\(\int (a+b x+c x^2) (1+(a x+\frac {b x^2}{2}+\frac {c x^3}{3})^5) \, dx\) [79]

Optimal result

Integrand size = 35, antiderivative size = 43 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=a x+\frac {b x^2}{2}+\frac {c x^3}{3}+\frac {\left (6 a x+3 b x^2+2 c x^3\right )^6}{279936} \] 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
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(244\) vs. \(2(43)=86\).

Time = 0.08 (sec) , antiderivative size = 244, normalized size of antiderivative = 5.67 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {a^6 x^6}{6}+\frac {1}{6} a^5 x^7 (3 b+2 c x)+\frac {5}{72} a^4 x^8 (3 b+2 c x)^2+\frac {5}{324} a^3 x^9 (3 b+2 c x)^3+\frac {5 a^2 x^{10} (3 b+2 c x)^4}{2592}+a \left (x+\frac {b^5 x^{11}}{32}+\frac {5}{48} b^4 c x^{12}+\frac {5}{36} b^3 c^2 x^{13}+\frac {5}{54} b^2 c^3 x^{14}+\frac {5}{162} b c^4 x^{15}+\frac {c^5 x^{16}}{243}\right )+\frac {x^2 \left (729 b^6 x^{10}+2916 b^5 c x^{11}+4860 b^4 c^2 x^{12}+4320 b^3 c^3 x^{13}+2160 b^2 c^4 x^{14}+576 b \left (243+c^5 x^{15}\right )+64 c x \left (1458+c^5 x^{15}\right )\right )}{279936} \] Input:

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

Output:

(a^6*x^6)/6 + (a^5*x^7*(3*b + 2*c*x))/6 + (5*a^4*x^8*(3*b + 2*c*x)^2)/72 + 
 (5*a^3*x^9*(3*b + 2*c*x)^3)/324 + (5*a^2*x^10*(3*b + 2*c*x)^4)/2592 + a*( 
x + (b^5*x^11)/32 + (5*b^4*c*x^12)/48 + (5*b^3*c^2*x^13)/36 + (5*b^2*c^3*x 
^14)/54 + (5*b*c^4*x^15)/162 + (c^5*x^16)/243) + (x^2*(729*b^6*x^10 + 2916 
*b^5*c*x^11 + 4860*b^4*c^2*x^12 + 4320*b^3*c^3*x^13 + 2160*b^2*c^4*x^14 + 
576*b*(243 + c^5*x^15) + 64*c*x*(1458 + c^5*x^15)))/279936
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, 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.

\(\displaystyle \int \left (a+b x+c x^2\right ) \left (\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5+1\right ) \, dx\)

\(\Big \downarrow \) 2024

\(\displaystyle \int \left (\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5+1\right )d\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} \left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6+a x+\frac {b x^2}{2}+\frac {c x^3}{3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2024

