\(\int (a+b x+c x^2) (1+(d+a x+\frac {b x^2}{2}+\frac {c x^3}{3})^5) \, dx\) [215]

Optimal result

Integrand size = 36, antiderivative size = 47 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {1}{6} \left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6 \]

[Out]

a*x+1/2*b*x^2+1/3*c*x^3+1/6*(d+a*x+1/2*b*x^2+1/3*c*x^3)^6

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1605} \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{6} \left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}+d\right )^6+a x+\frac {b x^2}{2}+\frac {c x^3}{3} \]

[In]

Int[(a + b*x + c*x^2)*(1 + (d + a*x + (b*x^2)/2 + (c*x^3)/3)^5),x]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3 + (d + a*x + (b*x^2)/2 + (c*x^3)/3)^6/6

Rule 1605

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[Pq, x], r = Expon[Qr, x]}, Dist[Coef
f[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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \left (1+x^5\right ) \, dx,x,d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right ) \\ & = a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {1}{6} \left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(248\) vs. \(2(47)=94\).

Time = 0.08 (sec) , antiderivative size = 248, normalized size of antiderivative = 5.28 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {x (6 a+x (3 b+2 c x)) \left (46656+46656 d^5+7776 a^5 x^5+243 b^5 x^{10}+810 b^4 c x^{11}+1080 b^3 c^2 x^{12}+720 b^2 c^3 x^{13}+240 b c^4 x^{14}+32 c^5 x^{15}+6480 a^4 x^6 (3 b+2 c x)+2160 a^3 x^7 (3 b+2 c x)^2+360 a^2 x^8 (3 b+2 c x)^3+30 a x^9 (3 b+2 c x)^4+19440 d^4 x (6 a+x (3 b+2 c x))+4320 d^3 x^2 (6 a+x (3 b+2 c x))^2+540 d^2 x^3 (6 a+x (3 b+2 c x))^3+36 d x^4 (6 a+x (3 b+2 c x))^4\right )}{279936} \]

[In]

Integrate[(a + b*x + c*x^2)*(1 + (d + a*x + (b*x^2)/2 + (c*x^3)/3)^5),x]

[Out]

(x*(6*a + x*(3*b + 2*c*x))*(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))/27
9936

Maple [A] (verified)

