3.180 \(\int (b+2 c x+3 d x^2) (a+b x+c x^2+d x^3)^n \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {1588} \[ \frac {\left (a+b x+c x^2+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.92 \[ \frac {(a+x (b+x (c+d x)))^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + x*(b + x*(c + d*x)))^(1 + n)/(1 + n)

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fricas [A]  time = 0.84, size = 38, normalized size = 1.52 \[ \frac {{\left (d x^{3} + c x^{2} + b x + a\right )} {\left (d x^{3} + c x^{2} + b x + a\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.29, size = 25, normalized size = 1.00 \[ \frac {{\left (d x^{3} + c x^{2} + b x + a\right )}^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 26, normalized size = 1.04 \[ \frac {\left (d \,x^{3}+c \,x^{2}+b x +a \right )^{n +1}}{n +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.67, size = 25, normalized size = 1.00 \[ \frac {{\left (d x^{3} + c x^{2} + b x + a\right )}^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.19, size = 54, normalized size = 2.16 \[ {\left (d\,x^3+c\,x^2+b\,x+a\right )}^n\,\left (\frac {a}{n+1}+\frac {b\,x}{n+1}+\frac {c\,x^2}{n+1}+\frac {d\,x^3}{n+1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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