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]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {\left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )^{6}}{6}+\frac {c \,x^{3}}{3}+\frac {b \,x^{2}}{2}+x a\)	\(37\)
norman	\(x a +\left (\frac {1}{243} a \,c^{5}+\frac {5}{648} b^{2} c^{4}\right ) x^{16}+\left (\frac {1}{3} a^{5} c +\frac {5}{8} a^{4} b^{2}\right ) x^{8}+\left (\frac {5}{162} a b \,c^{4}+\frac {5}{324} b^{3} c^{3}\right ) x^{15}+\left (\frac {5}{6} a^{4} b c +\frac {5}{12} b^{3} a^{3}\right ) x^{9}+\left (\frac {5}{162} a^{2} c^{4}+\frac {5}{54} a \,b^{2} c^{3}+\frac {5}{288} b^{4} c^{2}\right ) x^{14}+\left (\frac {5}{18} a^{4} c^{2}+\frac {5}{6} a^{3} b^{2} c +\frac {5}{32} b^{4} a^{2}\right ) x^{10}+\left (\frac {5}{27} a^{2} b \,c^{3}+\frac {5}{36} a \,b^{3} c^{2}+\frac {1}{96} b^{5} c \right ) x^{13}+\left (\frac {5}{9} a^{3} b \,c^{2}+\frac {5}{12} a^{2} b^{3} c +\frac {1}{32} b^{5} a \right ) x^{11}+\left (\frac {10}{81} a^{3} c^{3}+\frac {5}{12} a^{2} b^{2} c^{2}+\frac {5}{48} a \,b^{4} c +\frac {1}{384} b^{6}\right ) x^{12}+\frac {a^{6} x^{6}}{6}+\frac {b \,x^{2}}{2}+\frac {c \,x^{3}}{3}+\frac {c^{6} x^{18}}{4374}+\frac {a^{5} b \,x^{7}}{2}+\frac {b \,c^{5} x^{17}}{486}\)	\(283\)
risch	\(\frac {1}{32} x^{11} b^{5} a +\frac {5}{8} x^{8} a^{4} b^{2}+\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {1}{4374} c^{6} x^{18}+x a +\frac {1}{243} a \,c^{5} x^{16}+\frac {1}{384} b^{6} x^{12}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {5}{12} x^{9} b^{3} a^{3}+\frac {1}{6} a^{6} x^{6}+\frac {5}{32} x^{10} b^{4} a^{2}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {1}{2} a^{5} b \,x^{7}+\frac {1}{3} a^{5} c \,x^{8}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{162} x^{15} a b \,c^{4}+\frac {5}{54} x^{14} a \,b^{2} c^{3}+\frac {5}{27} x^{13} a^{2} b \,c^{3}+\frac {5}{36} x^{13} a \,b^{3} c^{2}+\frac {5}{12} x^{12} a^{2} b^{2} c^{2}+\frac {5}{48} x^{12} a \,b^{4} c +\frac {5}{9} x^{11} a^{3} b \,c^{2}+\frac {5}{12} x^{11} a^{2} b^{3} c +\frac {5}{6} x^{10} a^{3} b^{2} c +\frac {5}{6} x^{9} a^{4} b c +\frac {1}{96} b^{5} c \,x^{13}+\frac {10}{81} a^{3} c^{3} x^{12}\)	\(310\)
parallelrisch	\(\frac {1}{32} x^{11} b^{5} a +\frac {5}{8} x^{8} a^{4} b^{2}+\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {1}{4374} c^{6} x^{18}+x a +\frac {1}{243} a \,c^{5} x^{16}+\frac {1}{384} b^{6} x^{12}+\frac {5}{648} b^{2} c^{4} x^{16}+\frac {5}{12} x^{9} b^{3} a^{3}+\frac {1}{6} a^{6} x^{6}+\frac {5}{32} x^{10} b^{4} a^{2}+\frac {1}{486} b \,c^{5} x^{17}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {1}{2} a^{5} b \,x^{7}+\frac {1}{3} a^{5} c \,x^{8}+\frac {5}{288} b^{4} c^{2} x^{14}+\frac {5}{324} b^{3} c^{3} x^{15}+\frac {5}{162} x^{15} a b \,c^{4}+\frac {5}{54} x^{14} a \,b^{2} c^{3}+\frac {5}{27} x^{13} a^{2} b \,c^{3}+\frac {5}{36} x^{13} a \,b^{3} c^{2}+\frac {5}{12} x^{12} a^{2} b^{2} c^{2}+\frac {5}{48} x^{12} a \,b^{4} c +\frac {5}{9} x^{11} a^{3} b \,c^{2}+\frac {5}{12} x^{11} a^{2} b^{3} c +\frac {5}{6} x^{10} a^{3} b^{2} c +\frac {5}{6} x^{9} a^{4} b c +\frac {1}{96} b^{5} c \,x^{13}+\frac {10}{81} a^{3} c^{3} x^{12}\)	\(310\)
