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

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Rubi [A]  time = 0.046813, antiderivative size = 46, normalized size of antiderivative = 1., 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*}

Mathematica [B]  time = 0.0619625, size = 244, normalized size = 5.3 \[ \frac{5 a^2 x^{10} (3 b+2 c x)^4}{2592}+\frac{5}{324} a^3 x^9 (3 b+2 c x)^3+\frac{5}{72} a^4 x^8 (3 b+2 c x)^2+\frac{1}{6} a^5 x^7 (3 b+2 c x)+\frac{a^6 x^6}{6}+a \left (\frac{5}{54} b^2 c^3 x^{14}+\frac{5}{36} b^3 c^2 x^{13}+\frac{5}{48} b^4 c x^{12}+\frac{b^5 x^{11}}{32}+\frac{5}{162} b c^4 x^{15}+\frac{c^5 x^{16}}{243}+x\right )+\frac{x^2 \left (2160 b^2 c^4 x^{14}+4320 b^3 c^3 x^{13}+4860 b^4 c^2 x^{12}+2916 b^5 c x^{11}+729 b^6 x^{10}+576 b \left (c^5 x^{15}+243\right )+64 c x \left (c^5 x^{15}+1458\right )\right )}{279936} \]

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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Maple [B]  time = 0.002, size = 1523, normalized size = 33.1 \begin{align*} \text{result too large to display} \end{align*}

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*c^5*a+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*c^2*b^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*c^2*b^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*c^2*b^2)+1/3*c*(2/9*c^2*a*b+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*c^2*b^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*c^2*b^2)+1/3*c*(2/9*c^2*a*b+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*c^2*b^2)+1/2*b*(2/9*c^2*a*b+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*c^2*b^2)+1/3*c*(2/
9*c^2*a*b+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*c^2*b^2)+1/2*b*(2/9*c^2*a*b+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*c^2*a*b+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*c^2*b^2)+1/2*b*(2/9*c^2*a*b+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*c^2*a*b+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)+b^2*a^2)))*x^12+1/11*(
a*(a*(2/9*c^2*a*b+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)+b^2*a^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)+b^2*a^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)+b^2*a^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)+b^2*a^2)+2/3*c*a^3*b)+c*(a*(2*a^2*(2/3*a*c+1/4*b^2)+b^2*a^2)+a^3
*b^2+1/3*a^4*c))*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)+b^2*a^2)+
2/3*c*a^3*b)+b*(a*(2*a^2*(2/3*a*c+1/4*b^2)+b^2*a^2)+a^3*b^2+1/3*a^4*c)+5/2*c*a^4*b)*x^9+1/8*(a*(a*(2*a^2*(2/3*
a*c+1/4*b^2)+b^2*a^2)+a^3*b^2+1/3*a^4*c)+5/2*a^4*b^2+a^5*c)*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 = 1.05579, size = 390, normalized size = 8.48 \begin{align*} \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 \end{align*}

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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Fricas [B]  time = 1.09908, size = 782, normalized size = 17. \begin{align*} \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 \end{align*}

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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Sympy [B]  time = 0.145776, size = 323, normalized size = 7.02 \begin{align*} \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 ) \end{align*}

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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Giac [B]  time = 1.18154, size = 417, normalized size = 9.07 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{486} \, b c^{5} x^{17} + \frac{5}{648} \, b^{2} c^{4} x^{16} + \frac{1}{243} \, a c^{5} x^{16} + \frac{5}{324} \, b^{3} c^{3} x^{15} + \frac{5}{162} \, a b c^{4} x^{15} + \frac{5}{288} \, b^{4} c^{2} x^{14} + \frac{5}{54} \, a b^{2} c^{3} x^{14} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{1}{96} \, b^{5} c x^{13} + \frac{5}{36} \, a b^{3} c^{2} x^{13} + \frac{5}{27} \, a^{2} b c^{3} x^{13} + \frac{1}{384} \, b^{6} x^{12} + \frac{5}{48} \, a b^{4} c x^{12} + \frac{5}{12} \, a^{2} b^{2} c^{2} x^{12} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{1}{32} \, a b^{5} x^{11} + \frac{5}{12} \, a^{2} b^{3} c x^{11} + \frac{5}{9} \, a^{3} b c^{2} x^{11} + \frac{5}{32} \, a^{2} b^{4} x^{10} + \frac{5}{6} \, a^{3} b^{2} c x^{10} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{5}{12} \, a^{3} b^{3} x^{9} + \frac{5}{6} \, a^{4} b c x^{9} + \frac{5}{8} \, a^{4} b^{2} x^{8} + \frac{1}{3} \, a^{5} c x^{8} + \frac{1}{2} \, a^{5} b x^{7} + \frac{1}{6} \, a^{6} x^{6} + \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \end{align*}

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]

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