Integrand size = 17, antiderivative size = 203 \[ \int (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^{1+n}}{b^7 (1+n)}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)^{2+n}}{b^7 (2+n)}-\frac {3 a d \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{3+n}}{b^7 (3+n)}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{4+n}}{b^7 (4+n)}+\frac {15 a^2 d^2 (a+b x)^{5+n}}{b^7 (5+n)}-\frac {6 a d^2 (a+b x)^{6+n}}{b^7 (6+n)}+\frac {d^2 (a+b x)^{7+n}}{b^7 (7+n)} \]
(-a^3*d+b^3*c)^2*(b*x+a)^(1+n)/b^7/(1+n)+6*a^2*d*(-a^3*d+b^3*c)*(b*x+a)^(2 +n)/b^7/(2+n)-3*a*d*(-5*a^3*d+2*b^3*c)*(b*x+a)^(3+n)/b^7/(3+n)+2*d*(-10*a^ 3*d+b^3*c)*(b*x+a)^(4+n)/b^7/(4+n)+15*a^2*d^2*(b*x+a)^(5+n)/b^7/(5+n)-6*a* d^2*(b*x+a)^(6+n)/b^7/(6+n)+d^2*(b*x+a)^(7+n)/b^7/(7+n)
Time = 0.16 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.85 \[ \int (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {(a+b x)^{1+n} \left (\frac {\left (b^3 c-a^3 d\right )^2}{1+n}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)}{2+n}+\frac {3 a d \left (-2 b^3 c+5 a^3 d\right ) (a+b x)^2}{3+n}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^3}{4+n}+\frac {15 a^2 d^2 (a+b x)^4}{5+n}-\frac {6 a d^2 (a+b x)^5}{6+n}+\frac {d^2 (a+b x)^6}{7+n}\right )}{b^7} \]
((a + b*x)^(1 + n)*((b^3*c - a^3*d)^2/(1 + n) + (6*a^2*d*(b^3*c - a^3*d)*( a + b*x))/(2 + n) + (3*a*d*(-2*b^3*c + 5*a^3*d)*(a + b*x)^2)/(3 + n) + (2* d*(b^3*c - 10*a^3*d)*(a + b*x)^3)/(4 + n) + (15*a^2*d^2*(a + b*x)^4)/(5 + n) - (6*a*d^2*(a + b*x)^5)/(6 + n) + (d^2*(a + b*x)^6)/(7 + n)))/b^7
Time = 0.37 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2389, 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.
| \(\displaystyle \int \left (c+d x^3\right )^2 (a+b x)^n \, dx\) |
| \(\Big \downarrow \) 2389 |
| \(\displaystyle \int \left (\frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^n}{b^6}+\frac {3 a d \left (5 a^3 d-2 b^3 c\right ) (a+b x)^{n+2}}{b^6}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+3}}{b^6}+\frac {15 a^2 d^2 (a+b x)^{n+4}}{b^6}-\frac {6 a^2 d \left (a^3 d-b^3 c\right ) (a+b x)^{n+1}}{b^6}-\frac {6 a d^2 (a+b x)^{n+5}}{b^6}+\frac {d^2 (a+b x)^{n+6}}{b^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (b^3 c-a^3 d\right )^2 (a+b x)^{n+1}}{b^7 (n+1)}-\frac {3 a d \left (2 b^3 c-5 a^3 d\right ) (a+b x)^{n+3}}{b^7 (n+3)}+\frac {2 d \left (b^3 c-10 a^3 d\right ) (a+b x)^{n+4}}{b^7 (n+4)}+\frac {15 a^2 d^2 (a+b x)^{n+5}}{b^7 (n+5)}+\frac {6 a^2 d \left (b^3 c-a^3 d\right ) (a+b x)^{n+2}}{b^7 (n+2)}-\frac {6 a d^2 (a+b x)^{n+6}}{b^7 (n+6)}+\frac {d^2 (a+b x)^{n+7}}{b^7 (n+7)}\) |
((b^3*c - a^3*d)^2*(a + b*x)^(1 + n))/(b^7*(1 + n)) + (6*a^2*d*(b^3*c - a^ 3*d)*(a + b*x)^(2 + n))/(b^7*(2 + n)) - (3*a*d*(2*b^3*c - 5*a^3*d)*(a + b* x)^(3 + n))/(b^7*(3 + n)) + (2*d*(b^3*c - 10*a^3*d)*(a + b*x)^(4 + n))/(b^ 7*(4 + n)) + (15*a^2*d^2*(a + b*x)^(5 + n))/(b^7*(5 + n)) - (6*a*d^2*(a + b*x)^(6 + n))/(b^7*(6 + n)) + (d^2*(a + b*x)^(7 + n))/(b^7*(7 + n))
3.2.80.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(699\) vs. \(2(203)=406\).
