Integrand size = 36, antiderivative size = 46 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {\left (6 d+6 a x+3 b x^2+2 c x^3\right )^6}{279936} \] Output:
a*x+1/2*b*x^2+1/3*c*x^3+1/279936*(2*c*x^3+3*b*x^2+6*a*x+6*d)^6
Leaf count is larger than twice the leaf count of optimal. \(248\) vs. \(2(46)=92\).
Time = 0.13 (sec) , antiderivative size = 248, normalized size of antiderivative = 5.39 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {x (6 a+x (3 b+2 c x)) \left (46656+46656 d^5+7776 a^5 x^5+243 b^5 x^{10}+810 b^4 c x^{11}+1080 b^3 c^2 x^{12}+720 b^2 c^3 x^{13}+240 b c^4 x^{14}+32 c^5 x^{15}+6480 a^4 x^6 (3 b+2 c x)+2160 a^3 x^7 (3 b+2 c x)^2+360 a^2 x^8 (3 b+2 c x)^3+30 a x^9 (3 b+2 c x)^4+19440 d^4 x (6 a+x (3 b+2 c x))+4320 d^3 x^2 (6 a+x (3 b+2 c x))^2+540 d^2 x^3 (6 a+x (3 b+2 c x))^3+36 d x^4 (6 a+x (3 b+2 c x))^4\right )}{279936} \] Input:
Integrate[(a + b*x + c*x^2)*(1 + (d + a*x + (b*x^2)/2 + (c*x^3)/3)^5),x]
Output:
(x*(6*a + x*(3*b + 2*c*x))*(46656 + 46656*d^5 + 7776*a^5*x^5 + 243*b^5*x^1 0 + 810*b^4*c*x^11 + 1080*b^3*c^2*x^12 + 720*b^2*c^3*x^13 + 240*b*c^4*x^14 + 32*c^5*x^15 + 6480*a^4*x^6*(3*b + 2*c*x) + 2160*a^3*x^7*(3*b + 2*c*x)^2 + 360*a^2*x^8*(3*b + 2*c*x)^3 + 30*a*x^9*(3*b + 2*c*x)^4 + 19440*d^4*x*(6 *a + x*(3*b + 2*c*x)) + 4320*d^3*x^2*(6*a + x*(3*b + 2*c*x))^2 + 540*d^2*x ^3*(6*a + x*(3*b + 2*c*x))^3 + 36*d*x^4*(6*a + x*(3*b + 2*c*x))^4))/279936
Time = 0.54 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2024, 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.
| \(\displaystyle \int \left (a+b x+c x^2\right ) \left (\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^5+1\right ) \, dx\) |
| \(\Big \downarrow \) 2024 |
| \(\displaystyle \int \left (\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^5+1\right )d\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^6+a x+\frac {b x^2}{2}+\frac {c x^3}{3}+d\) |
Input:
Int[(a + b*x + c*x^2)*(1 + (d + a*x + (b*x^2)/2 + (c*x^3)/3)^5),x]
Output:
d + a*x + (b*x^2)/2 + (c*x^3)/3 + (d + a*x + (b*x^2)/2 + (c*x^3)/3)^6/6
Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[P q, x], r = Expon[Qr, x]}, Simp[Coeff[Qr, x, r]/(q*Coeff[Pq, x, q]) Subst[ Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D [Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] & & PolyQ[Qr, x]
Time = 0.65 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85
| method | result | size |
| default | \(\frac {\left (d +x a +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{6}}{6}+d +x a +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}\) | \(39\) |
