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

Optimal. Leaf size=46 \[ \frac {1}{6} \left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6+a x+\frac {b x^2}{2}+\frac {c x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {1591} \[ \frac {1}{6} \left (a x+\frac {b x^2}{2}+\frac {c x^3}{3}\right )^6+a x+\frac {b x^2}{2}+\frac {c x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1591

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

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Mathematica [B]  time = 0.06, size = 244, normalized size = 5.30 \[ \frac {a^6 x^6}{6}+\frac {1}{6} a^5 x^7 (3 b+2 c x)+\frac {5}{72} a^4 x^8 (3 b+2 c x)^2+\frac {5}{324} a^3 x^9 (3 b+2 c x)^3+\frac {5 a^2 x^{10} (3 b+2 c x)^4}{2592}+a \left (\frac {b^5 x^{11}}{32}+\frac {5}{48} b^4 c x^{12}+\frac {5}{36} b^3 c^2 x^{13}+\frac {5}{54} b^2 c^3 x^{14}+\frac {5}{162} b c^4 x^{15}+\frac {c^5 x^{16}}{243}+x\right )+\frac {x^2 \left (729 b^6 x^{10}+2916 b^5 c x^{11}+4860 b^4 c^2 x^{12}+4320 b^3 c^3 x^{13}+2160 b^2 c^4 x^{14}+576 b \left (c^5 x^{15}+243\right )+64 c x \left (c^5 x^{15}+1458\right )\right )}{279936} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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

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fricas [B]  time = 0.68, size = 309, normalized size = 6.72 \[ \frac {1}{4374} x^{18} c^{6} + \frac {1}{486} x^{17} c^{5} b + \frac {5}{648} x^{16} c^{4} b^{2} + \frac {1}{243} x^{16} c^{5} a + \frac {5}{324} x^{15} c^{3} b^{3} + \frac {5}{162} x^{15} c^{4} b a + \frac {5}{288} x^{14} c^{2} b^{4} + \frac {5}{54} x^{14} c^{3} b^{2} a + \frac {5}{162} x^{14} c^{4} a^{2} + \frac {1}{96} x^{13} c b^{5} + \frac {5}{36} x^{13} c^{2} b^{3} a + \frac {5}{27} x^{13} c^{3} b a^{2} + \frac {1}{384} x^{12} b^{6} + \frac {5}{48} x^{12} c b^{4} a + \frac {5}{12} x^{12} c^{2} b^{2} a^{2} + \frac {10}{81} x^{12} c^{3} a^{3} + \frac {1}{32} x^{11} b^{5} a + \frac {5}{12} x^{11} c b^{3} a^{2} + \frac {5}{9} x^{11} c^{2} b a^{3} + \frac {5}{32} x^{10} b^{4} a^{2} + \frac {5}{6} x^{10} c b^{2} a^{3} + \frac {5}{18} x^{10} c^{2} a^{4} + \frac {5}{12} x^{9} b^{3} a^{3} + \frac {5}{6} x^{9} c b a^{4} + \frac {5}{8} x^{8} b^{2} a^{4} + \frac {1}{3} x^{8} c a^{5} + \frac {1}{2} x^{7} b a^{5} + \frac {1}{6} x^{6} a^{6} + \frac {1}{3} x^{3} c + \frac {1}{2} x^{2} b + x a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*x^18*c^6 + 1/486*x^17*c^5*b + 5/648*x^16*c^4*b^2 + 1/243*x^16*c^5*a + 5/324*x^15*c^3*b^3 + 5/162*x^15*c
^4*b*a + 5/288*x^14*c^2*b^4 + 5/54*x^14*c^3*b^2*a + 5/162*x^14*c^4*a^2 + 1/96*x^13*c*b^5 + 5/36*x^13*c^2*b^3*a
+ 5/27*x^13*c^3*b*a^2 + 1/384*x^12*b^6 + 5/48*x^12*c*b^4*a + 5/12*x^12*c^2*b^2*a^2 + 10/81*x^12*c^3*a^3 + 1/3
2*x^11*b^5*a + 5/12*x^11*c*b^3*a^2 + 5/9*x^11*c^2*b*a^3 + 5/32*x^10*b^4*a^2 + 5/6*x^10*c*b^2*a^3 + 5/18*x^10*c
^2*a^4 + 5/12*x^9*b^3*a^3 + 5/6*x^9*c*b*a^4 + 5/8*x^8*b^2*a^4 + 1/3*x^8*c*a^5 + 1/2*x^7*b*a^5 + 1/6*x^6*a^6 +
1/3*x^3*c + 1/2*x^2*b + x*a

