Integrand size = 18, antiderivative size = 248 \[ \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx=-\frac {a \left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^8 (1+n)}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^8 (2+n)}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{3+n}}{b^8 (3+n)}-\frac {a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{4+n}}{b^8 (4+n)}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{5+n}}{b^8 (5+n)}+\frac {21 a^2 d^2 (a+b x)^{6+n}}{b^8 (6+n)}-\frac {7 a d^2 (a+b x)^{7+n}}{b^8 (7+n)}+\frac {d^2 (a+b x)^{8+n}}{b^8 (8+n)} \] Output:
-a*(-a^3*d+b^3*c)^2*(b*x+a)^(1+n)/b^8/(1+n)+(-7*a^3*d+b^3*c)*(-a^3*d+b^3*c )*(b*x+a)^(2+n)/b^8/(2+n)+3*a^2*d*(-7*a^3*d+4*b^3*c)*(b*x+a)^(3+n)/b^8/(3+ n)-a*d*(-35*a^3*d+8*b^3*c)*(b*x+a)^(4+n)/b^8/(4+n)+d*(-35*a^3*d+2*b^3*c)*( b*x+a)^(5+n)/b^8/(5+n)+21*a^2*d^2*(b*x+a)^(6+n)/b^8/(6+n)-7*a*d^2*(b*x+a)^ (7+n)/b^8/(7+n)+d^2*(b*x+a)^(8+n)/b^8/(8+n)
Time = 0.28 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.85 \[ \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {(a+b x)^{1+n} \left (-\frac {a \left (b^3 c-a^3 d\right )^2}{1+n}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)}{2+n}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^2}{3+n}+\frac {a d \left (-8 b^3 c+35 a^3 d\right ) (a+b x)^3}{4+n}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^4}{5+n}+\frac {21 a^2 d^2 (a+b x)^5}{6+n}-\frac {7 a d^2 (a+b x)^6}{7+n}+\frac {d^2 (a+b x)^7}{8+n}\right )}{b^8} \] Input:
Integrate[x*(a + b*x)^n*(c + d*x^3)^2,x]
Output:
((a + b*x)^(1 + n)*(-((a*(b^3*c - a^3*d)^2)/(1 + n)) + ((b^3*c - 7*a^3*d)* (b^3*c - a^3*d)*(a + b*x))/(2 + n) + (3*a^2*d*(4*b^3*c - 7*a^3*d)*(a + b*x )^2)/(3 + n) + (a*d*(-8*b^3*c + 35*a^3*d)*(a + b*x)^3)/(4 + n) + (d*(2*b^3 *c - 35*a^3*d)*(a + b*x)^4)/(5 + n) + (21*a^2*d^2*(a + b*x)^5)/(6 + n) - ( 7*a*d^2*(a + b*x)^6)/(7 + n) + (d^2*(a + b*x)^7)/(8 + n)))/b^8
Time = 0.76 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int x \left (c+d x^3\right )^2 (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (-\frac {a \left (a^3 d-b^3 c\right )^2 (a+b x)^n}{b^7}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^7}+\frac {a d \left (35 a^3 d-8 b^3 c\right ) (a+b x)^{n+3}}{b^7}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{n+4}}{b^7}+\frac {21 a^2 d^2 (a+b x)^{n+5}}{b^7}-\frac {3 a^2 d \left (7 a^3 d-4 b^3 c\right ) (a+b x)^{n+2}}{b^7}-\frac {7 a d^2 (a+b x)^{n+6}}{b^7}+\frac {d^2 (a+b x)^{n+7}}{b^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^8 (n+1)}+\frac {\left (b^3 c-7 a^3 d\right ) \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac {a d \left (8 b^3 c-35 a^3 d\right ) (a+b x)^{n+4}}{b^8 (n+4)}+\frac {d \left (2 b^3 c-35 a^3 d\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac {21 a^2 d^2 (a+b x)^{n+6}}{b^8 (n+6)}+\frac {3 a^2 d \left (4 b^3 c-7 a^3 d\right ) (a+b x)^{n+3}}{b^8 (n+3)}-\frac {7 a d^2 (a+b x)^{n+7}}{b^8 (n+7)}+\frac {d^2 (a+b x)^{n+8}}{b^8 (n+8)}\) |
Input:
Int[x*(a + b*x)^n*(c + d*x^3)^2,x]
Output:
-((a*(b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^3*c - 7*a^3 *d)*(b^3*c - a^3*d)*(a + b*x)^(2 + n))/(b^8*(2 + n)) + (3*a^2*d*(4*b^3*c - 7*a^3*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) - (a*d*(8*b^3*c - 35*a^3*d)*(a + b*x)^(4 + n))/(b^8*(4 + n)) + (d*(2*b^3*c - 35*a^3*d)*(a + b*x)^(5 + n)) /(b^8*(5 + n)) + (21*a^2*d^2*(a + b*x)^(6 + n))/(b^8*(6 + n)) - (7*a*d^2*( a + b*x)^(7 + n))/(b^8*(7 + n)) + (d^2*(a + b*x)^(8 + n))/(b^8*(8 + 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. \(892\) vs. \(2(248)=496\).