| norman | \(\left (\frac {1}{243} a \,c^{5}+\frac {5}{648} b^{2} c^{4}\right ) x^{16}+\left (\frac {5}{2} a^{2} d^{4}+\frac {1}{2} b \,d^{5}+\frac {1}{2} b \right ) x^{2}+\left (\frac {5}{162} a b \,c^{4}+\frac {5}{324} b^{3} c^{3}+\frac {1}{243} c^{5} d \right ) x^{15}+\left (\frac {5}{162} a^{2} c^{4}+\frac {5}{54} a \,b^{2} c^{3}+\frac {5}{288} b^{4} c^{2}+\frac {5}{162} b \,c^{4} d \right ) x^{14}+\left (\frac {10}{3} a^{3} d^{3}+\frac {5}{2} a b \,d^{4}+\frac {1}{3} c \,d^{5}+\frac {1}{3} c \right ) x^{3}+\left (\frac {5}{2} a^{4} d^{2}+5 d^{3} b \,a^{2}+\frac {5}{3} d^{4} c a +\frac {5}{8} b^{2} d^{4}\right ) x^{4}+\left (a^{5} d +5 a^{3} b \,d^{2}+\frac {10}{3} a^{2} c \,d^{3}+\frac {5}{2} a \,b^{2} d^{3}+\frac {5}{6} b \,d^{4} c \right ) x^{5}+\left (\frac {5}{27} a^{2} b \,c^{3}+\frac {5}{36} a \,b^{3} c^{2}+\frac {5}{81} a \,c^{4} d +\frac {1}{96} b^{5} c +\frac {5}{54} d \,b^{2} c^{3}\right ) x^{13}+\left (\frac {1}{6} a^{6}+\frac {5}{2} a^{4} b d +\frac {10}{3} a^{3} c \,d^{2}+\frac {15}{4} a^{2} b^{2} d^{2}+\frac {10}{3} a b c \,d^{3}+\frac {5}{12} b^{3} d^{3}+\frac {5}{18} c^{2} d^{4}\right ) x^{6}+\left (\frac {10}{81} a^{3} c^{3}+\frac {5}{12} a^{2} b^{2} c^{2}+\frac {5}{48} a \,b^{4} c +\frac {10}{27} a b \,c^{3} d +\frac {1}{384} b^{6}+\frac {5}{36} b^{3} c^{2} d +\frac {5}{162} c^{4} d^{2}\right ) x^{12}+\left (\frac {1}{2} a^{5} b +\frac {5}{3} a^{4} c d +\frac {5}{2} a^{3} b^{2} d +5 a^{2} b c \,d^{2}+\frac {5}{4} a \,b^{3} d^{2}+\frac {10}{9} a \,c^{2} d^{3}+\frac {5}{6} b^{2} c \,d^{3}\right ) x^{7}+\left (\frac {5}{9} a^{3} b \,c^{2}+\frac {5}{12} a^{2} b^{3} c +\frac {10}{27} a^{2} c^{3} d +\frac {1}{32} b^{5} a +\frac {5}{6} d \,c^{2} b^{2} a +\frac {5}{48} d c \,b^{4}+\frac {5}{27} b \,c^{3} d^{2}\right ) x^{11}+\left (\frac {5}{18} a^{4} c^{2}+\frac {5}{6} a^{3} b^{2} c +\frac {5}{32} b^{4} a^{2}+\frac {5}{3} a^{2} b \,c^{2} d +\frac {5}{6} a \,b^{3} c d +\frac {10}{27} a \,c^{3} d^{2}+\frac {1}{32} b^{5} d +\frac {5}{12} b^{2} c^{2} d^{2}\right ) x^{10}+\left (\frac {1}{3} a^{5} c +\frac {5}{8} a^{4} b^{2}+\frac {10}{3} a^{3} b c d +\frac {5}{4} a^{2} b^{3} d +\frac {5}{3} a^{2} c^{2} d^{2}+\frac {5}{2} a \,b^{2} c \,d^{2}+\frac {5}{32} b^{4} d^{2}+\frac {5}{9} b \,c^{2} d^{3}\right ) x^{8}+\left (\frac {5}{6} a^{4} b c +\frac {5}{12} b^{3} a^{3}+\frac {10}{9} a^{3} c^{2} d +\frac {5}{2} a^{2} b^{2} c d +\frac {5}{16} a \,b^{4} d +\frac {5}{3} a b \,c^{2} d^{2}+\frac {5}{12} b^{3} c \,d^{2}+\frac {10}{81} c^{3} d^{3}\right ) x^{9}+\left (a \,d^{5}+a \right ) x +\frac {c^{6} x^{18}}{4374}+\frac {b \,c^{5} x^{17}}{486}\) | \(758\) |
| gosper | \(\frac {x \left (64 c^{6} x^{17}+576 b \,c^{5} x^{16}+1152 a \,c^{5} x^{15}+2160 b^{2} c^{4} x^{15}+8640 a b \,c^{4} x^{14}+4320 b^{3} c^{3} x^{14}+1152 c^{5} d \,x^{14}+8640 a^{2} c^{4} x^{13}+25920 a \,b^{2} c^{3} x^{13}+4860 b^{4} c^{2} x^{13}+8640 b \,c^{4} d \,x^{13}+51840 a^{2} b \,c^{3} x^{12}+38880 a \,b^{3} c^{2} x^{12}+17280 a \,c^{4} d \,x^{12}+2916 b^{5} c \,x^{12}+25920 b^{2} c^{3} d \,x^{12}+34560 a^{3} c^{3} x^{11}+116640 x^{11} a^{2} b^{2} c^{2}+29160 x^{11} a \,b^{4} c +103680 x^{11} a b \,c^{3} d +729 b^{6} x^{11}+38880 x^{11} b^{3} c^{2} d +8640 x^{11} c^{4} d^{2}+155520 a^{3} b \,c^{2} x^{10}+116640 a^{2} b^{3} c \,x^{10}+103680 a^{2} c^{3} d \,x^{10}+8748 a \,b^{5} x^{10}+233280 a \,b^{2} c^{2} d \,x^{10}+29160 b^{4} c d \,x^{10}+51840 b \,c^{3} d^{2} x^{10}+77760 a^{4} c^{2} x^{9}+233280 a^{3} b^{2} c \,x^{9}+43740 a^{2} b^{4} x^{9}+466560 a^{2} b \,c^{2} d \,x^{9}+233280 a \,b^{3} c d \,x^{9}+103680 x^{9} a \,c^{3} d^{2}+8748 b^{5} d \,x^{9}+116640 b^{2} c^{2} d^{2} x^{9}+233280 a^{4} b c \,x^{8}+116640 a^{3} b^{3} x^{8}+311040 a^{3} c^{2} d \,x^{8}+699840 a^{2} b^{2} c d \,x^{8}+87480 a \,b^{4} d \,x^{8}+466560 a b \,c^{2} d^{2} x^{8}+116640 b^{3} c \,d^{2} x^{8}+34560 c^{3} d^{3} x^{8}+93312 a^{5} c \,x^{7}+174960 a^{4} b^{2} x^{7}+933120 a^{3} b c d \,x^{7}+349920 a^{2} b^{3} d \,x^{7}+466560 a^{2} c^{2} d^{2} x^{7}+699840 a \,b^{2} c \,d^{2} x^{7}+43740 b^{4} d^{2} x^{7}+155520 b \,c^{2} d^{3} x^{7}+139968 a^{5} b \,x^{6}+466560 a^{4} c d \,x^{6}+699840 a^{3} b^{2} d \,x^{6}+1399680 a^{2} b c \,d^{2} x^{6}+349920 a \,b^{3} d^{2} x^{6}+311040 a \,c^{2} d^{3} x^{6}+233280 b^{2} c \,d^{3} x^{6}+46656 a^{6} x^{5}+699840 a^{4} b d \,x^{5}+933120 x^{5} a^{3} c \,d^{2}+1049760 a^{2} b^{2} d^{2} x^{5}+933120 a b c \,d^{3} x^{5}+116640 b^{3} d^{3} x^{5}+77760 c^{2} d^{4} x^{5}+279936 a^{5} d \,x^{4}+1399680 a^{3} b \,d^{2} x^{4}+933120 a^{2} c \,d^{3} x^{4}+699840 a \,b^{2} d^{3} x^{4}+233280 b c \,d^{4} x^{4}+699840 x^{3} a^{4} d^{2}+1399680 a^{2} b \,d^{3} x^{3}+466560 x^{3} d^{4} c a +174960 b^{2} d^{4} x^{3}+933120 a^{3} d^{3} x^{2}+699840 a b \,d^{4} x^{2}+93312 c \,d^{5} x^{2}+699840 a^{2} d^{4} x +139968 b \,d^{5} x +279936 a \,d^{5}+93312 c \,x^{2}+139968 b x +279936 a \right )}{279936}\) | \(927\) |
| risch | \(\frac {1}{32} x^{11} b^{5} a +\frac {5}{18} x^{6} c^{2} d^{4}+\frac {10}{3} x^{3} a^{3} d^{3}+\frac {1}{3} x^{3} c \,d^{5}+\frac {5}{8} x^{8} a^{4} b^{2}+\frac {10}{27} x^{12} a b \,c^{3} d +\frac {5}{6} x^{11} d \,c^{2} b^{2} a +\frac {5}{3} x^{10} a^{2} b \,c^{2} d +\frac {5}{6} x^{10} a \,b^{3} c d +\frac {5}{2} x^{9} a^{2} b^{2} c d +\frac {5}{3} x^{9} a b \,c^{2} d^{2}+\frac {1}{32} x^{10} b^{5} d +\frac {1}{3} c \,x^{3}+\frac {5}{162} x^{14} b \,c^{4} d +\frac {5}{54} x^{13} d \,b^{2} c^{3}+\frac {1}{2} b \,x^{2}+\frac {5}{36} x^{12} b^{3} c^{2} d +\frac {5}{48} x^{11} d c \,b^{4}+\frac {5}{27} x^{11} b \,c^{3} d^{2}+\frac {5}{12} x^{10} b^{2} c^{2} d^{2}+\frac {5}{12} x^{9} b^{3} c \,d^{2}+\frac {5}{9} x^{8} b \,c^{2} d^{3}+\frac {5}{81} a \,c^{4} d \,x^{13}+\frac {10}{27} a^{2} c^{3} d \,x^{11}+\frac {5}{6} b^{2} c \,d^{3} x^{7}+\frac {5}{2} x^{6} a^{4} b d +\frac {15}{4} x^{6} a^{2} b^{2} d^{2}+5 x^{5} a^{3} b \,d^{2}+\frac {5}{2} x^{5} a \,b^{2} d^{3}+5 x^{4} d^{3} b \,a^{2}+\frac {5}{2} x^{3} a b \,d^{4}+\frac {10}{27} x^{10} a \,c^{3} d^{2}+\frac {10}{9} x^{9} a^{3} c^{2} d +\frac {5}{3} x^{8} a^{2} c^{2} d^{2}+\frac {5}{3} x^{7} a^{4} c d +\frac {10}{9} x^{7} a \,c^{2} d^{3}+\frac {10}{3} x^{6} a^{3} c \,d^{2}+\frac {10}{3} x^{5} a^{2} c \,d^{3}+\frac {5}{3} x^{4} d^{4} c a +\frac {5}{6} b \,d^{4} c \,x^{5}+\frac {5}{2} a^{2} d^{4} x^{2}+\frac {5}{32} x^{8} b^{4} d^{2}+\frac {5}{18} a^{4} c^{2} x^{10}+a \,d^{5} x +\frac {1}{4374} c^{6} x^{18}+x a +\frac {5}{162} x^{12} c^{4} d^{2}+\frac {1}{243} a \,c^{5} x^{16}+\frac {1}{384} b^{6} x^{12}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {5}{12} x^{9} b^{3} a^{3}+\frac {1}{6} a^{6} x^{6}+\frac {5}{32} x^{10} b^{4} a^{2}+\frac {1}{243} c^{5} d \,x^{15}+\frac {10}{81} x^{9} c^{3} d^{3}+\frac {1}{2} b \,d^{5} x^{2}+\frac {5}{2} x^{4} a^{4} d^{2}+\frac {5}{16} x^{9} a \,b^{4} d +\frac {5}{4} x^{8} a^{2} b^{3} d +\frac {5}{2} x^{7} a^{3} b^{2} d +\frac {5}{4} x^{7} a \,b^{3} d^{2}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {5}{12} x^{6} b^{3} d^{3}+\frac {1}{2} a^{5} b \,x^{7}+\frac {5}{8} b^{2} d^{4} x^{4}+\frac {1}{3} a^{5} c \,x^{8}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{162} x^{15} a b \,c^{4}+\frac {5}{54} x^{14} a \,b^{2} c^{3}+\frac {5}{27} x^{13} a^{2} b \,c^{3}+\frac {5}{36} x^{13} a \,b^{3} c^{2}+\frac {5}{12} x^{12} a^{2} b^{2} c^{2}+\frac {5}{48} x^{12} a \,b^{4} c +\frac {5}{9} x^{11} a^{3} b \,c^{2}+\frac {5}{12} x^{11} a^{2} b^{3} c +\frac {5}{6} x^{10} a^{3} b^{2} c +\frac {5}{6} x^{9} a^{4} b c +\frac {1}{96} b^{5} c \,x^{13}+x^{5} a^{5} d +\frac {10}{81} a^{3} c^{3} x^{12}+\frac {10}{3} x^{8} a^{3} b c d +\frac {5}{2} x^{8} a \,b^{2} c \,d^{2}+5 x^{7} a^{2} b c \,d^{2}+\frac {10}{3} x^{6} a b c \,d^{3}\) | \(929\) |
| parallelrisch | \(\frac {1}{32} x^{11} b^{5} a +\frac {5}{18} x^{6} c^{2} d^{4}+\frac {10}{3} x^{3} a^{3} d^{3}+\frac {1}{3} x^{3} c \,d^{5}+\frac {5}{8} x^{8} a^{4} b^{2}+\frac {10}{27} x^{12} a b \,c^{3} d +\frac {5}{6} x^{11} d \,c^{2} b^{2} a +\frac {5}{3} x^{10} a^{2} b \,c^{2} d +\frac {5}{6} x^{10} a \,b^{3} c d +\frac {5}{2} x^{9} a^{2} b^{2} c d +\frac {5}{3} x^{9} a b \,c^{2} d^{2}+\frac {1}{32} x^{10} b^{5} d +\frac {1}{3} c \,x^{3}+\frac {5}{162} x^{14} b \,c^{4} d +\frac {5}{54} x^{13} d \,b^{2} c^{3}+\frac {1}{2} b \,x^{2}+\frac {5}{36} x^{12} b^{3} c^{2} d +\frac {5}{48} x^{11} d c \,b^{4}+\frac {5}{27} x^{11} b \,c^{3} d^{2}+\frac {5}{12} x^{10} b^{2} c^{2} d^{2}+\frac {5}{12} x^{9} b^{3} c \,d^{2}+\frac {5}{9} x^{8} b \,c^{2} d^{3}+\frac {5}{81} a \,c^{4} d \,x^{13}+\frac {10}{27} a^{2} c^{3} d \,x^{11}+\frac {5}{6} b^{2} c \,d^{3} x^{7}+\frac {5}{2} x^{6} a^{4} b d +\frac {15}{4} x^{6} a^{2} b^{2} d^{2}+5 x^{5} a^{3} b \,d^{2}+\frac {5}{2} x^{5} a \,b^{2} d^{3}+5 x^{4} d^{3} b \,a^{2}+\frac {5}{2} x^{3} a b \,d^{4}+\frac {10}{27} x^{10} a \,c^{3} d^{2}+\frac {10}{9} x^{9} a^{3} c^{2} d +\frac {5}{3} x^{8} a^{2} c^{2} d^{2}+\frac {5}{3} x^{7} a^{4} c d +\frac {10}{9} x^{7} a \,c^{2} d^{3}+\frac {10}{3} x^{6} a^{3} c \,d^{2}+\frac {10}{3} x^{5} a^{2} c \,d^{3}+\frac {5}{3} x^{4} d^{4} c a +\frac {5}{6} b \,d^{4} c \,x^{5}+\frac {5}{2} a^{2} d^{4} x^{2}+\frac {5}{32} x^{8} b^{4} d^{2}+\frac {5}{18} a^{4} c^{2} x^{10}+a \,d^{5} x +\frac {1}{4374} c^{6} x^{18}+x a +\frac {5}{162} x^{12} c^{4} d^{2}+\frac {1}{243} a \,c^{5} x^{16}+\frac {1}{384} b^{6} x^{12}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {5}{12} x^{9} b^{3} a^{3}+\frac {1}{6} a^{6} x^{6}+\frac {5}{32} x^{10} b^{4} a^{2}+\frac {1}{243} c^{5} d \,x^{15}+\frac {10}{81} x^{9} c^{3} d^{3}+\frac {1}{2} b \,d^{5} x^{2}+\frac {5}{2} x^{4} a^{4} d^{2}+\frac {5}{16} x^{9} a \,b^{4} d +\frac {5}{4} x^{8} a^{2} b^{3} d +\frac {5}{2} x^{7} a^{3} b^{2} d +\frac {5}{4} x^{7} a \,b^{3} d^{2}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {5}{12} x^{6} b^{3} d^{3}+\frac {1}{2} a^{5} b \,x^{7}+\frac {5}{8} b^{2} d^{4} x^{4}+\frac {1}{3} a^{5} c \,x^{8}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{162} x^{15} a b \,c^{4}+\frac {5}{54} x^{14} a \,b^{2} c^{3}+\frac {5}{27} x^{13} a^{2} b \,c^{3}+\frac {5}{36} x^{13} a \,b^{3} c^{2}+\frac {5}{12} x^{12} a^{2} b^{2} c^{2}+\frac {5}{48} x^{12} a \,b^{4} c +\frac {5}{9} x^{11} a^{3} b \,c^{2}+\frac {5}{12} x^{11} a^{2} b^{3} c +\frac {5}{6} x^{10} a^{3} b^{2} c +\frac {5}{6} x^{9} a^{4} b c +\frac {1}{96} b^{5} c \,x^{13}+x^{5} a^{5} d +\frac {10}{81} a^{3} c^{3} x^{12}+\frac {10}{3} x^{8} a^{3} b c d +\frac {5}{2} x^{8} a \,b^{2} c \,d^{2}+5 x^{7} a^{2} b c \,d^{2}+\frac {10}{3} x^{6} a b c \,d^{3}\) | \(929\) |
| orering | \(\text {Expression too large to display}\) | \(1560\) |
Input:
int((c*x^2+b*x+a)*(1+(d+x*a+1/2*b*x^2+1/3*c*x^3)^5),x,method=_RETURNVERBOS E)
Output:
1/6*(d+x*a+1/2*b*x^2+1/3*c*x^3)^6+d+x*a+1/2*b*x^2+1/3*c*x^3
Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (40) = 80\).
Time = 0.08 (sec) , antiderivative size = 773, normalized size of antiderivative = 16.80 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="fr icas")
Output:
1/4374*c^6*x^18 + 1/486*b*c^5*x^17 + 1/1944*(15*b^2*c^4 + 8*a*c^5)*x^16 + 1/972*(15*b^3*c^3 + 30*a*b*c^4 + 4*c^5*d)*x^15 + 5/2592*(9*b^4*c^2 + 48*a* b^2*c^3 + 16*a^2*c^4 + 16*b*c^4*d)*x^14 + 1/2592*(27*b^5*c + 360*a*b^3*c^2 + 480*a^2*b*c^3 + 80*(3*b^2*c^3 + 2*a*c^4)*d)*x^13 + 1/10368*(27*b^6 + 10 80*a*b^4*c + 4320*a^2*b^2*c^2 + 1280*a^3*c^3 + 320*c^4*d^2 + 480*(3*b^3*c^ 2 + 8*a*b*c^3)*d)*x^12 + 1/864*(27*a*b^5 + 360*a^2*b^3*c + 480*a^3*b*c^2 + 160*b*c^3*d^2 + 10*(9*b^4*c + 72*a*b^2*c^2 + 32*a^2*c^3)*d)*x^11 + 1/864* (135*a^2*b^4 + 720*a^3*b^2*c + 240*a^4*c^2 + 40*(9*b^2*c^2 + 8*a*c^3)*d^2 + 9*(3*b^5 + 80*a*b^3*c + 160*a^2*b*c^2)*d)*x^10 + 5/1296*(108*a^3*b^3 + 2 16*a^4*b*c + 32*c^3*d^3 + 108*(b^3*c + 4*a*b*c^2)*d^2 + 9*(9*a*b^4 + 72*a^ 2*b^2*c + 32*a^3*c^2)*d)*x^9 + 1/288*(180*a^4*b^2 + 96*a^5*c + 160*b*c^2*d ^3 + 15*(3*b^4 + 48*a*b^2*c + 32*a^2*c^2)*d^2 + 120*(3*a^2*b^3 + 8*a^3*b*c )*d)*x^8 + 1/36*(18*a^5*b + 10*(3*b^2*c + 4*a*c^2)*d^3 + 45*(a*b^3 + 4*a^2 *b*c)*d^2 + 30*(3*a^3*b^2 + 2*a^4*c)*d)*x^7 + 1/36*(6*a^6 + 90*a^4*b*d + 1 0*c^2*d^4 + 15*(b^3 + 8*a*b*c)*d^3 + 15*(9*a^2*b^2 + 8*a^3*c)*d^2)*x^6 + 1 /6*(6*a^5*d + 30*a^3*b*d^2 + 5*b*c*d^4 + 5*(3*a*b^2 + 4*a^2*c)*d^3)*x^5 + 5/24*(12*a^4*d^2 + 24*a^2*b*d^3 + (3*b^2 + 8*a*c)*d^4)*x^4 + 1/6*(20*a^3*d ^3 + 15*a*b*d^4 + 2*c*d^5 + 2*c)*x^3 + 1/2*(5*a^2*d^4 + b*d^5 + b)*x^2 + ( a*d^5 + a)*x
Leaf count of result is larger than twice the leaf count of optimal. 930 vs. \(2 (41) = 82\).