Time = 0.53 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {\left (d +a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\right )^{6}}{6}+d +a x +\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}\)	\(39\)
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 a^{2} b \,d^{3}+\frac {5}{3} a c \,d^{4}+\frac {5}{8} b^{2} d^{4}\right ) x^{4}+\left (d \,a^{5}+5 a^{3} d^{2} b +\frac {10}{3} a^{2} c \,d^{3}+\frac {5}{2} a \,b^{2} d^{3}+\frac {5}{6} b c \,d^{4}\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} b^{2} c^{3} d \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} b^{2} d^{2} a^{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 {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 {10}{81} c^{3} a^{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} d^{2} c^{4}\right ) x^{12}+\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} a^{2} b^{4}+\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} d^{2} c^{2} b^{2}\right ) x^{10}+\left (\frac {1}{3} c \,a^{5}+\frac {5}{8} b^{2} a^{4}+\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} a^{3} b^{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} b \,c^{2} d^{2} a +\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 x^{11} c^{3} a^{3}+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} d^{2} c^{4}+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 a \,c^{3} d^{2} x^{9}+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 c \,a^{5} 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 a c \,d^{4} x^{3}+174960 b^{2} d^{4} x^{3}+933120 a^{3} d^{3} x^{2}+699840 x^{2} a b \,d^{4}+93312 x^{2} c \,d^{5}+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 {5}{162} x^{14} b \,c^{4} d +\frac {5}{54} x^{13} b^{2} c^{3} d +\frac {5}{2} a^{2} d^{4} x^{2}+\frac {5}{12} x^{6} b^{3} d^{3}+\frac {1}{2} b \,d^{5} x^{2}+\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} a c \,d^{4}+\frac {5}{81} a \,c^{4} d \,x^{13}+\frac {1}{2} a^{5} b \,x^{7}+\frac {5}{12} a^{3} b^{3} x^{9}+\frac {1}{32} a \,b^{5} x^{11}+\frac {1}{384} b^{6} x^{12}+\frac {1}{6} a^{6} x^{6}+\frac {1}{3} c \,x^{3}+a x +\frac {1}{2} b \,x^{2}+\frac {5}{162} x^{12} d^{2} c^{4}+\frac {10}{81} x^{9} c^{3} d^{3}+\frac {5}{18} x^{6} c^{2} d^{4}+x^{5} d \,a^{5}+\frac {5}{2} x^{4} a^{4} d^{2}+\frac {10}{3} x^{3} a^{3} d^{3}+\frac {1}{3} x^{3} c \,d^{5}+\frac {5}{6} b c \,d^{4} x^{5}+\frac {5}{6} b^{2} c \,d^{3} x^{7}+\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} b \,c^{2} d^{2} a +\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}+\frac {1}{96} b^{5} c \,x^{13}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{32} x^{8} b^{4} d^{2}+\frac {1}{243} c^{5} d \,x^{15}+\frac {5}{8} x^{4} b^{2} d^{4}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {1}{486} b \,c^{5} x^{17}+\frac {1}{243} a \,c^{5} x^{16}+\frac {1}{32} b^{5} d \,x^{10}+\frac {5}{8} a^{4} b^{2} x^{8}+\frac {1}{3} c \,a^{5} x^{8}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {10}{81} c^{3} a^{3} x^{12}+\frac {5}{162} a^{2} c^{4} x^{14}+\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} d^{2} c^{2} b^{2}+\frac {5}{12} x^{9} b^{3} c \,d^{2}+\frac {5}{9} x^{8} b \,c^{2} d^{3}+\frac {10}{27} x^{10} a \,c^{3} d^{2}+\frac {5}{3} x^{8} a^{2} c^{2} d^{2}+\frac {10}{9} x^{7} a \,c^{2} d^{3}+\frac {5}{32} a^{2} b^{4} x^{10}+\frac {1}{4374} c^{6} x^{18}+\frac {5}{3} a^{4} c d \,x^{7}+\frac {10}{9} a^{3} c^{2} d \,x^{9}+\frac {10}{27} a^{2} c^{3} d \,x^{11}+\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 +5 x^{4} a^{2} b \,d^{3}+\frac {5}{2} x^{3} a b \,d^{4}+\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 {5}{2} x^{6} a^{4} b d +\frac {15}{4} x^{6} b^{2} d^{2} a^{2}+5 x^{5} a^{3} d^{2} b +\frac {5}{2} x^{5} a \,b^{2} d^{3}+a \,d^{5} x\)	\(929\)
parallelrisch	\(\frac {5}{162} x^{14} b \,c^{4} d +\frac {5}{54} x^{13} b^{2} c^{3} d +\frac {5}{2} a^{2} d^{4} x^{2}+\frac {5}{12} x^{6} b^{3} d^{3}+\frac {1}{2} b \,d^{5} x^{2}+\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} a c \,d^{4}+\frac {5}{81} a \,c^{4} d \,x^{13}+\frac {1}{2} a^{5} b \,x^{7}+\frac {5}{12} a^{3} b^{3} x^{9}+\frac {1}{32} a \,b^{5} x^{11}+\frac {1}{384} b^{6} x^{12}+\frac {1}{6} a^{6} x^{6}+\frac {1}{3} c \,x^{3}+a x +\frac {1}{2} b \,x^{2}+\frac {5}{162} x^{12} d^{2} c^{4}+\frac {10}{81} x^{9} c^{3} d^{3}+\frac {5}{18} x^{6} c^{2} d^{4}+x^{5} d \,a^{5}+\frac {5}{2} x^{4} a^{4} d^{2}+\frac {10}{3} x^{3} a^{3} d^{3}+\frac {1}{3} x^{3} c \,d^{5}+\frac {5}{6} b c \,d^{4} x^{5}+\frac {5}{6} b^{2} c \,d^{3} x^{7}+\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} b \,c^{2} d^{2} a +\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}+\frac {1}{96} b^{5} c \,x^{13}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{32} x^{8} b^{4} d^{2}+\frac {1}{243} c^{5} d \,x^{15}+\frac {5}{8} x^{4} b^{2} d^{4}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {1}{486} b \,c^{5} x^{17}+\frac {1}{243} a \,c^{5} x^{16}+\frac {1}{32} b^{5} d \,x^{10}+\frac {5}{8} a^{4} b^{2} x^{8}+\frac {1}{3} c \,a^{5} x^{8}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {10}{81} c^{3} a^{3} x^{12}+\frac {5}{162} a^{2} c^{4} x^{14}+\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} d^{2} c^{2} b^{2}+\frac {5}{12} x^{9} b^{3} c \,d^{2}+\frac {5}{9} x^{8} b \,c^{2} d^{3}+\frac {10}{27} x^{10} a \,c^{3} d^{2}+\frac {5}{3} x^{8} a^{2} c^{2} d^{2}+\frac {10}{9} x^{7} a \,c^{2} d^{3}+\frac {5}{32} a^{2} b^{4} x^{10}+\frac {1}{4374} c^{6} x^{18}+\frac {5}{3} a^{4} c d \,x^{7}+\frac {10}{9} a^{3} c^{2} d \,x^{9}+\frac {10}{27} a^{2} c^{3} d \,x^{11}+\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 +5 x^{4} a^{2} b \,d^{3}+\frac {5}{2} x^{3} a b \,d^{4}+\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 {5}{2} x^{6} a^{4} b d +\frac {15}{4} x^{6} b^{2} d^{2} a^{2}+5 x^{5} a^{3} d^{2} b +\frac {5}{2} x^{5} a \,b^{2} d^{3}+a \,d^{5} x\)	\(929\)