Time = 0.26 (sec) , antiderivative size = 893, normalized size of antiderivative = 3.60
| method | result | size |
| norman | \(\frac {d^{2} x^{8} {\mathrm e}^{n \ln \left (b x +a \right )}}{8+n}+\frac {n a \left (b^{6} c^{2} n^{6}+33 b^{6} c^{2} n^{5}+445 b^{6} c^{2} n^{4}-48 a^{3} b^{3} c d \,n^{3}+3135 b^{6} c^{2} n^{3}-1008 a^{3} b^{3} c d \,n^{2}+12154 b^{6} c^{2} n^{2}-7008 a^{3} b^{3} c d n +24552 b^{6} c^{2} n +5040 a^{6} d^{2}-16128 a^{3} b^{3} c d +20160 b^{6} c^{2}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{7} \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 )}+\frac {d^{2} a n \,x^{7} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+15 n +56\right )}-\frac {a^{2} \left (b^{6} c^{2} n^{6}+33 b^{6} c^{2} n^{5}+445 b^{6} c^{2} n^{4}-48 a^{3} b^{3} c d \,n^{3}+3135 b^{6} c^{2} n^{3}-1008 a^{3} b^{3} c d \,n^{2}+12154 b^{6} c^{2} n^{2}-7008 a^{3} b^{3} c d n +24552 b^{6} c^{2} n +5040 a^{6} d^{2}-16128 a^{3} b^{3} c d +20160 b^{6} c^{2}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{8} \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 )}-\frac {\left (-b^{6} c^{2} n^{6}-33 b^{6} c^{2} n^{5}-24 a^{3} b^{3} c d \,n^{4}-445 b^{6} c^{2} n^{4}-504 a^{3} b^{3} c d \,n^{3}-3135 b^{6} c^{2} n^{3}-3504 a^{3} b^{3} c d \,n^{2}-12154 b^{6} c^{2} n^{2}+2520 a^{6} d^{2} n -8064 a^{3} b^{3} c d n -24552 b^{6} c^{2} n -20160 b^{6} c^{2}\right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{6} \left (n^{7}+35 n^{6}+511 n^{5}+4025 n^{4}+18424 n^{3}+48860 n^{2}+69264 n +40320\right )}+\frac {2 d \left (b^{3} c \,n^{3}+21 b^{3} c \,n^{2}+21 a^{3} d n +146 b^{3} c n +336 b^{3} c \right ) x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+26 n^{3}+251 n^{2}+1066 n +1680\right )}-\frac {7 n \,a^{2} d^{2} x^{6} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+21 n^{2}+146 n +336\right )}-\frac {2 n a d \left (-b^{3} c \,n^{3}-21 b^{3} c \,n^{2}-146 b^{3} c n +105 a^{3} d -336 b^{3} c \right ) x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+30 n^{4}+355 n^{3}+2070 n^{2}+5944 n +6720\right )}+\frac {8 \left (-b^{3} c \,n^{3}-21 b^{3} c \,n^{2}-146 b^{3} c n +105 a^{3} d -336 b^{3} c \right ) a^{2} d n \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{6}+33 n^{5}+445 n^{4}+3135 n^{3}+12154 n^{2}+24552 n +20160\right )}\) | \(893\) |
