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

Optimal. Leaf size=21 \[ x \left (a+b x+c x^2+d x^3\right )^{p+1} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {1588} \[ x \left (a+b x+c x^2+d x^3\right )^{p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.12, size = 19, normalized size = 0.90 \[ x (a+x (b+x (c+d x)))^{p+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(a + x*(b + x*(c + d*x)))^(1 + p)

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fricas [A]  time = 0.87, size = 37, normalized size = 1.76 \[ {\left (d x^{4} + c x^{3} + b x^{2} + a x\right )} {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [B]  time = 1.08, size = 87, normalized size = 4.14 \[ {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} d x^{4} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} c x^{3} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} b x^{2} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 22, normalized size = 1.05 \[ x \left (d \,x^{3}+c \,x^{2}+b x +a \right )^{p +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 37, normalized size = 1.76 \[ {\left (d x^{4} + c x^{3} + b x^{2} + a x\right )} {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.27, size = 37, normalized size = 1.76 \[ {\left (d\,x^3+c\,x^2+b\,x+a\right )}^p\,\left (d\,x^4+c\,x^3+b\,x^2+a\,x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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