3.236 \(\int x (a+b x+c x^2+d x^3)^p (2 a+b (3+p) x+c (4+2 p) x^2+d (5+3 p) x^3) \, dx\)

Optimal. Leaf size=23 \[ x^2 \left (a+b x+c x^2+d x^3\right )^{p+1} \]

[Out]

x^2*(a + b*x + c*x^2 + d*x^3)^(1 + p)

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Rubi [A]  time = 0.0590147, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02, Rules used = {1588} \[ x^2 \left (a+b x+c x^2+d x^3\right )^{p+1} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x + c*x^2 + d*x^3)^p*(2*a + b*(3 + p)*x + c*(4 + 2*p)*x^2 + d*(5 + 3*p)*x^3),x]

[Out]

x^2*(a + b*x + c*x^2 + d*x^3)^(1 + p)

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x \left (a+b x+c x^2+d x^3\right )^p \left (2 a+b (3+p) x+c (4+2 p) x^2+d (5+3 p) x^3\right ) \, dx &=x^2 \left (a+b x+c x^2+d x^3\right )^{1+p}\\ \end{align*}

Mathematica [A]  time = 0.177702, size = 21, normalized size = 0.91 \[ x^2 (a+x (b+x (c+d x)))^{p+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x + c*x^2 + d*x^3)^p*(2*a + b*(3 + p)*x + c*(4 + 2*p)*x^2 + d*(5 + 3*p)*x^3),x]

[Out]

x^2*(a + x*(b + x*(c + d*x)))^(1 + p)

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Maple [A]  time = 0.006, size = 24, normalized size = 1. \begin{align*}{x}^{2} \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x^3+c*x^2+b*x+a)^p*(2*a+b*(3+p)*x+c*(4+2*p)*x^2+d*(5+3*p)*x^3),x)

[Out]

x^2*(d*x^3+c*x^2+b*x+a)^(1+p)

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Maxima [A]  time = 1.20393, size = 53, normalized size = 2.3 \begin{align*}{\left (d x^{5} + c x^{4} + b x^{3} + a x^{2}\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x^3+c*x^2+b*x+a)^p*(2*a+b*(3+p)*x+c*(4+2*p)*x^2+d*(5+3*p)*x^3),x, algorithm="maxima")

[Out]

(d*x^5 + c*x^4 + b*x^3 + a*x^2)*(d*x^3 + c*x^2 + b*x + a)^p

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Fricas [A]  time = 1.42424, size = 82, normalized size = 3.57 \begin{align*}{\left (d x^{5} + c x^{4} + b x^{3} + a x^{2}\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x^3+c*x^2+b*x+a)^p*(2*a+b*(3+p)*x+c*(4+2*p)*x^2+d*(5+3*p)*x^3),x, algorithm="fricas")

[Out]

(d*x^5 + c*x^4 + b*x^3 + a*x^2)*(d*x^3 + c*x^2 + b*x + a)^p

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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*(d*x**3+c*x**2+b*x+a)**p*(2*a+b*(3+p)*x+c*(4+2*p)*x**2+d*(5+3*p)*x**3),x)

[Out]

Timed out

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Giac [B]  time = 1.36567, size = 120, normalized size = 5.22 \begin{align*}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} d x^{5} +{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} c x^{4} +{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} b x^{3} +{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x^3+c*x^2+b*x+a)^p*(2*a+b*(3+p)*x+c*(4+2*p)*x^2+d*(5+3*p)*x^3),x, algorithm="giac")

[Out]

(d*x^3 + c*x^2 + b*x + a)^p*d*x^5 + (d*x^3 + c*x^2 + b*x + a)^p*c*x^4 + (d*x^3 + c*x^2 + b*x + a)^p*b*x^3 + (d
*x^3 + c*x^2 + b*x + a)^p*a*x^2