| gosper | \(\text {Expression too large to display}\) | \(1142\) |
| orering | \(\text {Expression too large to display}\) | \(1145\) |
| risch | \(\text {Expression too large to display}\) | \(1369\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2009\) |
Input:
int(x*(b*x+a)^n*(d*x^3+c)^2,x,method=_RETURNVERBOSE)
Output:
d^2/(8+n)*x^8*exp(n*ln(b*x+a))+1/b^7*n*a*(b^6*c^2*n^6+33*b^6*c^2*n^5+445*b ^6*c^2*n^4-48*a^3*b^3*c*d*n^3+3135*b^6*c^2*n^3-1008*a^3*b^3*c*d*n^2+12154* b^6*c^2*n^2-7008*a^3*b^3*c*d*n+24552*b^6*c^2*n+5040*a^6*d^2-16128*a^3*b^3* c*d+20160*b^6*c^2)/(n^8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3+118124 *n^2+109584*n+40320)*x*exp(n*ln(b*x+a))+d^2*a/b*n/(n^2+15*n+56)*x^7*exp(n* ln(b*x+a))-a^2*(b^6*c^2*n^6+33*b^6*c^2*n^5+445*b^6*c^2*n^4-48*a^3*b^3*c*d* n^3+3135*b^6*c^2*n^3-1008*a^3*b^3*c*d*n^2+12154*b^6*c^2*n^2-7008*a^3*b^3*c *d*n+24552*b^6*c^2*n+5040*a^6*d^2-16128*a^3*b^3*c*d+20160*b^6*c^2)/b^8/(n^ 8+36*n^7+546*n^6+4536*n^5+22449*n^4+67284*n^3+118124*n^2+109584*n+40320)*e xp(n*ln(b*x+a))-(-b^6*c^2*n^6-33*b^6*c^2*n^5-24*a^3*b^3*c*d*n^4-445*b^6*c^ 2*n^4-504*a^3*b^3*c*d*n^3-3135*b^6*c^2*n^3-3504*a^3*b^3*c*d*n^2-12154*b^6* c^2*n^2+2520*a^6*d^2*n-8064*a^3*b^3*c*d*n-24552*b^6*c^2*n-20160*b^6*c^2)/b ^6/(n^7+35*n^6+511*n^5+4025*n^4+18424*n^3+48860*n^2+69264*n+40320)*x^2*exp (n*ln(b*x+a))+2*d*(b^3*c*n^3+21*b^3*c*n^2+21*a^3*d*n+146*b^3*c*n+336*b^3*c )/b^3/(n^4+26*n^3+251*n^2+1066*n+1680)*x^5*exp(n*ln(b*x+a))-7*n*a^2/b^2*d^ 2/(n^3+21*n^2+146*n+336)*x^6*exp(n*ln(b*x+a))-2*n*a*d*(-b^3*c*n^3-21*b^3*c *n^2-146*b^3*c*n+105*a^3*d-336*b^3*c)/b^4/(n^5+30*n^4+355*n^3+2070*n^2+594 4*n+6720)*x^4*exp(n*ln(b*x+a))+8*(-b^3*c*n^3-21*b^3*c*n^2-146*b^3*c*n+105* a^3*d-336*b^3*c)*a^2/b^5*d*n/(n^6+33*n^5+445*n^4+3135*n^3+12154*n^2+24552* n+20160)*x^3*exp(n*ln(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (248) = 496\).