Time = 0.95 (sec) , antiderivative size = 700, normalized size of antiderivative = 3.45
| method | result | size |
| norman | \(\frac {d^{2} x^{7} {\mathrm e}^{n \ln \left (b x +a \right )}}{7+n}+\frac {a \left (b^{6} c^{2} n^{6}+27 b^{6} c^{2} n^{5}+295 b^{6} c^{2} n^{4}-12 a^{3} b^{3} c d \,n^{3}+1665 b^{6} c^{2} n^{3}-216 a^{3} b^{3} c d \,n^{2}+5104 b^{6} c^{2} n^{2}-1284 a^{3} b^{3} c d n +8028 b^{6} c^{2} n +720 a^{6} d^{2}-2520 a^{3} b^{3} c d +5040 c^{2} b^{6}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{7} \left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right )}+\frac {d^{2} a n \,x^{6} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+13 n +42\right )}-\frac {\left (-b^{6} c^{2} n^{6}-27 b^{6} c^{2} n^{5}-12 a^{3} b^{3} c d \,n^{4}-295 b^{6} c^{2} n^{4}-216 a^{3} b^{3} c d \,n^{3}-1665 b^{6} c^{2} n^{3}-1284 a^{3} b^{3} c d \,n^{2}-5104 b^{6} c^{2} n^{2}+720 a^{6} d^{2} n -2520 a^{3} b^{3} c d n -8028 b^{6} c^{2} n -5040 c^{2} b^{6}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{6} \left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right )}+\frac {2 \left (b^{3} c \,n^{3}+18 b^{3} c \,n^{2}+15 a^{3} d n +107 b^{3} c n +210 b^{3} c \right ) d \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+22 n^{3}+179 n^{2}+638 n +840\right )}-\frac {6 n \,a^{2} d^{2} x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+18 n^{2}+107 n +210\right )}-\frac {2 n a d \left (-b^{3} c \,n^{3}-18 b^{3} c \,n^{2}-107 b^{3} c n +60 a^{3} d -210 b^{3} c \right ) x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+25 n^{4}+245 n^{3}+1175 n^{2}+2754 n +2520\right )}+\frac {6 \left (-b^{3} c \,n^{3}-18 b^{3} c \,n^{2}-107 b^{3} c n +60 a^{3} d -210 b^{3} c \right ) d \,a^{2} n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{6}+27 n^{5}+295 n^{4}+1665 n^{3}+5104 n^{2}+8028 n +5040\right )}\) | \(700\) |
| gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{6} d^{2} n^{6} x^{6}+21 b^{6} d^{2} n^{5} x^{6}-6 a \,b^{5} d^{2} n^{5} x^{5}+175 b^{6} d^{2} n^{4} x^{6}-90 a \,b^{5} d^{2} n^{4} x^{5}+2 b^{6} c d \,n^{6} x^{3}+735 b^{6} d^{2} n^{3} x^{6}+30 a^{2} b^{4} d^{2} n^{4} x^{4}-510 a \,b^{5} d^{2} n^{3} x^{5}+48 b^{6} c d \,n^{5} x^{3}+1624 b^{6} d^{2} n^{2} x^{6}+300 a^{2} b^{4} d^{2} n^{3} x^{4}-6 a \,b^{5} c d \,n^{5} x^{2}-1350 a \,b^{5} d^{2} n^{2} x^{5}+452 