Time = 0.13 (sec) , antiderivative size = 930, normalized size of antiderivative = 20.22 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx =\text {Too large to display} \] Input:
integrate((c*x**2+b*x+a)*(1+(d+a*x+1/2*b*x**2+1/3*c*x**3)**5),x)
Output:
b*c**5*x**17/486 + c**6*x**18/4374 + x**16*(a*c**5/243 + 5*b**2*c**4/648) + x**15*(5*a*b*c**4/162 + 5*b**3*c**3/324 + c**5*d/243) + x**14*(5*a**2*c* *4/162 + 5*a*b**2*c**3/54 + 5*b**4*c**2/288 + 5*b*c**4*d/162) + x**13*(5*a **2*b*c**3/27 + 5*a*b**3*c**2/36 + 5*a*c**4*d/81 + b**5*c/96 + 5*b**2*c**3 *d/54) + x**12*(10*a**3*c**3/81 + 5*a**2*b**2*c**2/12 + 5*a*b**4*c/48 + 10 *a*b*c**3*d/27 + b**6/384 + 5*b**3*c**2*d/36 + 5*c**4*d**2/162) + x**11*(5 *a**3*b*c**2/9 + 5*a**2*b**3*c/12 + 10*a**2*c**3*d/27 + a*b**5/32 + 5*a*b* *2*c**2*d/6 + 5*b**4*c*d/48 + 5*b*c**3*d**2/27) + x**10*(5*a**4*c**2/18 + 5*a**3*b**2*c/6 + 5*a**2*b**4/32 + 5*a**2*b*c**2*d/3 + 5*a*b**3*c*d/6 + 10 *a*c**3*d**2/27 + b**5*d/32 + 5*b**2*c**2*d**2/12) + x**9*(5*a**4*b*c/6 + 5*a**3*b**3/12 + 10*a**3*c**2*d/9 + 5*a**2*b**2*c*d/2 + 5*a*b**4*d/16 + 5* a*b*c**2*d**2/3 + 5*b**3*c*d**2/12 + 10*c**3*d**3/81) + x**8*(a**5*c/3 + 5 *a**4*b**2/8 + 10*a**3*b*c*d/3 + 5*a**2*b**3*d/4 + 5*a**2*c**2*d**2/3 + 5* a*b**2*c*d**2/2 + 5*b**4*d**2/32 + 5*b*c**2*d**3/9) + x**7*(a**5*b/2 + 5*a **4*c*d/3 + 5*a**3*b**2*d/2 + 5*a**2*b*c*d**2 + 5*a*b**3*d**2/4 + 10*a*c** 2*d**3/9 + 5*b**2*c*d**3/6) + x**6*(a**6/6 + 5*a**4*b*d/2 + 10*a**3*c*d**2 /3 + 15*a**2*b**2*d**2/4 + 10*a*b*c*d**3/3 + 5*b**3*d**3/12 + 5*c**2*d**4/ 18) + x**5*(a**5*d + 5*a**3*b*d**2 + 10*a**2*c*d**3/3 + 5*a*b**2*d**3/2 + 5*b*c*d**4/6) + x**4*(5*a**4*d**2/2 + 5*a**2*b*d**3 + 5*a*c*d**4/3 + 5*b** 2*d**4/8) + x**3*(10*a**3*d**3/3 + 5*a*b*d**4/2 + c*d**5/3 + c/3) + x**...
Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (40) = 80\).