[In]

int((c*x^2+b*x+a)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x,method=_RETURNVERBOSE)

[Out]

1/6*(d+a*x+1/2*b*x^2+1/3*c*x^3)^6+d+a*x+1/2*b*x^2+1/3*c*x^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (40) = 80\).

Time = 0.25 (sec) , antiderivative size = 773, normalized size of antiderivative = 16.45 \[ \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}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {1}{1944} \, {\left (15 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 30 \, a b c^{4} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} + 16 \, a^{2} c^{4} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{2592} \, {\left (27 \, b^{5} c + 360 \, a b^{3} c^{2} + 480 \, a^{2} b c^{3} + 80 \, {\left (3 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d\right )} x^{13} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1080 \, a b^{4} c + 4320 \, a^{2} b^{2} c^{2} + 1280 \, a^{3} c^{3} + 320 \, c^{4} d^{2} + 480 \, {\left (3 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d\right )} x^{12} + \frac {1}{864} \, {\left (27 \, a b^{5} + 360 \, a^{2} b^{3} c + 480 \, a^{3} b c^{2} + 160 \, b c^{3} d^{2} + 10 \, {\left (9 \, b^{4} c + 72 \, a b^{2} c^{2} + 32 \, a^{2} c^{3}\right )} d\right )} x^{11} + \frac {1}{864} \, {\left (135 \, a^{2} b^{4} + 720 \, a^{3} b^{2} c + 240 \, a^{4} c^{2} + 40 \, {\left (9 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} + 9 \, {\left (3 \, b^{5} + 80 \, a b^{3} c + 160 \, a^{2} b c^{2}\right )} d\right )} x^{10} + \frac {5}{1296} \, {\left (108 \, a^{3} b^{3} + 216 \, a^{4} b c + 32 \, c^{3} d^{3} + 108 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2} + 9 \, {\left (9 \, a b^{4} + 72 \, a^{2} b^{2} c + 32 \, a^{3} c^{2}\right )} d\right )} x^{9} + \frac {1}{288} \, {\left (180 \, a^{4} b^{2} + 96 \, a^{5} c + 160 \, b c^{2} d^{3} + 15 \, {\left (3 \, b^{4} + 48 \, a b^{2} c + 32 \, a^{2} c^{2}\right )} d^{2} + 120 \, {\left (3 \, a^{2} b^{3} + 8 \, a^{3} b c\right )} d\right )} x^{8} + \frac {1}{36} \, {\left (18 \, a^{5} b + 10 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{3} + 45 \, {\left (a b^{3} + 4 \, a^{2} b c\right )} d^{2} + 30 \, {\left (3 \, a^{3} b^{2} + 2 \, a^{4} c\right )} d\right )} x^{7} + \frac {1}{36} \, {\left (6 \, a^{6} + 90 \, a^{4} b d + 10 \, c^{2} d^{4} + 15 \, {\left (b^{3} + 8 \, a b c\right )} d^{3} + 15 \, {\left (9 \, a^{2} b^{2} + 8 \, a^{3} c\right )} d^{2}\right )} x^{6} + \frac {1}{6} \, {\left (6 \, a^{5} d + 30 \, a^{3} b d^{2} + 5 \, b c d^{4} + 5 \, {\left (3 \, a b^{2} + 4 \, a^{2} c\right )} d^{3}\right )} x^{5} + \frac {5}{24} \, {\left (12 \, a^{4} d^{2} + 24 \, a^{2} b d^{3} + {\left (3 \, b^{2} + 8 \, a c\right )} d^{4}\right )} x^{4} + \frac {1}{6} \, {\left (20 \, a^{3} d^{3} + 15 \, a b d^{4} + 2 \, c d^{5} + 2 \, c\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{2} d^{4} + b d^{5} + b\right )} x^{2} + {\left (a d^{5} + a\right )} x \]

