3.182 \(\int x^n (b+c x+d x^2)^n (b+2 c x+3 d x^2) \, dx\)

Optimal. Leaf size=25 \[ \frac{x^{n+1} \left (b+c x+d x^2\right )^{n+1}}{n+1} \]

[Out]

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

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Rubi [A]  time = 0.0212676, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1590} \[ \frac{x^{n+1} \left (b+c x+d x^2\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[x^n*(b + c*x + d*x^2)^n*(b + 2*c*x + 3*d*x^2),x]

[Out]

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

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

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

Mathematica [A]  time = 0.0204268, size = 24, normalized size = 0.96 \[ \frac{x^{n+1} (b+x (c+d x))^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^n*(b + c*x + d*x^2)^n*(b + 2*c*x + 3*d*x^2),x]

[Out]

(x^(1 + n)*(b + x*(c + d*x))^(1 + n))/(1 + n)

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Maple [A]  time = 0.003, size = 26, normalized size = 1. \begin{align*}{\frac{{x}^{1+n} \left ( d{x}^{2}+cx+b \right ) ^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n*(d*x^2+c*x+b)^n*(3*d*x^2+2*c*x+b),x)

[Out]

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

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Maxima [A]  time = 1.20091, size = 53, normalized size = 2.12 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + b x\right )} e^{\left (n \log \left (d x^{2} + c x + b\right ) + n \log \left (x\right )\right )}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n*(d*x^2+c*x+b)^n*(3*d*x^2+2*c*x+b),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.35457, size = 74, normalized size = 2.96 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + b x\right )}{\left (d x^{2} + c x + b\right )}^{n} x^{n}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n*(d*x^2+c*x+b)^n*(3*d*x^2+2*c*x+b),x, algorithm="fricas")

[Out]

(d*x^3 + c*x^2 + b*x)*(d*x^2 + c*x + b)^n*x^n/(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(x**n*(d*x**2+c*x+b)**n*(3*d*x**2+2*c*x+b),x)

[Out]

Timed out

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Giac [B]  time = 1.19264, size = 88, normalized size = 3.52 \begin{align*} \frac{{\left (d x^{2} + c x + b\right )}^{n} d x^{3} x^{n} +{\left (d x^{2} + c x + b\right )}^{n} c x^{2} x^{n} +{\left (d x^{2} + c x + b\right )}^{n} b x x^{n}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^n*(d*x^2+c*x+b)^n*(3*d*x^2+2*c*x+b),x, algorithm="giac")

[Out]

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