b^{6} c d \,n^{4} x^{3}+1764 b^{6} d^{2} n \,x^{6}-120 a^{3} b^{3} d^{2} n^{3} x^{3}+1050 a^{2} b^{4} d^{2} n^{2} x^{4}-126 a \,b^{5} c d \,n^{4} x^{2}-1644 a \,b^{5} d^{2} n \,x^{5}+b^{6} c^{2} n^{6}+2112 b^{6} c d \,n^{3} x^{3}+720 d^{2} x^{6} b^{6}-720 a^{3} b^{3} d^{2} n^{2} x^{3}+12 a^{2} b^{4} c d \,n^{4} x +1500 a^{2} b^{4} d^{2} n \,x^{4}-978 a \,b^{5} c d \,n^{3} x^{2}-720 a \,d^{2} x^{5} b^{5}+27 b^{6} c^{2} n^{5}+5090 b^{6} c d \,n^{2} x^{3}+360 a^{4} b^{2} d^{2} n^{2} x^{2}-1320 a^{3} b^{3} d^{2} n \,x^{3}+228 a^{2} b^{4} c d \,n^{3} x +720 a^{2} d^{2} x^{4} b^{4}-3402 a \,b^{5} c d \,n^{2} x^{2}+295 b^{6} c^{2} n^{4}+5904 b^{6} c d n \,x^{3}+1080 a^{4} b^{2} d^{2} n \,x^{2}-12 a^{3} b^{3} c d \,n^{3}-720 a^{3} b^{3} d^{2} x^{3}+1500 a^{2} b^{4} c d \,n^{2} x -5064 a \,b^{5} c d n \,x^{2}+1665 b^{6} c^{2} n^{3}+2520 b^{6} c d \,x^{3}-720 a^{5} b \,d^{2} n x +720 a^{4} b^{2} d^{2} x^{2}-216 a^{3} b^{3} c d \,n^{2}+3804 a^{2} b^{4} c d n x -2520 a \,b^{5} c d \,x^{2}+5104 b^{6} c^{2} n^{2}-720 a^{5} b \,d^{2} x -1284 a^{3} b^{3} c d n +2520 a^{2} b^{4} c d x +8028 b^{6} c^{2} n +720 a^{6} d^{2}-2520 a^{3} b^{3} c d +5040 c^{2} b^{6}\right )}{b^{7} \left (n^{7}+28 n^{6}+322 n^{5}+1960 n^{4}+6769 n^{3}+13132 n^{2}+13068 n +5040\right )}\) | \(793\) |
| risch | \(\frac {\left (b^{7} d^{2} n^{6} x^{7}+a \,b^{6} d^{2} n^{6} x^{6}+21 b^{7} d^{2} n^{5} x^{7}+15 a \,b^{6} d^{2} n^{5} x^{6}+175 b^{7} d^{2} n^{4} x^{7}-6 a^{2} b^{5} d^{2} n^{5} x^{5}+85 a \,b^{6} d^{2} n^{4} x^{6}+2 b^{7} c d \,n^{6} x^{4}+735 b^{7} d^{2} n^{3} x^{7}-60 a^{2} b^{5} d^{2} n^{4} x^{5}+2 a \,b^{6} c d \,n^{6} x^{3}+225 a \,b^{6} d^{2} n^{3} x^{6}+48 b^{7} c d \,n^{5} x^{4}+1624 b^{7} d^{2} n^{2} x^{7}+30 a^{3} b^{4} d^{2} n^{4} x^{4}-210 a^{2} b^{5} d^{2} n^{3} x^{5}+42 a \,b^{6} c d \,n^{5} x^{3}+274 a \,b^{6} d^{2} n^{2} x^{6}+452 b^{7} c d \,n^{4} x^{4}+1764 b^{7} d^{2} n \,x^{7}+180 a^{3} b^{4} d^{2} n^{3} x^{4}-6 a^{2} b^{5} c d \,n^{5} x^{2}-300 a^{2} b^{5} d^{2} n^{2} x^{5}+326 a \,b^{6} c d \,n^{4} x^{3}+120 a \,d^{2} n \,x^{6} b^{6}+b^{7} c^{2} n^{6} x +2112 b^{7} c d \,n^{3} x^{4}+720 d^{2} x^{7} b^{7}-120 a^{4} b^{3} d^{2} n^{3} x^{3}+330 a^{3} b^{4} d^{2} n^{2} x^{4}-114 a^{2} b^{5} c d \,n^{4} x^{2}-144 a^{2} d^{2} n \,x^{5} b^{5}+a \,b^{6} c^{2} n^{6}+1134 a \,b^{6} c d \,n^{3} x^{3}+27 