*b^9*c*d + 70*a^3*b^6*d^2)*n^5 + (29469*b^9*c*d + 2380*a^3*b^6*d^2)*n^4 + 9*(10279*b^9*c*d + 700*a^3*b^6 *d^2)*n^3 + 2*(84307*b^9*c*d + 3836*a^3*b^6*d^2)*n^2 + 24*(6709*b^9*c*d + 140*a^3*b^6*d^2)*n)*x^6 + 2*(a*b^8*c*d*n^8 + 34*a*b^8*c*d*n^7 + 466*a*b^8* c*d*n^6 + 56*(59*a*b^8*c*d - 3*a^4*b^5*d^2)*n^5 + (12949*a*b^8*c*d - 1680* a^4*b^5*d^2)*n^4 + 2*(13883*a*b^8*c*d - 2940*a^4*b^5*d^2)*n^3 + 24*(1241*a *b^8*c*d - 350*a^4*b^5*d^2)*n^2 + 4032*(3*a*b^8*c*d - a^4*b^5*d^2)*n)*x^5 - 10*(a^2*b^7*c*d*n^7 + 30*a^2*b^7*c*d*n^6 + 346*a^2*b^7*c*d*n^5 + 24*(80* a^2*b^7*c*d - 7*a^5*b^4*d^2)*n^4 + (5269*a^2*b^7*c*d - 1008*a^5*b^4*d^2)*n ^3 + 6*(1115*a^2*b^7*c*d - 308*a^5*b^4*d^2)*n^2 + 1008*(3*a^2*b^7*c*d - a^ 5*b^4*d^2)*n)*x^4 + 30*(351*a^3*b^6*c^2 - 8*a^6*b^3*c*d)*n^3 + (b^9*c^2...
Leaf count of result is larger than twice the leaf count of optimal. 26746 vs. \(2 (275) = 550\).
Time = 8.28 (sec) , antiderivative size = 26746, normalized size of antiderivative = 90.97 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x**2*(b*x+a)**n*(d*x**3+c)**2,x)
Output:
Piecewise((a**n*(c**2*x**3/3 + c*d*x**6/3 + d**2*x**9/9), Eq(b, 0)), (840* a**8*d**2*log(a/b + x)/(840*a**8*b**9 + 6720*a**7*b**10*x + 23520*a**6*b** 11*x**2 + 47040*a**5*b**12*x**3 + 58800*a**4*b**13*x**4 + 47040*a**3*b**14 *x**5 + 23520*a**2*b**15*x**6 + 6720*a*b**16*x**7 + 840*b**17*x**8) + 2283 *a**8*d**2/(840*a**8*b**9 + 6720*a**7*b**10*x + 23520*a**6*b**11*x**2 + 47 040*a**5*b**12*x**3 + 58800*a**4*b**13*x**4 + 47040*a**3*b**14*x**5 + 2352 0*a**2*b**15*x**6 + 6720*a*b**16*x**7 + 840*b**17*x**8) + 6720*a**7*b*d**2 *x*log(a/b + x)/(840*a**8*b**9 + 6720*a**7*b**10*x + 23520*a**6*b**11*x**2 + 47040*a**5*b**12*x**3 + 58800*a**4*b**13*x**4 + 47040*a**3*b**14*x**5 + 23520*a**2*b**15*x**6 + 6720*a*b**16*x**7 + 840*b**17*x**8) + 17424*a**7* b*d**2*x/(840*a**8*b**9 + 6720*a**7*b**10*x + 23520*a**6*b**11*x**2 + 4704 0*a**5*b**12*x**3 + 58800*a**4*b**13*x**4 + 47040*a**3*b**14*x**5 + 23520* a**2*b**15*x**6 + 6720*a*b**16*x**7 + 840*b**17*x**8) + 23520*a**6*b**2*d* *2*x**2*log(a/b + x)/(840*a**8*b**9 + 6720*a**7*b**10*x + 23520*a**6*b**11 *x**2 + 47040*a**5*b**12*x**3 + 58800*a**4*b**13*x**4 + 47040*a**3*b**14*x **5 + 23520*a**2*b**15*x**6 + 6720*a*b**16*x**7 + 840*b**17*x**8) + 57624* a**6*b**2*d**2*x**2/(840*a**8*b**9 + 6720*a**7*b**10*x + 23520*a**6*b**11* x**2 + 47040*a**5*b**12*x**3 + 58800*a**4*b**13*x**4 + 47040*a**3*b**14*x* *5 + 23520*a**2*b**15*x**6 + 6720*a*b**16*x**7 + 840*b**17*x**8) - 10*a**5 *b**3*c*d/(840*a**8*b**9 + 6720*a**7*b**10*x + 23520*a**6*b**11*x**2 + ...
Leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (294) = 588\).
Time = 0.05 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.04 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {2 \, {\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} + \frac {{\left ({\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{9} x^{9} + {\left (n^{8} + 28 \, n^{7} + 322 \, n^{6} + 1960 \, n^{5} + 6769 \, n^{4} + 13132 \, n^{3} + 13068 \, n^{2} + 5040 \, n\right )} a b^{8} x^{8} - 8 \, {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a^{2} b^{7} x^{7} + 56 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{3} b^{6} x^{6} - 336 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{4} b^{5} x^{5} + 1680 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{5} b^{4} x^{4} - 6720 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{6} b^{3} x^{3} + 20160 \, {\left (n^{2} + n\right )} a^{7} b^{2} x^{2} - 40320 \, a^{8} b n x + 40320 \, a^{9}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{9} + 45 \, n^{8} + 870 \, n^{7} + 9450 \, n^{6} + 63273 \, n^{5} + 269325 \, n^{4} + 723680 \, n^{3} + 1172700 \, n^{2} + 1026576 \, n + 362880\right )} b^{9}} \] Input:
integrate(x^2*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="maxima")
Output:
((n^2 + 3*n + 2)*b^3*x^3 + (n^2 + n)*a*b^2*x^2 - 2*a^2*b*n*x + 2*a^3)*(b*x + a)^n*c^2/((n^3 + 6*n^2 + 11*n + 6)*b^3) + 2*((n^5 + 15*n^4 + 85*n^3 + 2 25*n^2 + 274*n + 120)*b^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a* b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2*b^4*x^4 + 20*(n^3 + 3*n^2 + 2 *n)*a^3*b^3*x^3 - 60*(n^2 + n)*a^4*b^2*x^2 + 120*a^5*b*n*x - 120*a^6)*(b*x + a)^n*c*d/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)* b^6) + ((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 11812 4*n^2 + 109584*n + 40320)*b^9*x^9 + (n^8 + 28*n^7 + 322*n^6 + 1960*n^5 + 6 769*n^4 + 13132*n^3 + 13068*n^2 + 5040*n)*a*b^8*x^8 - 8*(n^7 + 21*n^6 + 17 5*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a^2*b^7*x^7 + 56*(n^6 + 15* n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a^3*b^6*x^6 - 336*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^4*b^5*x^5 + 1680*(n^4 + 6*n^3 + 11*n^2 + 6*n)* a^5*b^4*x^4 - 6720*(n^3 + 3*n^2 + 2*n)*a^6*b^3*x^3 + 20160*(n^2 + n)*a^7*b ^2*x^2 - 40320*a^8*b*n*x + 40320*a^9)*(b*x + a)^n*d^2/((n^9 + 45*n^8 + 870 *n^7 + 9450*n^6 + 63273*n^5 + 269325*n^4 + 723680*n^3 + 1172700*n^2 + 1026 576*n + 362880)*b^9)
Leaf count of result is larger than twice the leaf count of optimal. 2660 vs. \(2 (294) = 588\).