[In]

integrate((c*x^2+b*x+a)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="fricas")

[Out]

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 + 1080*a*b^4*c + 4320*a^2*b^2*c^2 + 12
80*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 + 2
40*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 + 216*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 + 10*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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 930 vs. \(2 (37) = 74\).

Time = 0.12 (sec) , antiderivative size = 930, normalized size of antiderivative = 19.79 \[ \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 {b c^{5} x^{17}}{486} + \frac {c^{6} x^{18}}{4374} + x^{16} \left (\frac {a c^{5}}{243} + \frac {5 b^{2} c^{4}}{648}\right ) + x^{15} \cdot \left (\frac {5 a b c^{4}}{162} + \frac {5 b^{3} c^{3}}{324} + \frac {c^{5} d}{243}\right ) + x^{14} \cdot \left (\frac {5 a^{2} c^{4}}{162} + \frac {5 a b^{2} c^{3}}{54} + \frac {5 b^{4} c^{2}}{288} + \frac {5 b c^{4} d}{162}\right ) + x^{13} \cdot \left (\frac {5 a^{2} b c^{3}}{27} + \frac {5 a b^{3} c^{2}}{36} + \frac {5 a c^{4} d}{81} + \frac {b^{5} c}{96} + \frac {5 b^{2} c^{3} d}{54}\right ) + x^{12} \cdot \left (\frac {10 a^{3} c^{3}}{81} + \frac {5 a^{2} b^{2} c^{2}}{12} + \frac {5 a b^{4} c}{48} + \frac {10 a b c^{3} d}{27} + \frac {b^{6}}{384} + \frac {5 b^{3} c^{2} d}{36} + \frac {5 c^{4} d^{2}}{162}\right ) + x^{11} \cdot \left (\frac {5 a^{3} b c^{2}}{9} + \frac {5 a^{2} b^{3} c}{12} + \frac {10 a^{2} c^{3} d}{27} + \frac {a b^{5}}{32} + \frac {5 a b^{2} c^{2} d}{6} + \frac {5 b^{4} c d}{48} + \frac {5 b c^{3} d^{2}}{27}\right ) + x^{10} \cdot \left (\frac {5 a^{4} c^{2}}{18} + \frac {5 a^{3} b^{2} c}{6} + \frac {5 a^{2} b^{4}}{32} + \frac {5 a^{2} b c^{2} d}{3} + \frac {5 a b^{3} c d}{6} + \frac {10 a c^{3} d^{2}}{27} + \frac {b^{5} d}{32} + \frac {5 b^{2} c^{2} d^{2}}{12}\right ) + x^{9} \cdot \left (\frac {5 a^{4} b c}{6} + \frac {5 a^{3} b^{3}}{12} + \frac {10 a^{3} c^{2} d}{9} + \frac {5 a^{2} b^{2} c d}{2} + \frac {5 a b^{4} d}{16} + \frac {5 a b c^{2} d^{2}}{3} + \frac {5 b^{3} c d^{2}}{12} + \frac {10 c^{3} d^{3}}{81}\right ) + x^{8} \left (\frac {a^{5} c}{3} + \frac {5 a^{4} b^{2}}{8} + \frac {10 a^{3} b c d}{3} + \frac {5 a^{2} b^{3} d}{4} + \frac {5 a^{2} c^{2} d^{2}}{3} + \frac {5 a b^{2} c d^{2}}{2} + \frac {5 b^{4} d^{2}}{32} + \frac {5 b c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac {a^{5} b}{2} + \frac {5 a^{4} c d}{3} + \frac {5 a^{3} b^{2} d}{2} + 5 a^{2} b c d^{2} + \frac {5 a b^{3} d^{2}}{4} + \frac {10 a c^{2} d^{3}}{9} + \frac {5 b^{2} c d^{3}}{6}\right ) + x^{6} \left (\frac {a^{6}}{6} + \frac {5 a^{4} b d}{2} + \frac {10 a^{3} c d^{2}}{3} + \frac {15 a^{2} b^{2} d^{2}}{4} + \frac {10 a b c d^{3}}{3} + \frac {5 b^{3} d^{3}}{12} + \frac {5 c^{2} d^{4}}{18}\right ) + x^{5} \left (a^{5} d + 5 a^{3} b d^{2} + \frac {10 a^{2} c d^{3}}{3} + \frac {5 a b^{2} d^{3}}{2} + \frac {5 b c d^{4}}{6}\right ) + x^{4} \cdot \left (\frac {5 a^{4} d^{2}}{2} + 5 a^{2} b d^{3} + \frac {5 a c d^{4}}{3} + \frac {5 b^{2} d^{4}}{8}\right ) + x^{3} \cdot \left (\frac {10 a^{3} d^{3}}{3} + \frac {5 a b d^{4}}{2} + \frac {c d^{5}}{3} + \frac {c}{3}\right ) + x^{2} \cdot \left (\frac {5 a^{2} d^{4}}{2} + \frac {b d^{5}}{2} + \frac {b}{2}\right ) + x \left (a d^{5} + a\right ) \]