Time = 0.04 (sec) , antiderivative size = 773, normalized size of antiderivative = 16.80 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="ma xima")
Output:
1/4374*c^6*x^18 + 1/486*b*c^5*x^17 + 1/1944*(15*b^2*c^4 + 8*a*c^5)*x^16 + 1/972*(15*b^3*c^3 + 30*a*b*c^4 + 4*c^5*d)*x^15 + 5/2592*(9*b^4*c^2 + 48*a* b^2*c^3 + 16*a^2*c^4 + 16*b*c^4*d)*x^14 + 1/2592*(27*b^5*c + 360*a*b^3*c^2 + 480*a^2*b*c^3 + 80*(3*b^2*c^3 + 2*a*c^4)*d)*x^13 + 1/10368*(27*b^6 + 10 80*a*b^4*c + 4320*a^2*b^2*c^2 + 1280*a^3*c^3 + 320*c^4*d^2 + 480*(3*b^3*c^ 2 + 8*a*b*c^3)*d)*x^12 + 1/864*(27*a*b^5 + 360*a^2*b^3*c + 480*a^3*b*c^2 + 160*b*c^3*d^2 + 10*(9*b^4*c + 72*a*b^2*c^2 + 32*a^2*c^3)*d)*x^11 + 1/864* (135*a^2*b^4 + 720*a^3*b^2*c + 240*a^4*c^2 + 40*(9*b^2*c^2 + 8*a*c^3)*d^2 + 9*(3*b^5 + 80*a*b^3*c + 160*a^2*b*c^2)*d)*x^10 + 5/1296*(108*a^3*b^3 + 2 16*a^4*b*c + 32*c^3*d^3 + 108*(b^3*c + 4*a*b*c^2)*d^2 + 9*(9*a*b^4 + 72*a^ 2*b^2*c + 32*a^3*c^2)*d)*x^9 + 1/288*(180*a^4*b^2 + 96*a^5*c + 160*b*c^2*d ^3 + 15*(3*b^4 + 48*a*b^2*c + 32*a^2*c^2)*d^2 + 120*(3*a^2*b^3 + 8*a^3*b*c )*d)*x^8 + 1/36*(18*a^5*b + 10*(3*b^2*c + 4*a*c^2)*d^3 + 45*(a*b^3 + 4*a^2 *b*c)*d^2 + 30*(3*a^3*b^2 + 2*a^4*c)*d)*x^7 + 1/36*(6*a^6 + 90*a^4*b*d + 1 0*c^2*d^4 + 15*(b^3 + 8*a*b*c)*d^3 + 15*(9*a^2*b^2 + 8*a^3*c)*d^2)*x^6 + 1 /6*(6*a^5*d + 30*a^3*b*d^2 + 5*b*c*d^4 + 5*(3*a*b^2 + 4*a^2*c)*d^3)*x^5 + 5/24*(12*a^4*d^2 + 24*a^2*b*d^3 + (3*b^2 + 8*a*c)*d^4)*x^4 + 1/6*(20*a^3*d ^3 + 15*a*b*d^4 + 2*c*d^5 + 2*c)*x^3 + 1/2*(5*a^2*d^4 + b*d^5 + b)*x^2 + ( a*d^5 + a)*x
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (40) = 80\).
Time = 0.12 (sec) , antiderivative size = 153, normalized size of antiderivative = 3.33 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{6} + \frac {1}{7776} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{5} d + \frac {5}{2592} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{4} d^{2} + \frac {5}{324} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{3} d^{3} + \frac {5}{72} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{2} d^{4} + \frac {1}{6} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} d^{5} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \] Input:
integrate((c*x^2+b*x+a)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="gi ac")
Output:
1/279936*(2*c*x^3 + 3*b*x^2 + 6*a*x)^6 + 1/7776*(2*c*x^3 + 3*b*x^2 + 6*a*x )^5*d + 5/2592*(2*c*x^3 + 3*b*x^2 + 6*a*x)^4*d^2 + 5/324*(2*c*x^3 + 3*b*x^ 2 + 6*a*x)^3*d^3 + 5/72*(2*c*x^3 + 3*b*x^2 + 6*a*x)^2*d^4 + 1/6*(2*c*x^3 + 3*b*x^2 + 6*a*x)*d^5 + 1/3*c*x^3 + 1/2*b*x^2 + a*x
Time = 22.29 (sec) , antiderivative size = 753, normalized size of antiderivative = 16.37 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx =\text {Too large to display} \] Input:
int(((d + a*x + (b*x^2)/2 + (c*x^3)/3)^5 + 1)*(a + b*x + c*x^2),x)
Output:
x^10*((b^5*d)/32 + (5*a^2*b^4)/32 + (5*a^4*c^2)/18 + (5*a^3*b^2*c)/6 + (10 *a*c^3*d^2)/27 + (5*b^2*c^2*d^2)/12 + (5*a*b^3*c*d)/6 + (5*a^2*b*c^2*d)/3) + x^8*((a^5*c)/3 + (5*a^4*b^2)/8 + (5*b^4*d^2)/32 + (5*a^2*b^3*d)/4 + (5* b*c^2*d^3)/9 + (5*a^2*c^2*d^2)/3 + (10*a^3*b*c*d)/3 + (5*a*b^2*c*d^2)/2) + x^9*((5*a^3*b^3)/12 + (10*c^3*d^3)/81 + (10*a^3*c^2*d)/9 + (5*b^3*c*d^2)/ 12 + (5*a^4*b*c)/6 + (5*a*b^4*d)/16 + (5*a*b*c^2*d^2)/3 + (5*a^2*b^2*c*d)/ 2) + x^14*((5*a^2*c^4)/162 + (5*b^4*c^2)/288 + (5*a*b^2*c^3)/54 + (5*b*c^4 *d)/162) + x^12*(b^6/384 + (10*a^3*c^3)/81 + (5*c^4*d^2)/162 + (5*b^3*c^2* d)/36 + (5*a^2*b^2*c^2)/12 + (5*a*b^4*c)/48 + (10*a*b*c^3*d)/27) + x^6*(a^ 6/6 + (5*b^3*d^3)/12 + (5*c^2*d^4)/18 + (10*a^3*c*d^2)/3 + (15*a^2*b^2*d^2 )/4 + (5*a^4*b*d)/2 + (10*a*b*c*d^3)/3) + x^3*(c/3 + (c*d^5)/3 + (10*a^3*d ^3)/3 + (5*a*b*d^4)/2) + x^11*((a*b^5)/32 + (5*a^2*b^3*c)/12 + (5*a^3*b*c^ 2)/9 + (10*a^2*c^3*d)/27 + (5*b*c^3*d^2)/27 + (5*b^4*c*d)/48 + (5*a*b^2*c^ 2*d)/6) + x^7*((a^5*b)/2 + (5*a*b^3*d^2)/4 + (5*a^3*b^2*d)/2 + (10*a*c^2*d ^3)/9 + (5*b^2*c*d^3)/6 + (5*a^4*c*d)/3 + 5*a^2*b*c*d^2) + x^2*(b/2 + (b*d ^5)/2 + (5*a^2*d^4)/2) + x^13*((b^5*c)/96 + (5*a*b^3*c^2)/36 + (5*a^2*b*c^ 3)/27 + (5*b^2*c^3*d)/54 + (5*a*c^4*d)/81) + x^5*(a^5*d + (5*a*b^2*d^3)/2 + 5*a^3*b*d^2 + (10*a^2*c*d^3)/3 + (5*b*c*d^4)/6) + (c^6*x^18)/4374 + (5*d ^2*x^4*(12*a^4 + 3*b^2*d^2 + 24*a^2*b*d + 8*a*c*d^2))/24 + a*x*(d^5 + 1) + (b*c^5*x^17)/486 + (c^3*x^15*(4*c^2*d + 15*b^3 + 30*a*b*c))/972 + (c^4...
Time = 0.15 (sec) , antiderivative size = 926, normalized size of antiderivative = 20.13 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x+a)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x)
Output:
(x*(46656*a**6*x**5 + 139968*a**5*b*x**6 + 93312*a**5*c*x**7 + 279936*a**5 *d*x**4 + 174960*a**4*b**2*x**7 + 233280*a**4*b*c*x**8 + 699840*a**4*b*d*x **5 + 77760*a**4*c**2*x**9 + 466560*a**4*c*d*x**6 + 699840*a**4*d**2*x**3 + 116640*a**3*b**3*x**8 + 233280*a**3*b**2*c*x**9 + 699840*a**3*b**2*d*x** 6 + 155520*a**3*b*c**2*x**10 + 933120*a**3*b*c*d*x**7 + 1399680*a**3*b*d** 2*x**4 + 34560*a**3*c**3*x**11 + 311040*a**3*c**2*d*x**8 + 933120*a**3*c*d **2*x**5 + 933120*a**3*d**3*x**2 + 43740*a**2*b**4*x**9 + 116640*a**2*b**3 *c*x**10 + 349920*a**2*b**3*d*x**7 + 116640*a**2*b**2*c**2*x**11 + 699840* a**2*b**2*c*d*x**8 + 1049760*a**2*b**2*d**2*x**5 + 51840*a**2*b*c**3*x**12 + 466560*a**2*b*c**2*d*x**9 + 1399680*a**2*b*c*d**2*x**6 + 1399680*a**2*b *d**3*x**3 + 8640*a**2*c**4*x**13 + 103680*a**2*c**3*d*x**10 + 466560*a**2 *c**2*d**2*x**7 + 933120*a**2*c*d**3*x**4 + 699840*a**2*d**4*x + 8748*a*b* *5*x**10 + 29160*a*b**4*c*x**11 + 87480*a*b**4*d*x**8 + 38880*a*b**3*c**2* x**12 + 233280*a*b**3*c*d*x**9 + 349920*a*b**3*d**2*x**6 + 25920*a*b**2*c* *3*x**13 + 233280*a*b**2*c**2*d*x**10 + 699840*a*b**2*c*d**2*x**7 + 699840 *a*b**2*d**3*x**4 + 8640*a*b*c**4*x**14 + 103680*a*b*c**3*d*x**11 + 466560 *a*b*c**2*d**2*x**8 + 933120*a*b*c*d**3*x**5 + 699840*a*b*d**4*x**2 + 1152 *a*c**5*x**15 + 17280*a*c**4*d*x**12 + 103680*a*c**3*d**2*x**9 + 311040*a* c**2*d**3*x**6 + 466560*a*c*d**4*x**3 + 279936*a*d**5 + 279936*a + 729*b** 6*x**11 + 2916*b**5*c*x**12 + 8748*b**5*d*x**9 + 4860*b**4*c**2*x**13 +...