gosper	\(\frac {x \left (64 c^{6} x^{17}+576 b \,c^{5} x^{16}+1152 a \,c^{5} x^{15}+2160 b^{2} c^{4} x^{15}+8640 a b \,c^{4} x^{14}+4320 b^{3} c^{3} x^{14}+8640 a^{2} c^{4} x^{13}+25920 a \,b^{2} c^{3} x^{13}+4860 b^{4} c^{2} x^{13}+51840 a^{2} b \,c^{3} x^{12}+38880 a \,b^{3} c^{2} x^{12}+2916 b^{5} c \,x^{12}+34560 a^{3} c^{3} x^{11}+116640 x^{11} a^{2} b^{2} c^{2}+29160 x^{11} a \,b^{4} c +729 b^{6} x^{11}+155520 a^{3} b \,c^{2} x^{10}+116640 a^{2} b^{3} c \,x^{10}+8748 a \,b^{5} x^{10}+77760 a^{4} c^{2} x^{9}+233280 a^{3} b^{2} c \,x^{9}+43740 a^{2} b^{4} x^{9}+233280 a^{4} b c \,x^{8}+116640 a^{3} b^{3} x^{8}+93312 a^{5} c \,x^{7}+174960 a^{4} b^{2} x^{7}+139968 a^{5} b \,x^{6}+46656 a^{6} x^{5}+93312 c \,x^{2}+139968 b x +279936 a \right )}{279936}\)	\(311\)
orering	\(\frac {x \left (64 c^{6} x^{17}+576 b \,c^{5} x^{16}+1152 a \,c^{5} x^{15}+2160 b^{2} c^{4} x^{15}+8640 a b \,c^{4} x^{14}+4320 b^{3} c^{3} x^{14}+8640 a^{2} c^{4} x^{13}+25920 a \,b^{2} c^{3} x^{13}+4860 b^{4} c^{2} x^{13}+51840 a^{2} b \,c^{3} x^{12}+38880 a \,b^{3} c^{2} x^{12}+2916 b^{5} c \,x^{12}+34560 a^{3} c^{3} x^{11}+116640 x^{11} a^{2} b^{2} c^{2}+29160 x^{11} a \,b^{4} c +729 b^{6} x^{11}+155520 a^{3} b \,c^{2} x^{10}+116640 a^{2} b^{3} c \,x^{10}+8748 a \,b^{5} x^{10}+77760 a^{4} c^{2} x^{9}+233280 a^{3} b^{2} c \,x^{9}+43740 a^{2} b^{4} x^{9}+233280 a^{4} b c \,x^{8}+116640 a^{3} b^{3} x^{8}+93312 a^{5} c \,x^{7}+174960 a^{4} b^{2} x^{7}+139968 a^{5} b \,x^{6}+46656 a^{6} x^{5}+93312 c \,x^{2}+139968 b x +279936 a \right ) \left (1+\left (\frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+x a \right )^{5}\right )}{36 \left (16 c^{4} x^{12}+96 b \,x^{11} c^{3}+192 x^{10} a \,c^{3}+216 b^{2} x^{10} c^{2}+864 c^{2} x^{9} a b +216 b^{3} x^{9} c +864 x^{8} a^{2} c^{2}+1296 c \,x^{8} b^{2} a +81 b^{4} x^{8}-48 c^{3} x^{9}+2592 c \,x^{7} a^{2} b +648 b^{3} x^{7} a -216 b \,x^{8} c^{2}+1728 x^{6} a^{3} c +1944 b^{2} x^{6} a^{2}-432 x^{7} a \,c^{2}-324 b^{2} x^{7} c +2592 a^{3} b \,x^{5}-1296 c \,x^{6} a b -162 b^{3} x^{6}+1296 x^{4} a^{4}-1296 x^{5} a^{2} c -972 b^{2} a \,x^{5}+144 c^{2} x^{6}-1944 a^{2} b \,x^{4}+432 b c \,x^{5}-1296 a^{3} x^{3}+864 x^{4} a c +324 b^{2} x^{4}+1296 a \,x^{3} b +1296 a^{2} x^{2}-432 c \,x^{3}-648 b \,x^{2}-1296 x a +1296\right ) \left (2 c \,x^{3}+3 b \,x^{2}+6 x a +6\right )}\)	\(643\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (37) = 74\).