Time = 0.10 (sec) , antiderivative size = 1216, normalized size of antiderivative = 4.90 \[ \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="fricas")
Output:
-(a^2*b^6*c^2*n^6 + 33*a^2*b^6*c^2*n^5 + 445*a^2*b^6*c^2*n^4 + 20160*a^2*b ^6*c^2 - 16128*a^5*b^3*c*d + 5040*a^8*d^2 - (b^8*d^2*n^7 + 28*b^8*d^2*n^6 + 322*b^8*d^2*n^5 + 1960*b^8*d^2*n^4 + 6769*b^8*d^2*n^3 + 13132*b^8*d^2*n^ 2 + 13068*b^8*d^2*n + 5040*b^8*d^2)*x^8 - (a*b^7*d^2*n^7 + 21*a*b^7*d^2*n^ 6 + 175*a*b^7*d^2*n^5 + 735*a*b^7*d^2*n^4 + 1624*a*b^7*d^2*n^3 + 1764*a*b^ 7*d^2*n^2 + 720*a*b^7*d^2*n)*x^7 + 7*(a^2*b^6*d^2*n^6 + 15*a^2*b^6*d^2*n^5 + 85*a^2*b^6*d^2*n^4 + 225*a^2*b^6*d^2*n^3 + 274*a^2*b^6*d^2*n^2 + 120*a^ 2*b^6*d^2*n)*x^6 - 2*(b^8*c*d*n^7 + 31*b^8*c*d*n^6 + 8064*b^8*c*d + (391*b ^8*c*d + 21*a^3*b^5*d^2)*n^5 + (2581*b^8*c*d + 210*a^3*b^5*d^2)*n^4 + (954 4*b^8*c*d + 735*a^3*b^5*d^2)*n^3 + 2*(9782*b^8*c*d + 525*a^3*b^5*d^2)*n^2 + 72*(282*b^8*c*d + 7*a^3*b^5*d^2)*n)*x^5 - 2*(a*b^7*c*d*n^7 + 27*a*b^7*c* d*n^6 + 283*a*b^7*c*d*n^5 + 21*(69*a*b^7*c*d - 5*a^4*b^4*d^2)*n^4 + 2*(187 4*a*b^7*c*d - 315*a^4*b^4*d^2)*n^3 + 3*(1524*a*b^7*c*d - 385*a^4*b^4*d^2)* n^2 + 126*(16*a*b^7*c*d - 5*a^4*b^4*d^2)*n)*x^4 + 3*(1045*a^2*b^6*c^2 - 16 *a^5*b^3*c*d)*n^3 + 8*(a^2*b^6*c*d*n^6 + 24*a^2*b^6*c*d*n^5 + 211*a^2*b^6* c*d*n^4 + 3*(272*a^2*b^6*c*d - 35*a^5*b^3*d^2)*n^3 + 5*(260*a^2*b^6*c*d - 63*a^5*b^3*d^2)*n^2 + 42*(16*a^2*b^6*c*d - 5*a^5*b^3*d^2)*n)*x^3 + 2*(6077 *a^2*b^6*c^2 - 504*a^5*b^3*c*d)*n^2 - (b^8*c^2*n^7 + 34*b^8*c^2*n^6 + 2016 0*b^8*c^2 + 2*(239*b^8*c^2 + 12*a^3*b^5*c*d)*n^5 + 4*(895*b^8*c^2 + 132*a^ 3*b^5*c*d)*n^4 + (15289*b^8*c^2 + 4008*a^3*b^5*c*d)*n^3 + 2*(18353*b^8*...
Leaf count of result is larger than twice the leaf count of optimal. 18328 vs. \(2 (228) = 456\).