Time = 0.15 (sec) , antiderivative size = 2660, normalized size of antiderivative = 9.05 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x^2*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="giac")
Output:
((b*x + a)^n*b^9*d^2*n^8*x^9 + (b*x + a)^n*a*b^8*d^2*n^8*x^8 + 36*(b*x + a )^n*b^9*d^2*n^7*x^9 + 28*(b*x + a)^n*a*b^8*d^2*n^7*x^8 + 546*(b*x + a)^n*b ^9*d^2*n^6*x^9 + 2*(b*x + a)^n*b^9*c*d*n^8*x^6 - 8*(b*x + a)^n*a^2*b^7*d^2 *n^7*x^7 + 322*(b*x + a)^n*a*b^8*d^2*n^6*x^8 + 4536*(b*x + a)^n*b^9*d^2*n^ 5*x^9 + 2*(b*x + a)^n*a*b^8*c*d*n^8*x^5 + 78*(b*x + a)^n*b^9*c*d*n^7*x^6 - 168*(b*x + a)^n*a^2*b^7*d^2*n^6*x^7 + 1960*(b*x + a)^n*a*b^8*d^2*n^5*x^8 + 22449*(b*x + a)^n*b^9*d^2*n^4*x^9 + 68*(b*x + a)^n*a*b^8*c*d*n^7*x^5 + 1 272*(b*x + a)^n*b^9*c*d*n^6*x^6 + 56*(b*x + a)^n*a^3*b^6*d^2*n^6*x^6 - 140 0*(b*x + a)^n*a^2*b^7*d^2*n^5*x^7 + 6769*(b*x + a)^n*a*b^8*d^2*n^4*x^8 + 6 7284*(b*x + a)^n*b^9*d^2*n^3*x^9 + (b*x + a)^n*b^9*c^2*n^8*x^3 - 10*(b*x + a)^n*a^2*b^7*c*d*n^7*x^4 + 932*(b*x + a)^n*a*b^8*c*d*n^6*x^5 + 11268*(b*x + a)^n*b^9*c*d*n^5*x^6 + 840*(b*x + a)^n*a^3*b^6*d^2*n^5*x^6 - 5880*(b*x + a)^n*a^2*b^7*d^2*n^4*x^7 + 13132*(b*x + a)^n*a*b^8*d^2*n^3*x^8 + 118124* (b*x + a)^n*b^9*d^2*n^2*x^9 + (b*x + a)^n*a*b^8*c^2*n^8*x^2 + 42*(b*x + a) ^n*b^9*c^2*n^7*x^3 - 300*(b*x + a)^n*a^2*b^7*c*d*n^6*x^4 + 6608*(b*x + a)^ n*a*b^8*c*d*n^5*x^5 - 336*(b*x + a)^n*a^4*b^5*d^2*n^5*x^5 + 58938*(b*x + a )^n*b^9*c*d*n^4*x^6 + 4760*(b*x + a)^n*a^3*b^6*d^2*n^4*x^6 - 12992*(b*x + a)^n*a^2*b^7*d^2*n^3*x^7 + 13068*(b*x + a)^n*a*b^8*d^2*n^2*x^8 + 109584*(b *x + a)^n*b^9*d^2*n*x^9 + 40*(b*x + a)^n*a*b^8*c^2*n^7*x^2 + 744*(b*x + a) ^n*b^9*c^2*n^6*x^3 + 40*(b*x + a)^n*a^3*b^6*c*d*n^6*x^3 - 3460*(b*x + a...
Time = 23.77 (sec) , antiderivative size = 1410, normalized size of antiderivative = 4.80 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \] Input:
int(x^2*(c + d*x^3)^2*(a + b*x)^n,x)
Output:
(d^2*x^9*(a + b*x)^n*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536 *n^5 + 546*n^6 + 36*n^7 + n^8 + 40320))/(1026576*n + 1172700*n^2 + 723680* n^3 + 269325*n^4 + 63273*n^5 + 9450*n^6 + 870*n^7 + 45*n^8 + n^9 + 362880) + (2*a^3*(a + b*x)^n*(20160*a^6*d^2 + 60480*b^6*c^2 + 60216*b^6*c^2*n + 2 4574*b^6*c^2*n^2 + 5265*b^6*c^2*n^3 + 625*b^6*c^2*n^4 + 39*b^6*c^2*n^5 + b ^6*c^2*n^6 - 60480*a^3*b^3*c*d - 22920*a^3*b^3*c*d*n - 2880*a^3*b^3*c*d*n^ 2 - 120*a^3*b^3*c*d*n^3))/(b^9*(1026576*n + 1172700*n^2 + 723680*n^3 + 269 325*n^4 + 63273*n^5 + 9450*n^6 + 870*n^7 + 45*n^8 + n^9 + 362880)) + (x^3* (a + b*x)^n*(3*n + n^2 + 2)*(60480*b^6*c^2 - 6720*a^6*d^2*n + 60216*b^6*c^ 2*n + 24574*b^6*c^2*n^2 + 5265*b^6*c^2*n^3 + 625*b^6*c^2*n^4 + 39*b^6*c^2* n^5 + b^6*c^2*n^6 + 20160*a^3*b^3*c*d*n + 7640*a^3*b^3*c*d*n^2 + 960*a^3*b ^3*c*d*n^3 + 40*a^3*b^3*c*d*n^4))/(b^6*(1026576*n + 1172700*n^2 + 723680*n ^3 + 269325*n^4 + 63273*n^5 + 9450*n^6 + 870*n^7 + 45*n^8 + n^9 + 362880)) + (2*d*x^6*(a + b*x)^n*(504*b^3*c + 24*b^3*c*n^2 + b^3*c*n^3 + 28*a^3*d*n + 191*b^3*c*n)*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(b^3*(102 6576*n + 1172700*n^2 + 723680*n^3 + 269325*n^4 + 63273*n^5 + 9450*n^6 + 87 0*n^7 + 45*n^8 + n^9 + 362880)) - (2*a^2*n*x*(a + b*x)^n*(20160*a^6*d^2 + 60480*b^6*c^2 + 60216*b^6*c^2*n + 24574*b^6*c^2*n^2 + 5265*b^6*c^2*n^3 + 6 25*b^6*c^2*n^4 + 39*b^6*c^2*n^5 + b^6*c^2*n^6 - 60480*a^3*b^3*c*d - 22920* a^3*b^3*c*d*n - 2880*a^3*b^3*c*d*n^2 - 120*a^3*b^3*c*d*n^3))/(b^8*(1026...