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giac [A]  time = 0.31, size = 37, normalized size = 0.80 \[ \frac {1}{279936} \, {\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}^{6} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [B]  time = 0.00, size = 1523, normalized size = 33.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4374*c^6*x^18+1/486*b*c^5*x^17+1/16*(1/243*a*c^5+5/162*b^2*c^4+c*(1/81*a*c^4+1/27*b^2*c^3+1/3*c*(2/9*(2/3*a*
c+1/4*b^2)*c^2+1/9*b^2*c^2)))*x^16+1/15*(5/162*a*b*c^4+b*(1/81*a*c^4+1/27*b^2*c^3+1/3*c*(2/9*(2/3*a*c+1/4*b^2)
*c^2+1/9*b^2*c^2))+c*(2/27*a*b*c^3+1/2*b*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*b^2*c^2)+1/3*c*(2/9*a*b*c^2+2/3*(2/3*a
*c+1/4*b^2)*b*c)))*x^15+1/14*(a*(1/81*a*c^4+1/27*b^2*c^3+1/3*c*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*b^2*c^2))+b*(2/2
7*a*b*c^3+1/2*b*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*b^2*c^2)+1/3*c*(2/9*a*b*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c))+c*(a*(2
/9*(2/3*a*c+1/4*b^2)*c^2+1/9*b^2*c^2)+1/2*b*(2/9*a*b*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/3*c*(2/9*a^2*c^2+2/3*a*b
^2*c+(2/3*a*c+1/4*b^2)^2)))*x^14+1/13*(a*(2/27*a*b*c^3+1/2*b*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*b^2*c^2)+1/3*c*(2/
9*a*b*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c))+b*(a*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*b^2*c^2)+1/2*b*(2/9*a*b*c^2+2/3*(2/3
*a*c+1/4*b^2)*b*c)+1/3*c*(2/9*a^2*c^2+2/3*a*b^2*c+(2/3*a*c+1/4*b^2)^2))+c*(a*(2/9*a*b*c^2+2/3*(2/3*a*c+1/4*b^2
)*b*c)+1/2*b*(2/9*a^2*c^2+2/3*a*b^2*c+(2/3*a*c+1/4*b^2)^2)+1/3*c*(2/3*a^2*b*c+2*a*b*(2/3*a*c+1/4*b^2))))*x^13+
1/12*(a*(a*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*b^2*c^2)+1/2*b*(2/9*a*b*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/3*c*(2/9*a^
2*c^2+2/3*a*b^2*c+(2/3*a*c+1/4*b^2)^2))+b*(a*(2/9*a*b*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/2*b*(2/9*a^2*c^2+2/3*a*
b^2*c+(2/3*a*c+1/4*b^2)^2)+1/3*c*(2/3*a^2*b*c+2*a*b*(2/3*a*c+1/4*b^2)))+c*(a*(2/9*a^2*c^2+2/3*a*b^2*c+(2/3*a*c
+1/4*b^2)^2)+1/2*b*(2/3*a^2*b*c+2*a*b*(2/3*a*c+1/4*b^2))+1/3*c*(2*a^2*(2/3*a*c+1/4*b^2)+a^2*b^2)))*x^12+1/11*(
a*(a*(2/9*a*b*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/2*b*(2/9*a^2*c^2+2/3*a*b^2*c+(2/3*a*c+1/4*b^2)^2)+1/3*c*(2/3*a^
2*b*c+2*a*b*(2/3*a*c+1/4*b^2)))+b*(a*(2/9*a^2*c^2+2/3*a*b^2*c+(2/3*a*c+1/4*b^2)^2)+1/2*b*(2/3*a^2*b*c+2*a*b*(2
/3*a*c+1/4*b^2))+1/3*c*(2*a^2*(2/3*a*c+1/4*b^2)+a^2*b^2))+c*(a*(2/3*a^2*b*c+2*a*b*(2/3*a*c+1/4*b^2))+1/2*b*(2*
a^2*(2/3*a*c+1/4*b^2)+a^2*b^2)+2/3*c*a^3*b))*x^11+1/10*(a*(a*(2/9*a^2*c^2+2/3*a*b^2*c+(2/3*a*c+1/4*b^2)^2)+1/2
*b*(2/3*a^2*b*c+2*a*b*(2/3*a*c+1/4*b^2))+1/3*c*(2*a^2*(2/3*a*c+1/4*b^2)+a^2*b^2))+b*(a*(2/3*a^2*b*c+2*a*b*(2/3
*a*c+1/4*b^2))+1/2*b*(2*a^2*(2/3*a*c+1/4*b^2)+a^2*b^2)+2/3*c*a^3*b)+c*(a*(2*a^2*(2/3*a*c+1/4*b^2)+a^2*b^2)+b^2
*a^3+1/3*c*a^4))*x^10+1/9*(a*(a*(2/3*a^2*b*c+2*a*b*(2/3*a*c+1/4*b^2))+1/2*b*(2*a^2*(2/3*a*c+1/4*b^2)+a^2*b^2)+
2/3*c*a^3*b)+b*(a*(2*a^2*(2/3*a*c+1/4*b^2)+a^2*b^2)+b^2*a^3+1/3*c*a^4)+5/2*c*a^4*b)*x^9+1/8*(a*(a*(2*a^2*(2/3*
a*c+1/4*b^2)+a^2*b^2)+b^2*a^3+1/3*c*a^4)+5/2*b^2*a^4+c*a^5)*x^8+1/2*a^5*b*x^7+1/6*a^6*x^6+1/3*c*x^3+1/2*b*x^2+
a*x