Time = 5.38 (sec) , antiderivative size = 18328, normalized size of antiderivative = 73.90 \[ \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x*(b*x+a)**n*(d*x**3+c)**2,x)
Output:
Piecewise((a**n*(c**2*x**2/2 + 2*c*d*x**5/5 + d**2*x**8/8), Eq(b, 0)), (42 0*a**7*d**2*log(a/b + x)/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b** 10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13* x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 1089*a**7*d**2/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700 *a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x* *7) + 2940*a**6*b*d**2*x*log(a/b + x)/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 882 0*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) + 7203*a**6*b*d**2 *x/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b **11*x**3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x* *6 + 420*b**15*x**7) + 8820*a**5*b**2*d**2*x**2*log(a/b + x)/(420*a**7*b** 8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 1470 0*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x **7) + 20139*a**5*b**2*d**2*x**2/(420*a**7*b**8 + 2940*a**6*b**9*x + 8820* a**5*b**10*x**2 + 14700*a**4*b**11*x**3 + 14700*a**3*b**12*x**4 + 8820*a** 2*b**13*x**5 + 2940*a*b**14*x**6 + 420*b**15*x**7) - 8*a**4*b**3*c*d/(420* a**7*b**8 + 2940*a**6*b**9*x + 8820*a**5*b**10*x**2 + 14700*a**4*b**11*x** 3 + 14700*a**3*b**12*x**4 + 8820*a**2*b**13*x**5 + 2940*a*b**14*x**6 + 420 *b**15*x**7) + 14700*a**4*b**3*d**2*x**3*log(a/b + x)/(420*a**7*b**8 + ...
Time = 0.05 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.91 \[ \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {2 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} + \frac {{\left ({\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{8} x^{8} + {\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a b^{7} x^{7} - 7 \, {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{2} b^{6} x^{6} + 42 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{3} b^{5} x^{5} - 210 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{4} b^{4} x^{4} + 840 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{5} b^{3} x^{3} - 2520 \, {\left (n^{2} + n\right )} a^{6} b^{2} x^{2} + 5040 \, a^{7} b n x - 5040 \, a^{8}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{8}} \] Input:
integrate(x*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="maxima")
Output:
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^2/((n^2 + 3*n + 2)*b^2) + 2*((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^5*x^5 + (n^4 + 6*n^3 + 11*n^2 + 6 *n)*a*b^4*x^4 - 4*(n^3 + 3*n^2 + 2*n)*a^2*b^3*x^3 + 12*(n^2 + n)*a^3*b^2*x ^2 - 24*a^4*b*n*x + 24*a^5)*(b*x + a)^n*c*d/((n^5 + 15*n^4 + 85*n^3 + 225* n^2 + 274*n + 120)*b^5) + ((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 + 1764*n^2 + 720*n)*a*b^7*x^7 - 7*(n^6 + 15*n^5 + 85*n^4 + 225*n ^3 + 274*n^2 + 120*n)*a^2*b^6*x^6 + 42*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 2 4*n)*a^3*b^5*x^5 - 210*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^4*b^4*x^4 + 840*(n^3 + 3*n^2 + 2*n)*a^5*b^3*x^3 - 2520*(n^2 + n)*a^6*b^2*x^2 + 5040*a^7*b*n*x - 5040*a^8)*(b*x + a)^n*d^2/((n^8 + 36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^ 4 + 67284*n^3 + 118124*n^2 + 109584*n + 40320)*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 2034 vs. \(2 (248) = 496\).