Time = 0.27 (sec) , antiderivative size = 1798, normalized size of antiderivative = 6.12 \[ \int x^2 (a+b x)^n \left (c+d x^3\right )^2 \, dx =\text {Too large to display} \] Input:
int(x^2*(b*x+a)^n*(d*x^3+c)^2,x)
Output:
((a + b*x)**n*(40320*a**9*d**2 - 40320*a**8*b*d**2*n*x + 20160*a**7*b**2*d **2*n**2*x**2 + 20160*a**7*b**2*d**2*n*x**2 - 240*a**6*b**3*c*d*n**3 - 576 0*a**6*b**3*c*d*n**2 - 45840*a**6*b**3*c*d*n - 120960*a**6*b**3*c*d - 6720 *a**6*b**3*d**2*n**3*x**3 - 20160*a**6*b**3*d**2*n**2*x**3 - 13440*a**6*b* *3*d**2*n*x**3 + 240*a**5*b**4*c*d*n**4*x + 5760*a**5*b**4*c*d*n**3*x + 45 840*a**5*b**4*c*d*n**2*x + 120960*a**5*b**4*c*d*n*x + 1680*a**5*b**4*d**2* n**4*x**4 + 10080*a**5*b**4*d**2*n**3*x**4 + 18480*a**5*b**4*d**2*n**2*x** 4 + 10080*a**5*b**4*d**2*n*x**4 - 120*a**4*b**5*c*d*n**5*x**2 - 3000*a**4* b**5*c*d*n**4*x**2 - 25800*a**4*b**5*c*d*n**3*x**2 - 83400*a**4*b**5*c*d*n **2*x**2 - 60480*a**4*b**5*c*d*n*x**2 - 336*a**4*b**5*d**2*n**5*x**5 - 336 0*a**4*b**5*d**2*n**4*x**5 - 11760*a**4*b**5*d**2*n**3*x**5 - 16800*a**4*b **5*d**2*n**2*x**5 - 8064*a**4*b**5*d**2*n*x**5 + 2*a**3*b**6*c**2*n**6 + 78*a**3*b**6*c**2*n**5 + 1250*a**3*b**6*c**2*n**4 + 10530*a**3*b**6*c**2*n **3 + 49148*a**3*b**6*c**2*n**2 + 120432*a**3*b**6*c**2*n + 120960*a**3*b* *6*c**2 + 40*a**3*b**6*c*d*n**6*x**3 + 1080*a**3*b**6*c*d*n**5*x**3 + 1060 0*a**3*b**6*c*d*n**4*x**3 + 45000*a**3*b**6*c*d*n**3*x**3 + 75760*a**3*b** 6*c*d*n**2*x**3 + 40320*a**3*b**6*c*d*n*x**3 + 56*a**3*b**6*d**2*n**6*x**6 + 840*a**3*b**6*d**2*n**5*x**6 + 4760*a**3*b**6*d**2*n**4*x**6 + 12600*a* *3*b**6*d**2*n**3*x**6 + 15344*a**3*b**6*d**2*n**2*x**6 + 6720*a**3*b**6*d **2*n*x**6 - 2*a**2*b**7*c**2*n**7*x - 78*a**2*b**7*c**2*n**6*x - 1250*...