Time = 0.09 (sec) , antiderivative size = 289, normalized size of antiderivative = 6.72 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {1}{1944} \, {\left (15 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{16} + \frac {5}{324} \, {\left (b^{3} c^{3} + 2 \, a b c^{4}\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{14} + \frac {1}{864} \, {\left (9 \, b^{5} c + 120 \, a b^{3} c^{2} + 160 \, a^{2} b c^{3}\right )} x^{13} + \frac {1}{2} \, a^{5} b x^{7} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1080 \, a b^{4} c + 4320 \, a^{2} b^{2} c^{2} + 1280 \, a^{3} c^{3}\right )} x^{12} + \frac {1}{6} \, a^{6} x^{6} + \frac {1}{288} \, {\left (9 \, a b^{5} + 120 \, a^{2} b^{3} c + 160 \, a^{3} b c^{2}\right )} x^{11} + \frac {5}{288} \, {\left (9 \, a^{2} b^{4} + 48 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} x^{10} + \frac {5}{12} \, {\left (a^{3} b^{3} + 2 \, a^{4} b c\right )} x^{9} + \frac {1}{24} \, {\left (15 \, a^{4} b^{2} + 8 \, a^{5} c\right )} x^{8} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \] Input:

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

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 + 
5/324*(b^3*c^3 + 2*a*b*c^4)*x^15 + 5/2592*(9*b^4*c^2 + 48*a*b^2*c^3 + 16*a 
^2*c^4)*x^14 + 1/864*(9*b^5*c + 120*a*b^3*c^2 + 160*a^2*b*c^3)*x^13 + 1/2* 
a^5*b*x^7 + 1/10368*(27*b^6 + 1080*a*b^4*c + 4320*a^2*b^2*c^2 + 1280*a^3*c 
^3)*x^12 + 1/6*a^6*x^6 + 1/288*(9*a*b^5 + 120*a^2*b^3*c + 160*a^3*b*c^2)*x 
^11 + 5/288*(9*a^2*b^4 + 48*a^3*b^2*c + 16*a^4*c^2)*x^10 + 5/12*(a^3*b^3 + 
 2*a^4*b*c)*x^9 + 1/24*(15*a^4*b^2 + 8*a^5*c)*x^8 + 1/3*c*x^3 + 1/2*b*x^2 
+ a*x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (37) = 74\).

Time = 0.10 (sec) , antiderivative size = 323, normalized size of antiderivative = 7.51 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {a^{6} x^{6}}{6} + \frac {a^{5} b x^{7}}{2} + a x + \frac {b c^{5} x^{17}}{486} + \frac {b x^{2}}{2} + \frac {c^{6} x^{18}}{4374} + \frac {c x^{3}}{3} + x^{16} \left (\frac {a c^{5}}{243} + \frac {5 b^{2} c^{4}}{648}\right ) + x^{15} \cdot \left (\frac {5 a b c^{4}}{162} + \frac {5 b^{3} c^{3}}{324}\right ) + x^{14} \cdot \left (\frac {5 a^{2} c^{4}}{162} + \frac {5 a b^{2} c^{3}}{54} + \frac {5 b^{4} c^{2}}{288}\right ) + x^{13} \cdot \left (\frac {5 a^{2} b c^{3}}{27} + \frac {5 a b^{3} c^{2}}{36} + \frac {b^{5} c}{96}\right ) + x^{12} \cdot \left (\frac {10 a^{3} c^{3}}{81} + \frac {5 a^{2} b^{2} c^{2}}{12} + \frac {5 a b^{4} c}{48} + \frac {b^{6}}{384}\right ) + x^{11} \cdot \left (\frac {5 a^{3} b c^{2}}{9} + \frac {5 a^{2} b^{3} c}{12} + \frac {a b^{5}}{32}\right ) + x^{10} \cdot \left (\frac {5 a^{4} c^{2}}{18} + \frac {5 a^{3} b^{2} c}{6} + \frac {5 a^{2} b^{4}}{32}\right ) + x^{9} \cdot \left (\frac {5 a^{4} b c}{6} + \frac {5 a^{3} b^{3}}{12}\right ) + x^{8} \left (\frac {a^{5} c}{3} + \frac {5 a^{4} b^{2}}{8}\right ) \] Input:

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

Output:

a**6*x**6/6 + a**5*b*x**7/2 + a*x + b*c**5*x**17/486 + b*x**2/2 + c**6*x** 
18/4374 + c*x**3/3 + 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) + x**14*(5*a**2*c**4/162 + 5*a*b**2*c**3/54 + 5 
*b**4*c**2/288) + x**13*(5*a**2*b*c**3/27 + 5*a*b**3*c**2/36 + b**5*c/96) 
+ x**12*(10*a**3*c**3/81 + 5*a**2*b**2*c**2/12 + 5*a*b**4*c/48 + b**6/384) 
 + x**11*(5*a**3*b*c**2/9 + 5*a**2*b**3*c/12 + a*b**5/32) + x**10*(5*a**4* 
c**2/18 + 5*a**3*b**2*c/6 + 5*a**2*b**4/32) + x**9*(5*a**4*b*c/6 + 5*a**3* 
b**3/12) + x**8*(a**5*c/3 + 5*a**4*b**2/8)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (37) = 74\).