Time = 0.13 (sec) , antiderivative size = 2034, normalized size of antiderivative = 8.20 \[ \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x*(b*x+a)^n*(d*x^3+c)^2,x, algorithm="giac")
Output:
((b*x + a)^n*b^8*d^2*n^7*x^8 + (b*x + a)^n*a*b^7*d^2*n^7*x^7 + 28*(b*x + a )^n*b^8*d^2*n^6*x^8 + 21*(b*x + a)^n*a*b^7*d^2*n^6*x^7 + 322*(b*x + a)^n*b ^8*d^2*n^5*x^8 + 2*(b*x + a)^n*b^8*c*d*n^7*x^5 - 7*(b*x + a)^n*a^2*b^6*d^2 *n^6*x^6 + 175*(b*x + a)^n*a*b^7*d^2*n^5*x^7 + 1960*(b*x + a)^n*b^8*d^2*n^ 4*x^8 + 2*(b*x + a)^n*a*b^7*c*d*n^7*x^4 + 62*(b*x + a)^n*b^8*c*d*n^6*x^5 - 105*(b*x + a)^n*a^2*b^6*d^2*n^5*x^6 + 735*(b*x + a)^n*a*b^7*d^2*n^4*x^7 + 6769*(b*x + a)^n*b^8*d^2*n^3*x^8 + 54*(b*x + a)^n*a*b^7*c*d*n^6*x^4 + 782 *(b*x + a)^n*b^8*c*d*n^5*x^5 + 42*(b*x + a)^n*a^3*b^5*d^2*n^5*x^5 - 595*(b *x + a)^n*a^2*b^6*d^2*n^4*x^6 + 1624*(b*x + a)^n*a*b^7*d^2*n^3*x^7 + 13132 *(b*x + a)^n*b^8*d^2*n^2*x^8 + (b*x + a)^n*b^8*c^2*n^7*x^2 - 8*(b*x + a)^n *a^2*b^6*c*d*n^6*x^3 + 566*(b*x + a)^n*a*b^7*c*d*n^5*x^4 + 5162*(b*x + a)^ n*b^8*c*d*n^4*x^5 + 420*(b*x + a)^n*a^3*b^5*d^2*n^4*x^5 - 1575*(b*x + a)^n *a^2*b^6*d^2*n^3*x^6 + 1764*(b*x + a)^n*a*b^7*d^2*n^2*x^7 + 13068*(b*x + a )^n*b^8*d^2*n*x^8 + (b*x + a)^n*a*b^7*c^2*n^7*x + 34*(b*x + a)^n*b^8*c^2*n ^6*x^2 - 192*(b*x + a)^n*a^2*b^6*c*d*n^5*x^3 + 2898*(b*x + a)^n*a*b^7*c*d* n^4*x^4 - 210*(b*x + a)^n*a^4*b^4*d^2*n^4*x^4 + 19088*(b*x + a)^n*b^8*c*d* n^3*x^5 + 1470*(b*x + a)^n*a^3*b^5*d^2*n^3*x^5 - 1918*(b*x + a)^n*a^2*b^6* d^2*n^2*x^6 + 720*(b*x + a)^n*a*b^7*d^2*n*x^7 + 5040*(b*x + a)^n*b^8*d^2*x ^8 + 33*(b*x + a)^n*a*b^7*c^2*n^6*x + 478*(b*x + a)^n*b^8*c^2*n^5*x^2 + 24 *(b*x + a)^n*a^3*b^5*c*d*n^5*x^2 - 1688*(b*x + a)^n*a^2*b^6*c*d*n^4*x^3...
Time = 22.48 (sec) , antiderivative size = 1136, normalized size of antiderivative = 4.58 \[ \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx =\text {Too large to display} \] Input:
int(x*(c + d*x^3)^2*(a + b*x)^n,x)
Output:
(d^2*x^8*(a + b*x)^n*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040))/(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4 536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320) - (a^2*(a + b*x)^n*(5040*a^6*d^2 + 20160*b^6*c^2 + 24552*b^6*c^2*n + 12154*b^6*c^2*n^2 + 3135*b^6*c^2*n^3 + 445*b^6*c^2*n^4 + 33*b^6*c^2*n^5 + b^6*c^2*n^6 - 16128*a^3*b^3*c*d - 700 8*a^3*b^3*c*d*n - 1008*a^3*b^3*c*d*n^2 - 48*a^3*b^3*c*d*n^3))/(b^8*(109584 *n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^ 8 + 40320)) + (x^2*(n + 1)*(a + b*x)^n*(20160*b^6*c^2 - 2520*a^6*d^2*n + 2 4552*b^6*c^2*n + 12154*b^6*c^2*n^2 + 3135*b^6*c^2*n^3 + 445*b^6*c^2*n^4 + 33*b^6*c^2*n^5 + b^6*c^2*n^6 + 8064*a^3*b^3*c*d*n + 3504*a^3*b^3*c*d*n^2 + 504*a^3*b^3*c*d*n^3 + 24*a^3*b^3*c*d*n^4))/(b^6*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (a*n *x*(a + b*x)^n*(5040*a^6*d^2 + 20160*b^6*c^2 + 24552*b^6*c^2*n + 12154*b^6 *c^2*n^2 + 3135*b^6*c^2*n^3 + 445*b^6*c^2*n^4 + 33*b^6*c^2*n^5 + b^6*c^2*n ^6 - 16128*a^3*b^3*c*d - 7008*a^3*b^3*c*d*n - 1008*a^3*b^3*c*d*n^2 - 48*a^ 3*b^3*c*d*n^3))/(b^7*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 4536 *n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (2*d*x^5*(a + b*x)^n*(50*n + 35* n^2 + 10*n^3 + n^4 + 24)*(336*b^3*c + 21*b^3*c*n^2 + b^3*c*n^3 + 21*a^3*d* n + 146*b^3*c*n))/(b^3*(109584*n + 118124*n^2 + 67284*n^3 + 22449*n^4 + 45 36*n^5 + 546*n^6 + 36*n^7 + n^8 + 40320)) + (a*d^2*n*x^7*(a + b*x)^n*(1...