[In]

integrate((c*x**2+b*x+a)*(1+(d+a*x+1/2*b*x**2+1/3*c*x**3)**5),x)

[Out]

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**1
1*(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**2*(5*a**2*d**4/2 + b*d**5/2 + b/2) + x*(a*d**5 + a)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (40) = 80\).

Time = 0.20 (sec) , antiderivative size = 773, normalized size of antiderivative = 16.45 \[ \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}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {1}{1944} \, {\left (15 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{16} + \frac {1}{972} \, {\left (15 \, b^{3} c^{3} + 30 \, a b c^{4} + 4 \, c^{5} d\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} + 16 \, a^{2} c^{4} + 16 \, b c^{4} d\right )} x^{14} + \frac {1}{2592} \, {\left (27 \, b^{5} c + 360 \, a b^{3} c^{2} + 480 \, a^{2} b c^{3} + 80 \, {\left (3 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d\right )} x^{13} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1080 \, a b^{4} c + 4320 \, a^{2} b^{2} c^{2} + 1280 \, a^{3} c^{3} + 320 \, c^{4} d^{2} + 480 \, {\left (3 \, b^{3} c^{2} + 8 \, a b c^{3}\right )} d\right )} x^{12} + \frac {1}{864} \, {\left (27 \, a b^{5} + 360 \, a^{2} b^{3} c + 480 \, a^{3} b c^{2} + 160 \, b c^{3} d^{2} + 10 \, {\left (9 \, b^{4} c + 72 \, a b^{2} c^{2} + 32 \, a^{2} c^{3}\right )} d\right )} x^{11} + \frac {1}{864} \, {\left (135 \, a^{2} b^{4} + 720 \, a^{3} b^{2} c + 240 \, a^{4} c^{2} + 40 \, {\left (9 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} + 9 \, {\left (3 \, b^{5} + 80 \, a b^{3} c + 160 \, a^{2} b c^{2}\right )} d\right )} x^{10} + \frac {5}{1296} \, {\left (108 \, a^{3} b^{3} + 216 \, a^{4} b c + 32 \, c^{3} d^{3} + 108 \, {\left (b^{3} c + 4 \, a b c^{2}\right )} d^{2} + 9 \, {\left (9 \, a b^{4} + 72 \, a^{2} b^{2} c + 32 \, a^{3} c^{2}\right )} d\right )} x^{9} + \frac {1}{288} \, {\left (180 \, a^{4} b^{2} + 96 \, a^{5} c + 160 \, b c^{2} d^{3} + 15 \, {\left (3 \, b^{4} + 48 \, a b^{2} c + 32 \, a^{2} c^{2}\right )} d^{2} + 120 \, {\left (3 \, a^{2} b^{3} + 8 \, a^{3} b c\right )} d\right )} x^{8} + \frac {1}{36} \, {\left (18 \, a^{5} b + 10 \, {\left (3 \, b^{2} c + 4 \, a c^{2}\right )} d^{3} + 45 \, {\left (a b^{3} + 4 \, a^{2} b c\right )} d^{2} + 30 \, {\left (3 \, a^{3} b^{2} + 2 \, a^{4} c\right )} d\right )} x^{7} + \frac {1}{36} \, {\left (6 \, a^{6} + 90 \, a^{4} b d + 10 \, c^{2} d^{4} + 15 \, {\left (b^{3} + 8 \, a b c\right )} d^{3} + 15 \, {\left (9 \, a^{2} b^{2} + 8 \, a^{3} c\right )} d^{2}\right )} x^{6} + \frac {1}{6} \, {\left (6 \, a^{5} d + 30 \, a^{3} b d^{2} + 5 \, b c d^{4} + 5 \, {\left (3 \, a b^{2} + 4 \, a^{2} c\right )} d^{3}\right )} x^{5} + \frac {5}{24} \, {\left (12 \, a^{4} d^{2} + 24 \, a^{2} b d^{3} + {\left (3 \, b^{2} + 8 \, a c\right )} d^{4}\right )} x^{4} + \frac {1}{6} \, {\left (20 \, a^{3} d^{3} + 15 \, a b d^{4} + 2 \, c d^{5} + 2 \, c\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{2} d^{4} + b d^{5} + b\right )} x^{2} + {\left (a d^{5} + a\right )} x \]

[In]

integrate((c*x^2+b*x+a)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="maxima")

[Out]

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 + 1080*a*b^4*c + 4320*a^2*b^2*c^2 + 12
80*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 + 2
40*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 + 216*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 + 10*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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (40) = 80\).

Time = 0.44 (sec) , antiderivative size = 153, normalized size of antiderivative = 3.26 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (d+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{6} + \frac {1}{7776} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{5} d + \frac {5}{2592} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{4} d^{2} + \frac {5}{324} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{3} d^{3} + \frac {5}{72} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{2} d^{4} + \frac {1}{6} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} d^{5} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \]

[In]