b^{7} c^{2} n^{5} x +5090 b^{7} c d \,n^{2} x^{4}-360 a^{4} b^{3} d^{2} n^{2} x^{3}+12 a^{3} b^{4} c d \,n^{4} x +180 a^{3} b^{4} d^{2} n \,x^{4}-750 a^{2} b^{5} c d \,n^{3} x^{2}+27 a \,b^{6} c^{2} n^{5}+1688 a \,b^{6} c d \,n^{2} x^{3}+295 b^{7} c^{2} n^{4} x +5904 b^{7} c d n \,x^{4}+360 a^{5} b^{2} d^{2} n^{2} x^{2}-240 a^{4} b^{3} d^{2} n \,x^{3}+216 a^{3} b^{4} c d \,n^{3} x -1902 a^{2} b^{5} c d \,n^{2} x^{2}+295 a \,b^{6} c^{2} n^{4}+840 a \,b^{6} c d n \,x^{3}+1665 b^{7} c^{2} n^{3} x +2520 b^{7} c d \,x^{4}+360 a^{5} b^{2} d^{2} n \,x^{2}-12 a^{4} b^{3} c d \,n^{3}+1284 a^{3} b^{4} c d \,n^{2} x -1260 a^{2} b^{5} c d n \,x^{2}+1665 a \,b^{6} c^{2} n^{3}+5104 b^{7} c^{2} n^{2} x -720 a^{6} b \,d^{2} n x -216 a^{4} b^{3} c d \,n^{2}+2520 a^{3} b^{4} c d n x +5104 a \,b^{6} c^{2} n^{2}+8028 b^{7} c^{2} n x -1284 a^{4} b^{3} c d n +8028 a \,b^{6} c^{2} n +5040 b^{7} c^{2} x +720 a^{7} d^{2}-2520 a^{4} b^{3} c d +5040 a \,b^{6} c^{2}\right ) \left (b x +a \right )^{n}}{\left (6+n \right ) \left (7+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{7}}\) | \(979\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1517\) |
d^2/(7+n)*x^7*exp(n*ln(b*x+a))+a*(b^6*c^2*n^6+27*b^6*c^2*n^5+295*b^6*c^2*n ^4-12*a^3*b^3*c*d*n^3+1665*b^6*c^2*n^3-216*a^3*b^3*c*d*n^2+5104*b^6*c^2*n^ 2-1284*a^3*b^3*c*d*n+8028*b^6*c^2*n+720*a^6*d^2-2520*a^3*b^3*c*d+5040*b^6* c^2)/b^7/(n^7+28*n^6+322*n^5+1960*n^4+6769*n^3+13132*n^2+13068*n+5040)*exp (n*ln(b*x+a))+d^2*a*n/b/(n^2+13*n+42)*x^6*exp(n*ln(b*x+a))-(-b^6*c^2*n^6-2 7*b^6*c^2*n^5-12*a^3*b^3*c*d*n^4-295*b^6*c^2*n^4-216*a^3*b^3*c*d*n^3-1665* b^6*c^2*n^3-1284*a^3*b^3*c*d*n^2-5104*b^6*c^2*n^2+720*a^6*d^2*n-2520*a^3*b ^3*c*d*n-8028*b^6*c^2*n-5040*b^6*c^2)/b^6/(n^7+28*n^6+322*n^5+1960*n^4+676 9*n^3+13132*n^2+13068*n+5040)*x*exp(n*ln(b*x+a))+2*(b^3*c*n^3+18*b^3*c*n^2 +15*a^3*d*n+107*b^3*c*n+210*b^3*c)*d/b^3/(n^4+22*n^3+179*n^2+638*n+840)*x^ 4*exp(n*ln(b*x+a))-6*n*a^2*d^2/b^2/(n^3+18*n^2+107*n+210)*x^5*exp(n*ln(b*x +a))-2*n*a*d*(-b^3*c*n^3-18*b^3*c*n^2-107*b^3*c*n+60*a^3*d-210*b^3*c)/b^4/ (n^5+25*n^4+245*n^3+1175*n^2+2754*n+2520)*x^3*exp(n*ln(b*x+a))+6*(-b^3*c*n ^3-18*b^3*c*n^2-107*b^3*c*n+60*a^3*d-210*b^3*c)*d*a^2/b^5*n/(n^6+27*n^5+29 5*n^4+1665*n^3+5104*n^2+8028*n+5040)*x^2*exp(n*ln(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 893 vs. \(2 (203) = 406\).
