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

Optimal. Leaf size=50 \[ \frac{\left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0091307, antiderivative size = 50, 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} \[ \frac{\left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.187863, size = 49, normalized size = 0.98 \[ \frac{x (6 a+x (3 b+2 c x)) \left (\left (a x+\frac{b x^2}{2}+\frac{c x^3}{3}\right )^n+n+1\right )}{6 (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 43, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}}+{\frac{1}{1+n} \left ( ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}} \right ) ^{1+n}} \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)^n),x)

[Out]

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

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Maxima [A]  time = 1.7359, size = 112, normalized size = 2.24 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x + \frac{{\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )} e^{\left (n \log \left (2 \, c x^{2} + 3 \, b x + 6 \, a\right ) + n \log \left (x\right )\right )}}{3^{n + 1} 2^{n + 1} n + 3^{n + 1} 2^{n + 1}} \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)^n),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.37342, size = 171, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (c n + c\right )} x^{3} + 3 \,{\left (b n + b\right )} x^{2} +{\left (2 \, c x^{3} + 3 \, b x^{2} + 6 \, a x\right )}{\left (\frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x\right )}^{n} + 6 \,{\left (a n + a\right )} x}{6 \,{\left (n + 1\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)^n),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**n),x)

[Out]

Timed out

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Giac [A]  time = 1.19638, size = 57, normalized size = 1.14 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x + \frac{{\left (\frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x\right )}^{n + 1}}{n + 1} \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)^n),x, algorithm="giac")

[Out]

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