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

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Rubi [A]  time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, 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*}

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Mathematica [A]  time = 0.18, 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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fricas [A]  time = 0.89, size = 39, normalized size = 1.70 \[ {\left (d x^{5} + c x^{4} + b x^{3} + a x^{2}\right )} {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

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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giac [B]  time = 0.73, size = 89, normalized size = 3.87 \[ {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} d x^{5} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} c x^{4} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} b x^{3} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} a x^{2} \]

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

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maple [A]  time = 0.01, size = 24, normalized size = 1.04 \[ x^{2} \left (d \,x^{3}+c \,x^{2}+b x +a \right )^{p +1} \]

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)^(p+1)

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maxima [A]  time = 0.63, size = 39, normalized size = 1.70 \[ {\left (d x^{5} + c x^{4} + b x^{3} + a x^{2}\right )} {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

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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mupad [B]  time = 2.32, size = 39, normalized size = 1.70 \[ {\left (d\,x^3+c\,x^2+b\,x+a\right )}^p\,\left (d\,x^5+c\,x^4+b\,x^3+a\,x^2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*(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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