Time = 0.28 (sec) , antiderivative size = 893, normalized size of antiderivative = 4.40 \[ \int (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {{\left (a b^{6} c^{2} n^{6} + 27 \, a b^{6} c^{2} n^{5} + 295 \, a b^{6} c^{2} n^{4} + 5040 \, a b^{6} c^{2} - 2520 \, a^{4} b^{3} c d + 720 \, a^{7} d^{2} + {\left (b^{7} d^{2} n^{6} + 21 \, b^{7} d^{2} n^{5} + 175 \, b^{7} d^{2} n^{4} + 735 \, b^{7} d^{2} n^{3} + 1624 \, b^{7} d^{2} n^{2} + 1764 \, b^{7} d^{2} n + 720 \, b^{7} d^{2}\right )} x^{7} + {\left (a b^{6} d^{2} n^{6} + 15 \, a b^{6} d^{2} n^{5} + 85 \, a b^{6} d^{2} n^{4} + 225 \, a b^{6} d^{2} n^{3} + 274 \, a b^{6} d^{2} n^{2} + 120 \, a b^{6} d^{2} n\right )} x^{6} - 6 \, {\left (a^{2} b^{5} d^{2} n^{5} + 10 \, a^{2} b^{5} d^{2} n^{4} + 35 \, a^{2} b^{5} d^{2} n^{3} + 50 \, a^{2} b^{5} d^{2} n^{2} + 24 \, a^{2} b^{5} d^{2} n\right )} x^{5} + 2 \, {\left (b^{7} c d n^{6} + 24 \, b^{7} c d n^{5} + 1260 \, b^{7} c d + {\left (226 \, b^{7} c d + 15 \, a^{3} b^{4} d^{2}\right )} n^{4} + 6 \, {\left (176 \, b^{7} c d + 15 \, a^{3} b^{4} d^{2}\right )} n^{3} + 5 \, {\left (509 \, b^{7} c d + 33 \, a^{3} b^{4} d^{2}\right )} n^{2} + 18 \, {\left (164 \, b^{7} c d + 5 \, a^{3} b^{4} d^{2}\right )} n\right )} x^{4} + 3 \, {\left (555 \, a b^{6} c^{2} - 4 \, a^{4} b^{3} c d\right )} n^{3} + 2 \, {\left (a b^{6} c d n^{6} + 21 \, a b^{6} c d n^{5} + 163 \, a b^{6} c d n^{4} + 3 \, {\left (189 \, a b^{6} c d - 20 \, a^{4} b^{3} d^{2}\right )} n^{3} + 4 \, {\left (211 \, a b^{6} c d - 45 \, a^{4} b^{3} d^{2}\right )} n^{2} + 60 \, {\left (7 \, a b^{6} c d - 2 \, a^{4} b^{3} d^{2}\right )} n\right )} x^{3} + 8 \, {\left (638 \, a b^{6} c^{2} - 27 \, a^{4} b^{3} c d\right )} n^{2} - 6 \, {\left (a^{2} b^{5} c d n^{5} + 19 \, a^{2} b^{5} c d n^{4} + 125 \, a^{2} b^{5} c d n^{3} + {\left (317 \, a^{2} b^{5} c d - 60 \, a^{5} b^{2} d^{2}\right )} n^{2} + 30 \, {\left (7 \, a^{2} b^{5} c d - 2 \, a^{5} b^{2} d^{2}\right )} n\right )} x^{2} + 12 \, {\left (669 \, a b^{6} c^{2} - 107 \, a^{4} b^{3} c d\right )} n + {\left (b^{7} c^{2} n^{6} + 27 \, b^{7} c^{2} n^{5} + 5040 \, b^{7} c^{2} + {\left (295 \, b^{7} c^{2} + 12 \, a^{3} b^{4} c d\right )} n^{4} + 9 \, {\left (185 \, b^{7} c^{2} + 24 \, a^{3} b^{4} c d\right )} n^{3} + 4 \, {\left (1276 \, b^{7} c^{2} + 321 \, a^{3} b^{4} c d\right )} n^{2} + 36 \, {\left (223 \, b^{7} c^{2} + 70 \, a^{3} b^{4} c d - 20 \, a^{6} b d^{2}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{7} n^{7} + 28 \, b^{7} n^{6} + 322 \, b^{7} n^{5} + 1960 \, b^{7} n^{4} + 6769 \, b^{7} n^{3} + 13132 \, b^{7} n^{2} + 13068 \, b^{7} n + 5040 \, b^{7}} \]
