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

Optimal. Leaf size=47 \[ \frac{1}{6} \left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}+d\right )^6+a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0908815, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {1591} \[ \frac{1}{6} \left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}+d\right )^6+a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[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 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 (d+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,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]  time = 0.117337, size = 248, normalized size = 5.28 \[ \frac{x (6 a+x (3 b+2 c x)) \left (360 a^2 x^8 (3 b+2 c x)^3+2160 a^3 x^7 (3 b+2 c x)^2+6480 a^4 x^6 (3 b+2 c x)+7776 a^5 x^5+540 d^2 x^3 (6 a+x (3 b+2 c x))^3+4320 d^3 x^2 (6 a+x (3 b+2 c x))^2+19440 d^4 x (6 a+x (3 b+2 c x))+36 d x^4 (6 a+x (3 b+2 c x))^4+30 a x^9 (3 b+2 c x)^4+720 b^2 c^3 x^{13}+1080 b^3 c^2 x^{12}+810 b^4 c x^{11}+243 b^5 x^{10}+240 b c^4 x^{14}+32 c^5 x^{15}+46656 d^5+46656\right )}{279936} \]

Antiderivative was successfully verified.

[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

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Maple [B]  time = 0.003, size = 4284, normalized size = 91.2 \begin{align*} \text{output 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+(d+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*(1/81*d*c^4+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*(2/3*c*
d+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*c^2*b^2))+b*(1/81*d*c^4+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*(2/3*c*d+a*
b)*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c))+c*(2/27*d*b*c^3+a*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*c^2*b^2)+1/2*b*(2/9*(2/3*c
*d+a*b)*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/3*c*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2)))*x
^14+1/13*(a*(1/81*d*c^4+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*(2/3*c*d+a*b)*c^
2+2/3*(2/3*a*c+1/4*b^2)*b*c))+b*(2/27*d*b*c^3+a*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*c^2*b^2)+1/2*b*(2/9*(2/3*c*d+a*
b)*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/3*c*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2))+c*(d*(2
/9*(2/3*a*c+1/4*b^2)*c^2+1/9*c^2*b^2)+a*(2/9*(2/3*c*d+a*b)*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/2*b*(2/9*(a^2+b*d)
*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2)+1/3*c*(4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+
1/4*b^2))))*x^13+1/12*(a*(2/27*d*b*c^3+a*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*c^2*b^2)+1/2*b*(2/9*(2/3*c*d+a*b)*c^2+
2/3*(2/3*a*c+1/4*b^2)*b*c)+1/3*c*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2))+b*(d*(2/9*(2/3
*a*c+1/4*b^2)*c^2+1/9*c^2*b^2)+a*(2/9*(2/3*c*d+a*b)*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/2*b*(2/9*(a^2+b*d)*c^2+2/
3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2)+1/3*c*(4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2
)))+c*(d*(2/9*(2/3*c*d+a*b)*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+a*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c
+1/4*b^2)^2)+1/2*b*(4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+1/3*c*(2/9*d^2*c^2+4/3*a*
b*c*d+2*(a^2+b*d)*(2/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2)))*x^12+1/11*(a*(d*(2/9*(2/3*a*c+1/4*b^2)*c^2+1/9*c^2*b^2)
+a*(2/9*(2/3*c*d+a*b)*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+1/2*b*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1
/4*b^2)^2)+1/3*c*(4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2)))+b*(d*(2/9*(2/3*c*d+a*b)*c^
2+2/3*(2/3*a*c+1/4*b^2)*b*c)+a*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2)+1/2*b*(4/9*a*c^2*
d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+1/3*c*(2/9*d^2*c^2+4/3*a*b*c*d+2*(a^2+b*d)*(2/3*a*c+1/4
*b^2)+(2/3*c*d+a*b)^2))+c*(d*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2)+a*(4/9*a*c^2*d+2/3*
(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+1/2*b*(2/9*d^2*c^2+4/3*a*b*c*d+2*(a^2+b*d)*(2/3*a*c+1/4*b^2)+
(2/3*c*d+a*b)^2)+1/3*c*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(2/3*c*d+a*b))))*x^11+1/10*(a*(d*(2/9*
(2/3*c*d+a*b)*c^2+2/3*(2/3*a*c+1/4*b^2)*b*c)+a*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2)+1
