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

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 1.02, size = 35, normalized size = 1.40 \[ \frac {{\left (d x^{3} + c x^{2} + b x\right )} {\left (d x^{2} + c x + b\right )}^{n} x^{n}}{n + 1} \]

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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giac [B]  time = 0.40, size = 65, normalized size = 2.60 \[ \frac {{\left (d x^{2} + c x + b\right )}^{n} d x^{3} x^{n} + {\left (d x^{2} + c x + b\right )}^{n} c x^{2} x^{n} + {\left (d x^{2} + c x + b\right )}^{n} b x x^{n}}{n + 1} \]

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)

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maple [A]  time = 0.01, size = 26, normalized size = 1.04 \[ \frac {x^{n +1} \left (d \,x^{2}+c x +b \right )^{n +1}}{n +1} \]

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

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maxima [A]  time = 0.86, size = 39, normalized size = 1.56 \[ \frac {{\left (d x^{3} + c x^{2} + b x\right )} e^{\left (n \log \left (d x^{2} + c x + b\right ) + n \log \relax (x)\right )}}{n + 1} \]

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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mupad [B]  time = 2.15, size = 51, normalized size = 2.04 \[ \left (\frac {c\,x^n\,x^2}{n+1}+\frac {d\,x^n\,x^3}{n+1}+\frac {b\,x\,x^n}{n+1}\right )\,{\left (d\,x^2+c\,x+b\right )}^n \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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