(a*b^6*c^2*n^6 + 27*a*b^6*c^2*n^5 + 295*a*b^6*c^2*n^4 + 5040*a*b^6*c^2 - 2 520*a^4*b^3*c*d + 720*a^7*d^2 + (b^7*d^2*n^6 + 21*b^7*d^2*n^5 + 175*b^7*d^ 2*n^4 + 735*b^7*d^2*n^3 + 1624*b^7*d^2*n^2 + 1764*b^7*d^2*n + 720*b^7*d^2) *x^7 + (a*b^6*d^2*n^6 + 15*a*b^6*d^2*n^5 + 85*a*b^6*d^2*n^4 + 225*a*b^6*d^ 2*n^3 + 274*a*b^6*d^2*n^2 + 120*a*b^6*d^2*n)*x^6 - 6*(a^2*b^5*d^2*n^5 + 10 *a^2*b^5*d^2*n^4 + 35*a^2*b^5*d^2*n^3 + 50*a^2*b^5*d^2*n^2 + 24*a^2*b^5*d^ 2*n)*x^5 + 2*(b^7*c*d*n^6 + 24*b^7*c*d*n^5 + 1260*b^7*c*d + (226*b^7*c*d + 15*a^3*b^4*d^2)*n^4 + 6*(176*b^7*c*d + 15*a^3*b^4*d^2)*n^3 + 5*(509*b^7*c *d + 33*a^3*b^4*d^2)*n^2 + 18*(164*b^7*c*d + 5*a^3*b^4*d^2)*n)*x^4 + 3*(55 5*a*b^6*c^2 - 4*a^4*b^3*c*d)*n^3 + 2*(a*b^6*c*d*n^6 + 21*a*b^6*c*d*n^5 + 1 63*a*b^6*c*d*n^4 + 3*(189*a*b^6*c*d - 20*a^4*b^3*d^2)*n^3 + 4*(211*a*b^6*c *d - 45*a^4*b^3*d^2)*n^2 + 60*(7*a*b^6*c*d - 2*a^4*b^3*d^2)*n)*x^3 + 8*(63 8*a*b^6*c^2 - 27*a^4*b^3*c*d)*n^2 - 6*(a^2*b^5*c*d*n^5 + 19*a^2*b^5*c*d*n^ 4 + 125*a^2*b^5*c*d*n^3 + (317*a^2*b^5*c*d - 60*a^5*b^2*d^2)*n^2 + 30*(7*a ^2*b^5*c*d - 2*a^5*b^2*d^2)*n)*x^2 + 12*(669*a*b^6*c^2 - 107*a^4*b^3*c*d)* n + (b^7*c^2*n^6 + 27*b^7*c^2*n^5 + 5040*b^7*c^2 + (295*b^7*c^2 + 12*a^3*b ^4*c*d)*n^4 + 9*(185*b^7*c^2 + 24*a^3*b^4*c*d)*n^3 + 4*(1276*b^7*c^2 + 321 *a^3*b^4*c*d)*n^2 + 36*(223*b^7*c^2 + 70*a^3*b^4*c*d - 20*a^6*b*d^2)*n)*x) *(b*x + a)^n/(b^7*n^7 + 28*b^7*n^6 + 322*b^7*n^5 + 1960*b^7*n^4 + 6769*b^7 *n^3 + 13132*b^7*n^2 + 13068*b^7*n + 5040*b^7)
Leaf count of result is larger than twice the leaf count of optimal. 11851 vs. \(2 (187) = 374\).
Time = 3.53 (sec) , antiderivative size = 11851, normalized size of antiderivative = 58.38 \[ \int (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \]
Piecewise((a**n*(c**2*x + c*d*x**4/2 + d**2*x**7/7), Eq(b, 0)), (60*a**6*d **2*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 12 00*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x** 6) + 147*a**6*d**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x **6) + 360*a**5*b*d**2*x*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 90 0*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**1 2*x**5 + 60*b**13*x**6) + 822*a**5*b*d**2*x/(60*a**6*b**7 + 360*a**5*b**8* x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360* a*b**12*x**5 + 60*b**13*x**6) + 900*a**4*b**2*d**2*x**2*log(a/b + x)/(60*a **6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 9 00*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1875*a**4*b**2*d* *2*x**2/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b **10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) - 2*a* *3*b**3*c*d/(60*a**6*b**7 + 360*a**5*b**8*x + 900*a**4*b**9*x**2 + 1200*a* *3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**12*x**5 + 60*b**13*x**6) + 1200*a**3*b**3*d**2*x**3*log(a/b + x)/(60*a**6*b**7 + 360*a**5*b**8*x + 90 0*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x**4 + 360*a*b**1 2*x**5 + 60*b**13*x**6) + 2200*a**3*b**3*d**2*x**3/(60*a**6*b**7 + 360*a** 5*b**8*x + 900*a**4*b**9*x**2 + 1200*a**3*b**10*x**3 + 900*a**2*b**11*x...