/2*b*(4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+1/3*c*(2/9*d^2*c^2+4/3*a*b*c*d+2*(a^2+b
*d)*(2/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2))+b*(d*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c+(2/3*a*c+1/4*b^2)^2)+a*(
4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+1/2*b*(2/9*d^2*c^2+4/3*a*b*c*d+2*(a^2+b*d)*(2
/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2)+1/3*c*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(2/3*c*d+a*b)))+c*(d*(
4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+a*(2/9*d^2*c^2+4/3*a*b*c*d+2*(a^2+b*d)*(2/3*a
*c+1/4*b^2)+(2/3*c*d+a*b)^2)+1/2*b*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(2/3*c*d+a*b))+1/3*c*(2*d^
2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2)))*x^10+1/9*(a*(d*(2/9*(a^2+b*d)*c^2+2/3*(2/3*c*d+a*b)*b*c
+(2/3*a*c+1/4*b^2)^2)+a*(4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+1/2*b*(2/9*d^2*c^2+4
/3*a*b*c*d+2*(a^2+b*d)*(2/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2)+1/3*c*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*
d)*(2/3*c*d+a*b)))+b*(d*(4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+a*(2/9*d^2*c^2+4/3*a
*b*c*d+2*(a^2+b*d)*(2/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2)+1/2*b*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(
2/3*c*d+a*b))+1/3*c*(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2))+c*(d*(2/9*d^2*c^2+4/3*a*b*c*d+2
*(a^2+b*d)*(2/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2)+a*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(2/3*c*d+a*b)
)+1/2*b*(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2)+1/3*c*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*d)))
)*x^9+1/8*(a*(d*(4/9*a*c^2*d+2/3*(a^2+b*d)*b*c+2*(2/3*c*d+a*b)*(2/3*a*c+1/4*b^2))+a*(2/9*d^2*c^2+4/3*a*b*c*d+2
*(a^2+b*d)*(2/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2)+1/2*b*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(2/3*c*d+
a*b))+1/3*c*(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2))+b*(d*(2/9*d^2*c^2+4/3*a*b*c*d+2*(a^2+b*
d)*(2/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2)+a*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(2/3*c*d+a*b))+1/2*b*
(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2)+1/3*c*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*d)))+c*(d*(2
/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(2/3*c*d+a*b))+a*(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(
a^2+b*d)^2)+1/2*b*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*d))+1/3*c*(2*d^2*(a^2+b*d)+4*a^2*d^2)))*x^8+1/7*(a*(d*(2/9
*d^2*c^2+4/3*a*b*c*d+2*(a^2+b*d)*(2/3*a*c+1/4*b^2)+(2/3*c*d+a*b)^2)+a*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(
a^2+b*d)*(2/3*c*d+a*b))+1/2*b*(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2)+1/3*c*(2*d^2*(2/3*c*d+
a*b)+4*a*d*(a^2+b*d)))+b*(d*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+b*d)*(2/3*c*d+a*b))+a*(2*d^2*(2/3*a*c+
1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2)+1/2*b*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*d))+1/3*c*(2*d^2*(a^2+b*d)+4
*a^2*d^2))+c*(d*(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2)+a*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*
d))+1/2*b*(2*d^2*(a^2+b*d)+4*a^2*d^2)+4/3*a*c*d^3))*x^7+1/6*(a*(d*(2/3*d^2*b*c+4*a*d*(2/3*a*c+1/4*b^2)+2*(a^2+
b*d)*(2/3*c*d+a*b))+a*(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2)+1/2*b*(2*d^2*(2/3*c*d+a*b)+4*a
*d*(a^2+b*d))+1/3*c*(2*d^2*(a^2+b*d)+4*a^2*d^2))+b*(d*(2*d^2*(2/3*a*c+1/4*b^2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2
)+a*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*d))+1/2*b*(2*d^2*(a^2+b*d)+4*a^2*d^2)+4/3*a*c*d^3)+c*(d*(2*d^2*(2/3*c*d+
a*b)+4*a*d*(a^2+b*d))+a*(2*d^2*(a^2+b*d)+4*a^2*d^2)+2*b*d^3*a+1/3*c*d^4))*x^6+1/5*(a*(d*(2*d^2*(2/3*a*c+1/4*b^
2)+4*a*d*(2/3*c*d+a*b)+(a^2+b*d)^2)+a*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*d))+1/2*b*(2*d^2*(a^2+b*d)+4*a^2*d^2)+
4/3*a*c*d^3)+b*(d*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*d))+a*(2*d^2*(a^2+b*d)+4*a^2*d^2)+2*b*d^3*a+1/3*c*d^4)+c*(
d*(2*d^2*(a^2+b*d)+4*a^2*d^2)+4*a^2*d^3+1/2*b*d^4))*x^5+1/4*(a*(d*(2*d^2*(2/3*c*d+a*b)+4*a*d*(a^2+b*d))+a*(2*d
^2*(a^2+b*d)+4*a^2*d^2)+2*b*d^3*a+1/3*c*d^4)+b*(d*(2*d^2*(a^2+b*d)+4*a^2*d^2)+4*a^2*d^3+1/2*b*d^4)+5*a*c*d^4)*
x^4+1/3*(a*(d*(2*d^2*(a^2+b*d)+4*a^2*d^2)+4*a^2*d^3+1/2*b*d^4)+5*b*d^4*a+c*(d^5+1))*x^3+1/2*(5*a^2*d^4+b*(d^5+
1))*x^2+a*(d^5+1)*x