Time = 0.03 (sec) , antiderivative size = 289, normalized size of antiderivative = 6.72 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {1}{1944} \, {\left (15 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{16} + \frac {5}{324} \, {\left (b^{3} c^{3} + 2 \, a b c^{4}\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{14} + \frac {1}{864} \, {\left (9 \, b^{5} c + 120 \, a b^{3} c^{2} + 160 \, a^{2} b c^{3}\right )} x^{13} + \frac {1}{2} \, a^{5} b x^{7} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1080 \, a b^{4} c + 4320 \, a^{2} b^{2} c^{2} + 1280 \, a^{3} c^{3}\right )} x^{12} + \frac {1}{6} \, a^{6} x^{6} + \frac {1}{288} \, {\left (9 \, a b^{5} + 120 \, a^{2} b^{3} c + 160 \, a^{3} b c^{2}\right )} x^{11} + \frac {5}{288} \, {\left (9 \, a^{2} b^{4} + 48 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} x^{10} + \frac {5}{12} \, {\left (a^{3} b^{3} + 2 \, a^{4} b c\right )} x^{9} + \frac {1}{24} \, {\left (15 \, a^{4} b^{2} + 8 \, a^{5} c\right )} x^{8} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \] Input:

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

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 + 
5/324*(b^3*c^3 + 2*a*b*c^4)*x^15 + 5/2592*(9*b^4*c^2 + 48*a*b^2*c^3 + 16*a 
^2*c^4)*x^14 + 1/864*(9*b^5*c + 120*a*b^3*c^2 + 160*a^2*b*c^3)*x^13 + 1/2* 
a^5*b*x^7 + 1/10368*(27*b^6 + 1080*a*b^4*c + 4320*a^2*b^2*c^2 + 1280*a^3*c 
^3)*x^12 + 1/6*a^6*x^6 + 1/288*(9*a*b^5 + 120*a^2*b^3*c + 160*a^3*b*c^2)*x 
^11 + 5/288*(9*a^2*b^4 + 48*a^3*b^2*c + 16*a^4*c^2)*x^10 + 5/12*(a^3*b^3 + 
 2*a^4*b*c)*x^9 + 1/24*(15*a^4*b^2 + 8*a^5*c)*x^8 + 1/3*c*x^3 + 1/2*b*x^2 
+ a*x
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{6} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \] Input:

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

Output:

1/279936*(2*c*x^3 + 3*b*x^2 + 6*a*x)^6 + 1/3*c*x^3 + 1/2*b*x^2 + a*x
 

Mupad [B] (verification not implemented)

Time = 21.96 (sec) , antiderivative size = 270, normalized size of antiderivative = 6.28 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=x^{12}\,\left (\frac {10\,a^3\,c^3}{81}+\frac {5\,a^2\,b^2\,c^2}{12}+\frac {5\,a\,b^4\,c}{48}+\frac {b^6}{384}\right )+a\,x+\frac {b\,x^2}{2}+\frac {c\,x^3}{3}+\frac {a^6\,x^6}{6}+\frac {c^6\,x^{18}}{4374}+\frac {5\,a^2\,x^{10}\,\left (16\,a^2\,c^2+48\,a\,b^2\,c+9\,b^4\right )}{288}+\frac {5\,c^2\,x^{14}\,\left (16\,a^2\,c^2+48\,a\,b^2\,c+9\,b^4\right )}{2592}+\frac {a^5\,b\,x^7}{2}+\frac {b\,c^5\,x^{17}}{486}+\frac {a^4\,x^8\,\left (15\,b^2+8\,a\,c\right )}{24}+\frac {c^4\,x^{16}\,\left (15\,b^2+8\,a\,c\right )}{1944}+\frac {a\,b\,x^{11}\,\left (160\,a^2\,c^2+120\,a\,b^2\,c+9\,b^4\right )}{288}+\frac {b\,c\,x^{13}\,\left (160\,a^2\,c^2+120\,a\,b^2\,c+9\,b^4\right )}{864}+\frac {5\,a^3\,b\,x^9\,\left (b^2+2\,a\,c\right )}{12}+\frac {5\,b\,c^3\,x^{15}\,\left (b^2+2\,a\,c\right )}{324} \] Input:

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

Output:

x^12*(b^6/384 + (10*a^3*c^3)/81 + (5*a^2*b^2*c^2)/12 + (5*a*b^4*c)/48) + a 
*x + (b*x^2)/2 + (c*x^3)/3 + (a^6*x^6)/6 + (c^6*x^18)/4374 + (5*a^2*x^10*( 
9*b^4 + 16*a^2*c^2 + 48*a*b^2*c))/288 + (5*c^2*x^14*(9*b^4 + 16*a^2*c^2 + 
48*a*b^2*c))/2592 + (a^5*b*x^7)/2 + (b*c^5*x^17)/486 + (a^4*x^8*(8*a*c + 1 
5*b^2))/24 + (c^4*x^16*(8*a*c + 15*b^2))/1944 + (a*b*x^11*(9*b^4 + 160*a^2 
*c^2 + 120*a*b^2*c))/288 + (b*c*x^13*(9*b^4 + 160*a^2*c^2 + 120*a*b^2*c))/ 
864 + (5*a^3*b*x^9*(2*a*c + b^2))/12 + (5*b*c^3*x^15*(2*a*c + b^2))/324
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 310, normalized size of antiderivative = 7.21 \[ \int \left (a+b x+c x^2\right ) \left (1+\left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^5\right ) \, dx=\frac {x \left (64 c^{6} x^{17}+576 b \,c^{5} x^{16}+1152 a \,c^{5} x^{15}+2160 b^{2} c^{4} x^{15}+8640 a b \,c^{4} x^{14}+4320 b^{3} c^{3} x^{14}+8640 a^{2} c^{4} x^{13}+25920 a \,b^{2} c^{3} x^{13}+4860 b^{4} c^{2} x^{13}+51840 a^{2} b \,c^{3} x^{12}+38880 a \,b^{3} c^{2} x^{12}+2916 b^{5} c \,x^{12}+34560 a^{3} c^{3} x^{11}+116640 a^{2} b^{2} c^{2} x^{11}+29160 a \,b^{4} c \,x^{11}+729 b^{6} x^{11}+155520 a^{3} b \,c^{2} x^{10}+116640 a^{2} b^{3} c \,x^{10}+8748 a \,b^{5} x^{10}+77760 a^{4} c^{2} x^{9}+233280 a^{3} b^{2} c \,x^{9}+43740 a^{2} b^{4} x^{9}+233280 a^{4} b c \,x^{8}+116640 a^{3} b^{3} x^{8}+93312 a^{5} c \,x^{7}+174960 a^{4} b^{2} x^{7}+139968 a^{5} b \,x^{6}+46656 a^{6} x^{5}+93312 c \,x^{2}+139968 b x +279936 a \right )}{279936} \] Input:

int((c*x^2+b*x+a)*(1+(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 + 174960*a**4 
*b**2*x**7 + 233280*a**4*b*c*x**8 + 77760*a**4*c**2*x**9 + 116640*a**3*b** 
3*x**8 + 233280*a**3*b**2*c*x**9 + 155520*a**3*b*c**2*x**10 + 34560*a**3*c 
**3*x**11 + 43740*a**2*b**4*x**9 + 116640*a**2*b**3*c*x**10 + 116640*a**2* 
b**2*c**2*x**11 + 51840*a**2*b*c**3*x**12 + 8640*a**2*c**4*x**13 + 8748*a* 
b**5*x**10 + 29160*a*b**4*c*x**11 + 38880*a*b**3*c**2*x**12 + 25920*a*b**2 
*c**3*x**13 + 8640*a*b*c**4*x**14 + 1152*a*c**5*x**15 + 279936*a + 729*b** 
6*x**11 + 2916*b**5*c*x**12 + 4860*b**4*c**2*x**13 + 4320*b**3*c**3*x**14 
+ 2160*b**2*c**4*x**15 + 576*b*c**5*x**16 + 139968*b*x + 64*c**6*x**17 + 9 
3312*c*x**2))/279936