Time = 0.21 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.77 \[ \int (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {{\left (b x + a\right )}^{n + 1} c^{2}}{b {\left (n + 1\right )}} + \frac {2 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} c d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac {{\left ({\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{7} x^{7} + {\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a b^{6} x^{6} - 6 \, {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{2} b^{5} x^{5} + 30 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{3} b^{4} x^{4} - 120 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{4} b^{3} x^{3} + 360 \, {\left (n^{2} + n\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b n x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{n} d^{2}}{{\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{7}} \]
(b*x + a)^(n + 1)*c^2/(b*(n + 1)) + 2*((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6* a^4)*(b*x + a)^n*c*d/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*b^4) + ((n^6 + 2 1*n^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*b^7*x^7 + (n^6 + 15*n ^5 + 85*n^4 + 225*n^3 + 274*n^2 + 120*n)*a*b^6*x^6 - 6*(n^5 + 10*n^4 + 35* n^3 + 50*n^2 + 24*n)*a^2*b^5*x^5 + 30*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^3*b^4 *x^4 - 120*(n^3 + 3*n^2 + 2*n)*a^4*b^3*x^3 + 360*(n^2 + n)*a^5*b^2*x^2 - 7 20*a^6*b*n*x + 720*a^7)*(b*x + a)^n*d^2/((n^7 + 28*n^6 + 322*n^5 + 1960*n^ 4 + 6769*n^3 + 13132*n^2 + 13068*n + 5040)*b^7)
Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (203) = 406\).
Time = 0.37 (sec) , antiderivative size = 1477, normalized size of antiderivative = 7.28 \[ \int (a+b x)^n \left (c+d x^3\right )^2 \, dx=\text {Too large to display} \]
((b*x + a)^n*b^7*d^2*n^6*x^7 + (b*x + a)^n*a*b^6*d^2*n^6*x^6 + 21*(b*x + a )^n*b^7*d^2*n^5*x^7 + 15*(b*x + a)^n*a*b^6*d^2*n^5*x^6 + 175*(b*x + a)^n*b ^7*d^2*n^4*x^7 + 2*(b*x + a)^n*b^7*c*d*n^6*x^4 - 6*(b*x + a)^n*a^2*b^5*d^2 *n^5*x^5 + 85*(b*x + a)^n*a*b^6*d^2*n^4*x^6 + 735*(b*x + a)^n*b^7*d^2*n^3* x^7 + 2*(b*x + a)^n*a*b^6*c*d*n^6*x^3 + 48*(b*x + a)^n*b^7*c*d*n^5*x^4 - 6 0*(b*x + a)^n*a^2*b^5*d^2*n^4*x^5 + 225*(b*x + a)^n*a*b^6*d^2*n^3*x^6 + 16 24*(b*x + a)^n*b^7*d^2*n^2*x^7 + 42*(b*x + a)^n*a*b^6*c*d*n^5*x^3 + 452*(b *x + a)^n*b^7*c*d*n^4*x^4 + 30*(b*x + a)^n*a^3*b^4*d^2*n^4*x^4 - 210*(b*x + a)^n*a^2*b^5*d^2*n^3*x^5 + 274*(b*x + a)^n*a*b^6*d^2*n^2*x^6 + 1764*(b*x + a)^n*b^7*d^2*n*x^7 + (b*x + a)^n*b^7*c^2*n^6*x - 6*(b*x + a)^n*a^2*b^5* c*d*n^5*x^2 + 326*(b*x + a)^n*a*b^6*c*d*n^4*x^3 + 2112*(b*x + a)^n*b^7*c*d *n^3*x^4 + 180*(b*x + a)^n*a^3*b^4*d^2*n^3*x^4 - 300*(b*x + a)^n*a^2*b^5*d ^2*n^2*x^5 + 120*(b*x + a)^n*a*b^6*d^2*n*x^6 + 720*(b*x + a)^n*b^7*d^2*x^7 + (b*x + a)^n*a*b^6*c^2*n^6 + 27*(b*x + a)^n*b^7*c^2*n^5*x - 114*(b*x + a )^n*a^2*b^5*c*d*n^4*x^2 + 1134*(b*x + a)^n*a*b^6*c*d*n^3*x^3 - 120*(b*x + a)^n*a^4*b^3*d^2*n^3*x^3 + 5090*(b*x + a)^n*b^7*c*d*n^2*x^4 + 330*(b*x + a )^n*a^3*b^4*d^2*n^2*x^4 - 144*(b*x + a)^n*a^2*b^5*d^2*n*x^5 + 27*(b*x + a) ^n*a*b^6*c^2*n^5 + 295*(b*x + a)^n*b^7*c^2*n^4*x + 12*(b*x + a)^n*a^3*b^4* c*d*n^4*x - 750*(b*x + a)^n*a^2*b^5*c*d*n^3*x^2 + 1688*(b*x + a)^n*a*b^6*c *d*n^2*x^3 - 360*(b*x + a)^n*a^4*b^3*d^2*n^2*x^3 + 5904*(b*x + a)^n*b^7...
