3.234 \(\int x^m (a+b x+c x^2+d x^3)^p (a (1+m)+x (b (2+m+p)+x (c (3+m+2 p)+d (4+m+3 p) x))) \, dx\)

Optimal. Leaf size=25 \[ x^{m+1} \left (a+b x+c x^2+d x^3\right )^{p+1} \]

[Out]

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

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Rubi [A]  time = 0.0247382, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.018, Rules used = {1590} \[ x^{m+1} \left (a+b x+c x^2+d x^3\right )^{p+1} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x + c*x^2 + d*x^3)^p*(a*(1 + m) + x*(b*(2 + m + p) + x*(c*(3 + m + 2*p) + d*(4 + m + 3*p)*x))),
x]

[Out]

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

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

Mathematica [A]  time = 0.336487, size = 23, normalized size = 0.92 \[ x^{m+1} (a+x (b+x (c+d x)))^{p+1} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x + c*x^2 + d*x^3)^p*(a*(1 + m) + x*(b*(2 + m + p) + x*(c*(3 + m + 2*p) + d*(4 + m + 3*p)
*x))),x]

[Out]

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

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Maple [A]  time = 0.007, size = 26, normalized size = 1. \begin{align*}{x}^{1+m} \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^m*(d*x^3+c*x^2+b*x+a)^p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x)

[Out]

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

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Maxima [A]  time = 1.3811, size = 59, normalized size = 2.36 \begin{align*}{\left (d x^{4} + c x^{3} + b x^{2} + a x\right )} e^{\left (p \log \left (d x^{3} + c x^{2} + b x + a\right ) + m \log \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(d*x^3+c*x^2+b*x+a)^p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x, algorithm="maxima
")

[Out]

(d*x^4 + c*x^3 + b*x^2 + a*x)*e^(p*log(d*x^3 + c*x^2 + b*x + a) + m*log(x))

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Fricas [A]  time = 3.31732, size = 85, normalized size = 3.4 \begin{align*}{\left (d x^{4} + c x^{3} + b x^{2} + a x\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} x^{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(d*x^3+c*x^2+b*x+a)^p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x, algorithm="fricas
")

[Out]

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

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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**m*(d*x**3+c*x**2+b*x+a)**p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x)

[Out]

Timed out

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Giac [B]  time = 1.61687, size = 134, normalized size = 5.36 \begin{align*}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} d x^{4} x^{m} +{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} c x^{3} x^{m} +{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} b x^{2} x^{m} +{\left (d x^{3} + c x^{2} + b x + a\right )}^{p} a x x^{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(d*x^3+c*x^2+b*x+a)^p*(a*(1+m)+x*(b*(2+m+p)+x*(c*(3+m+2*p)+d*(4+m+3*p)*x))),x, algorithm="giac")

[Out]

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