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Maxima [B]  time = 1.01338, size = 1044, normalized size = 22.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.06238, size = 2256, normalized size = 48. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [B]  time = 0.235683, size = 930, normalized size = 19.79 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.21755, size = 1253, normalized size = 26.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

1/4374*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 + 1/243*c^5*d*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 + 5/162*b*c^4*d*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 + 5/54*b^2*c^3*d*x^13 + 5/81*a*c^4*d*x^13 + 1/3
84*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 + 5/36*b^3*c^2*d*x^12 + 10/27*a*b
*c^3*d*x^12 + 5/162*c^4*d^2*x^12 + 1/32*a*b^5*x^11 + 5/12*a^2*b^3*c*x^11 + 5/9*a^3*b*c^2*x^11 + 5/48*b^4*c*d*x
^11 + 5/6*a*b^2*c^2*d*x^11 + 10/27*a^2*c^3*d*x^11 + 5/27*b*c^3*d^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 + 1/32*b^5*d*x^10 + 5/6*a*b^3*c*d*x^10 + 5/3*a^2*b*c^2*d*x^10 + 5/12*b^2*c^2*d^2*x^10 +
10/27*a*c^3*d^2*x^10 + 5/12*a^3*b^3*x^9 + 5/6*a^4*b*c*x^9 + 5/16*a*b^4*d*x^9 + 5/2*a^2*b^2*c*d*x^9 + 10/9*a^3
*c^2*d*x^9 + 5/12*b^3*c*d^2*x^9 + 5/3*a*b*c^2*d^2*x^9 + 10/81*c^3*d^3*x^9 + 5/8*a^4*b^2*x^8 + 1/3*a^5*c*x^8 +
5/4*a^2*b^3*d*x^8 + 10/3*a^3*b*c*d*x^8 + 5/32*b^4*d^2*x^8 + 5/2*a*b^2*c*d^2*x^8 + 5/3*a^2*c^2*d^2*x^8 + 5/9*b*
c^2*d^3*x^8 + 1/2*a^5*b*x^7 + 5/2*a^3*b^2*d*x^7 + 5/3*a^4*c*d*x^7 + 5/4*a*b^3*d^2*x^7 + 5*a^2*b*c*d^2*x^7 + 5/
6*b^2*c*d^3*x^7 + 10/9*a*c^2*d^3*x^7 + 1/6*a^6*x^6 + 5/2*a^4*b*d*x^6 + 15/4*a^2*b^2*d^2*x^6 + 10/3*a^3*c*d^2*x
^6 + 5/12*b^3*d^3*x^6 + 10/3*a*b*c*d^3*x^6 + 5/18*c^2*d^4*x^6 + a^5*d*x^5 + 5*a^3*b*d^2*x^5 + 5/2*a*b^2*d^3*x^
5 + 10/3*a^2*c*d^3*x^5 + 5/6*b*c*d^4*x^5 + 5/2*a^4*d^2*x^4 + 5*a^2*b*d^3*x^4 + 5/8*b^2*d^4*x^4 + 5/3*a*c*d^4*x
^4 + 10/3*a^3*d^3*x^3 + 5/2*a*b*d^4*x^3 + 1/3*c*d^5*x^3 + 5/2*a^2*d^4*x^2 + 1/2*b*d^5*x^2 + a*d^5*x + 1/3*c*x^
3 + 1/2*b*x^2 + a*x