Time = 0.24 (sec) , antiderivative size = 1364, normalized size of antiderivative = 5.50 \[ \int x (a+b x)^n \left (c+d x^3\right )^2 \, dx =\text {Too large to display} \] Input:
int(x*(b*x+a)^n*(d*x^3+c)^2,x)
Output:
((a + b*x)**n*( - 5040*a**8*d**2 + 5040*a**7*b*d**2*n*x - 2520*a**6*b**2*d **2*n**2*x**2 - 2520*a**6*b**2*d**2*n*x**2 + 48*a**5*b**3*c*d*n**3 + 1008* a**5*b**3*c*d*n**2 + 7008*a**5*b**3*c*d*n + 16128*a**5*b**3*c*d + 840*a**5 *b**3*d**2*n**3*x**3 + 2520*a**5*b**3*d**2*n**2*x**3 + 1680*a**5*b**3*d**2 *n*x**3 - 48*a**4*b**4*c*d*n**4*x - 1008*a**4*b**4*c*d*n**3*x - 7008*a**4* b**4*c*d*n**2*x - 16128*a**4*b**4*c*d*n*x - 210*a**4*b**4*d**2*n**4*x**4 - 1260*a**4*b**4*d**2*n**3*x**4 - 2310*a**4*b**4*d**2*n**2*x**4 - 1260*a**4 *b**4*d**2*n*x**4 + 24*a**3*b**5*c*d*n**5*x**2 + 528*a**3*b**5*c*d*n**4*x* *2 + 4008*a**3*b**5*c*d*n**3*x**2 + 11568*a**3*b**5*c*d*n**2*x**2 + 8064*a **3*b**5*c*d*n*x**2 + 42*a**3*b**5*d**2*n**5*x**5 + 420*a**3*b**5*d**2*n** 4*x**5 + 1470*a**3*b**5*d**2*n**3*x**5 + 2100*a**3*b**5*d**2*n**2*x**5 + 1 008*a**3*b**5*d**2*n*x**5 - a**2*b**6*c**2*n**6 - 33*a**2*b**6*c**2*n**5 - 445*a**2*b**6*c**2*n**4 - 3135*a**2*b**6*c**2*n**3 - 12154*a**2*b**6*c**2 *n**2 - 24552*a**2*b**6*c**2*n - 20160*a**2*b**6*c**2 - 8*a**2*b**6*c*d*n* *6*x**3 - 192*a**2*b**6*c*d*n**5*x**3 - 1688*a**2*b**6*c*d*n**4*x**3 - 652 8*a**2*b**6*c*d*n**3*x**3 - 10400*a**2*b**6*c*d*n**2*x**3 - 5376*a**2*b**6 *c*d*n*x**3 - 7*a**2*b**6*d**2*n**6*x**6 - 105*a**2*b**6*d**2*n**5*x**6 - 595*a**2*b**6*d**2*n**4*x**6 - 1575*a**2*b**6*d**2*n**3*x**6 - 1918*a**2*b **6*d**2*n**2*x**6 - 840*a**2*b**6*d**2*n*x**6 + a*b**7*c**2*n**7*x + 33*a *b**7*c**2*n**6*x + 445*a*b**7*c**2*n**5*x + 3135*a*b**7*c**2*n**4*x + ...