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maxima [B]  time = 0.45, size = 289, normalized size = 6.28 \[ \frac {1}{4374} \, c^{6} x^{18} + \frac {1}{486} \, b c^{5} x^{17} + \frac {1}{1944} \, {\left (15 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{16} + \frac {5}{324} \, {\left (b^{3} c^{3} + 2 \, a b c^{4}\right )} x^{15} + \frac {5}{2592} \, {\left (9 \, b^{4} c^{2} + 48 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{14} + \frac {1}{864} \, {\left (9 \, b^{5} c + 120 \, a b^{3} c^{2} + 160 \, a^{2} b c^{3}\right )} x^{13} + \frac {1}{2} \, a^{5} b x^{7} + \frac {1}{10368} \, {\left (27 \, b^{6} + 1080 \, a b^{4} c + 4320 \, a^{2} b^{2} c^{2} + 1280 \, a^{3} c^{3}\right )} x^{12} + \frac {1}{6} \, a^{6} x^{6} + \frac {1}{288} \, {\left (9 \, a b^{5} + 120 \, a^{2} b^{3} c + 160 \, a^{3} b c^{2}\right )} x^{11} + \frac {5}{288} \, {\left (9 \, a^{2} b^{4} + 48 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} x^{10} + \frac {5}{12} \, {\left (a^{3} b^{3} + 2 \, a^{4} b c\right )} x^{9} + \frac {1}{24} \, {\left (15 \, a^{4} b^{2} + 8 \, a^{5} c\right )} x^{8} + \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)*(1+(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 + 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

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mupad [B]  time = 2.24, size = 270, normalized size = 5.87 \[ x^{12}\,\left (\frac {10\,a^3\,c^3}{81}+\frac {5\,a^2\,b^2\,c^2}{12}+\frac {5\,a\,b^4\,c}{48}+\frac {b^6}{384}\right )+a\,x+\frac {b\,x^2}{2}+\frac {c\,x^3}{3}+\frac {a^6\,x^6}{6}+\frac {c^6\,x^{18}}{4374}+\frac {5\,a^2\,x^{10}\,\left (16\,a^2\,c^2+48\,a\,b^2\,c+9\,b^4\right )}{288}+\frac {5\,c^2\,x^{14}\,\left (16\,a^2\,c^2+48\,a\,b^2\,c+9\,b^4\right )}{2592}+\frac {a^5\,b\,x^7}{2}+\frac {b\,c^5\,x^{17}}{486}+\frac {a^4\,x^8\,\left (15\,b^2+8\,a\,c\right )}{24}+\frac {c^4\,x^{16}\,\left (15\,b^2+8\,a\,c\right )}{1944}+\frac {a\,b\,x^{11}\,\left (160\,a^2\,c^2+120\,a\,b^2\,c+9\,b^4\right )}{288}+\frac {b\,c\,x^{13}\,\left (160\,a^2\,c^2+120\,a\,b^2\,c+9\,b^4\right )}{864}+\frac {5\,a^3\,b\,x^9\,\left (b^2+2\,a\,c\right )}{12}+\frac {5\,b\,c^3\,x^{15}\,\left (b^2+2\,a\,c\right )}{324} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 15*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

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sympy [B]  time = 0.16, size = 323, normalized size = 7.02 \[ \frac {a^{6} x^{6}}{6} + \frac {a^{5} b x^{7}}{2} + a x + \frac {b c^{5} x^{17}}{486} + \frac {b x^{2}}{2} + \frac {c^{6} x^{18}}{4374} + \frac {c x^{3}}{3} + x^{16} \left (\frac {a c^{5}}{243} + \frac {5 b^{2} c^{4}}{648}\right ) + x^{15} \left (\frac {5 a b c^{4}}{162} + \frac {5 b^{3} c^{3}}{324}\right ) + x^{14} \left (\frac {5 a^{2} c^{4}}{162} + \frac {5 a b^{2} c^{3}}{54} + \frac {5 b^{4} c^{2}}{288}\right ) + x^{13} \left (\frac {5 a^{2} b c^{3}}{27} + \frac {5 a b^{3} c^{2}}{36} + \frac {b^{5} c}{96}\right ) + x^{12} \left (\frac {10 a^{3} c^{3}}{81} + \frac {5 a^{2} b^{2} c^{2}}{12} + \frac {5 a b^{4} c}{48} + \frac {b^{6}}{384}\right ) + x^{11} \left (\frac {5 a^{3} b c^{2}}{9} + \frac {5 a^{2} b^{3} c}{12} + \frac {a b^{5}}{32}\right ) + x^{10} \left (\frac {5 a^{4} c^{2}}{18} + \frac {5 a^{3} b^{2} c}{6} + \frac {5 a^{2} b^{4}}{32}\right ) + x^{9} \left (\frac {5 a^{4} b c}{6} + \frac {5 a^{3} b^{3}}{12}\right ) + x^{8} \left (\frac {a^{5} c}{3} + \frac {5 a^{4} b^{2}}{8}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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**5/2
43 + 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)

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