Time = 20.46 (sec) , antiderivative size = 878, normalized size of antiderivative = 4.33 \[ \int (a+b x)^n \left (c+d x^3\right )^2 \, dx=\frac {a\,{\left (a+b\,x\right )}^n\,\left (720\,a^6\,d^2-12\,a^3\,b^3\,c\,d\,n^3-216\,a^3\,b^3\,c\,d\,n^2-1284\,a^3\,b^3\,c\,d\,n-2520\,a^3\,b^3\,c\,d+b^6\,c^2\,n^6+27\,b^6\,c^2\,n^5+295\,b^6\,c^2\,n^4+1665\,b^6\,c^2\,n^3+5104\,b^6\,c^2\,n^2+8028\,b^6\,c^2\,n+5040\,b^6\,c^2\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {d^2\,x^7\,{\left (a+b\,x\right )}^n\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}{n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040}+\frac {x\,{\left (a+b\,x\right )}^n\,\left (-720\,a^6\,b\,d^2\,n+12\,a^3\,b^4\,c\,d\,n^4+216\,a^3\,b^4\,c\,d\,n^3+1284\,a^3\,b^4\,c\,d\,n^2+2520\,a^3\,b^4\,c\,d\,n+b^7\,c^2\,n^6+27\,b^7\,c^2\,n^5+295\,b^7\,c^2\,n^4+1665\,b^7\,c^2\,n^3+5104\,b^7\,c^2\,n^2+8028\,b^7\,c^2\,n+5040\,b^7\,c^2\right )}{b^7\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,d\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )\,\left (15\,d\,a^3\,n+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^3\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {a\,d^2\,n\,x^6\,{\left (a+b\,x\right )}^n\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{b\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {6\,a^2\,d^2\,n\,x^5\,{\left (a+b\,x\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b^2\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}+\frac {2\,a\,d\,n\,x^3\,{\left (a+b\,x\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-60\,d\,a^3+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^4\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )}-\frac {6\,a^2\,d\,n\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-60\,d\,a^3+c\,b^3\,n^3+18\,c\,b^3\,n^2+107\,c\,b^3\,n+210\,c\,b^3\right )}{b^5\,\left (n^7+28\,n^6+322\,n^5+1960\,n^4+6769\,n^3+13132\,n^2+13068\,n+5040\right )} \]
(a*(a + b*x)^n*(720*a^6*d^2 + 5040*b^6*c^2 + 8028*b^6*c^2*n + 5104*b^6*c^2 *n^2 + 1665*b^6*c^2*n^3 + 295*b^6*c^2*n^4 + 27*b^6*c^2*n^5 + b^6*c^2*n^6 - 2520*a^3*b^3*c*d - 1284*a^3*b^3*c*d*n - 216*a^3*b^3*c*d*n^2 - 12*a^3*b^3* c*d*n^3))/(b^7*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n ^6 + n^7 + 5040)) + (d^2*x^7*(a + b*x)^n*(1764*n + 1624*n^2 + 735*n^3 + 17 5*n^4 + 21*n^5 + n^6 + 720))/(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040) + (x*(a + b*x)^n*(5040*b^7*c^2 + 8028*b^7*c ^2*n + 5104*b^7*c^2*n^2 + 1665*b^7*c^2*n^3 + 295*b^7*c^2*n^4 + 27*b^7*c^2* n^5 + b^7*c^2*n^6 - 720*a^6*b*d^2*n + 2520*a^3*b^4*c*d*n + 1284*a^3*b^4*c* d*n^2 + 216*a^3*b^4*c*d*n^3 + 12*a^3*b^4*c*d*n^4))/(b^7*(13068*n + 13132*n ^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (2*d*x^4*(a + b*x)^n*(11*n + 6*n^2 + n^3 + 6)*(210*b^3*c + 18*b^3*c*n^2 + b^3*c*n^3 + 1 5*a^3*d*n + 107*b^3*c*n))/(b^3*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (a*d^2*n*x^6*(a + b*x)^n*(274*n + 225* n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(b*(13068*n + 13132*n^2 + 6769*n^3 + 1 960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) - (6*a^2*d^2*n*x^5*(a + b*x)^n*( 50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(b^2*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 + n^7 + 5040)) + (2*a*d*n*x^3*(a + b*x)^n*(3* n + n^2 + 2)*(210*b^3*c - 60*a^3*d + 18*b^3*c*n^2 + b^3*c*n^3 + 107*b^3*c* n))/(b^4*(13068*n + 13132*n^2 + 6769*n^3 + 1960*n^4 + 322*n^5 + 28*n^6 ...