integrate((c*x^2+b*x+a)*(1+(d+a*x+1/2*b*x^2+1/3*c*x^3)^5),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.34 (sec) , antiderivative size = 753, normalized size of antiderivative = 16.02 \[ \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=x^{10}\,\left (\frac {5\,a^4\,c^2}{18}+\frac {5\,a^3\,b^2\,c}{6}+\frac {5\,a^2\,b^4}{32}+\frac {5\,a^2\,b\,c^2\,d}{3}+\frac {5\,a\,b^3\,c\,d}{6}+\frac {10\,a\,c^3\,d^2}{27}+\frac {b^5\,d}{32}+\frac {5\,b^2\,c^2\,d^2}{12}\right )+x^8\,\left (\frac {a^5\,c}{3}+\frac {5\,a^4\,b^2}{8}+\frac {10\,a^3\,b\,c\,d}{3}+\frac {5\,a^2\,b^3\,d}{4}+\frac {5\,a^2\,c^2\,d^2}{3}+\frac {5\,a\,b^2\,c\,d^2}{2}+\frac {5\,b^4\,d^2}{32}+\frac {5\,b\,c^2\,d^3}{9}\right )+x^9\,\left (\frac {5\,a^4\,b\,c}{6}+\frac {5\,a^3\,b^3}{12}+\frac {10\,a^3\,c^2\,d}{9}+\frac {5\,a^2\,b^2\,c\,d}{2}+\frac {5\,a\,b^4\,d}{16}+\frac {5\,a\,b\,c^2\,d^2}{3}+\frac {5\,b^3\,c\,d^2}{12}+\frac {10\,c^3\,d^3}{81}\right )+x^{14}\,\left (\frac {5\,a^2\,c^4}{162}+\frac {5\,a\,b^2\,c^3}{54}+\frac {5\,b^4\,c^2}{288}+\frac {5\,d\,b\,c^4}{162}\right )+x^{12}\,\left (\frac {10\,a^3\,c^3}{81}+\frac {5\,a^2\,b^2\,c^2}{12}+\frac {5\,a\,b^4\,c}{48}+\frac {10\,a\,b\,c^3\,d}{27}+\frac {b^6}{384}+\frac {5\,b^3\,c^2\,d}{36}+\frac {5\,c^4\,d^2}{162}\right )+x^6\,\left (\frac {a^6}{6}+\frac {5\,a^4\,b\,d}{2}+\frac {10\,a^3\,c\,d^2}{3}+\frac {15\,a^2\,b^2\,d^2}{4}+\frac {10\,a\,b\,c\,d^3}{3}+\frac {5\,b^3\,d^3}{12}+\frac {5\,c^2\,d^4}{18}\right )+x^3\,\left (\frac {10\,a^3\,d^3}{3}+\frac {5\,b\,a\,d^4}{2}+\frac {c\,d^5}{3}+\frac {c}{3}\right )+x^{11}\,\left (\frac {5\,a^3\,b\,c^2}{9}+\frac {5\,a^2\,b^3\,c}{12}+\frac {10\,a^2\,c^3\,d}{27}+\frac {a\,b^5}{32}+\frac {5\,a\,b^2\,c^2\,d}{6}+\frac {5\,b^4\,c\,d}{48}+\frac {5\,b\,c^3\,d^2}{27}\right )+x^7\,\left (\frac {a^5\,b}{2}+\frac {5\,a^4\,c\,d}{3}+\frac {5\,a^3\,b^2\,d}{2}+5\,a^2\,b\,c\,d^2+\frac {5\,a\,b^3\,d^2}{4}+\frac {10\,a\,c^2\,d^3}{9}+\frac {5\,b^2\,c\,d^3}{6}\right )+x^2\,\left (\frac {5\,a^2\,d^4}{2}+\frac {b\,d^5}{2}+\frac {b}{2}\right )+x^{13}\,\left (\frac {5\,a^2\,b\,c^3}{27}+\frac {5\,a\,b^3\,c^2}{36}+\frac {5\,d\,a\,c^4}{81}+\frac {b^5\,c}{96}+\frac {5\,d\,b^2\,c^3}{54}\right )+x^5\,\left (a^5\,d+5\,a^3\,b\,d^2+\frac {10\,c\,a^2\,d^3}{3}+\frac {5\,a\,b^2\,d^3}{2}+\frac {5\,c\,b\,d^4}{6}\right )+\frac {c^6\,x^{18}}{4374}+\frac {5\,d^2\,x^4\,\left (12\,a^4+24\,a^2\,b\,d+8\,c\,a\,d^2+3\,b^2\,d^2\right )}{24}+a\,x\,\left (d^5+1\right )+\frac {b\,c^5\,x^{17}}{486}+\frac {c^3\,x^{15}\,\left (15\,b^3+30\,a\,b\,c+4\,d\,c^2\right )}{972}+\frac {c^4\,x^{16}\,\left (15\,b^2+8\,a\,c\right )}{1944} \]

[In]

int(((d + a*x + (b*x^2)/2 + (c*x^3)/3)^5 + 1)*(a + b*x + c*x^2),x)

[Out]

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*x^16*(8*a*c + 15*b^2))/1944