3.3.37 \(\int (a+b x+c x^2+d x^3)^p (a+b (2+p) x+c (3+2 p) x^2+d (4+3 p) x^3) \, dx\) [237]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, 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 = {1602} \begin {gather*} x \left (a+b x+c x^2+d x^3\right )^{p+1} \end {gather*}

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 1602

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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.74, size = 19, normalized size = 0.90 \begin {gather*} x (a+x (b+x (c+d x)))^{1+p} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 22, normalized size = 1.05

method	result	size
gosper	\(x \left (d \,x^{3}+c \,x^{2}+b x +a \right )^{1+p}\)	\(22\)
risch	\(\left (d \,x^{3}+c \,x^{2}+b x +a \right )^{p} x \left (d \,x^{3}+c \,x^{2}+b x +a \right )\)	\(35\)
norman	\(a x \,{\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+b \,x^{2} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+c \,x^{3} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+d \,x^{4} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 37, normalized size = 1.76 \begin {gather*} {\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} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.40, size = 37, normalized size = 1.76 \begin {gather*} {\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} \end {gather*}

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (21) = 42\).
time = 4.64, size = 87, normalized size = 4.14 \begin {gather*} {\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 \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 2.27, size = 37, normalized size = 1.76 \begin {gather*} {\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 ) \end {gather*}

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)

________________________________________________________________________________________