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

Optimal. Leaf size=22 \[ \frac {\left (a+c x^2+d x^3\right )^{n+1}}{n+1} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1588} \[ \frac {\left (a+c x^2+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.95 \[ \frac {\left (a+x^2 (c+d x)\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + x^2*(c + d*x))^(1 + n)/(1 + n)

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fricas [A]  time = 1.23, size = 32, normalized size = 1.45 \[ \frac {{\left (d x^{3} + c x^{2} + a\right )} {\left (d x^{3} + c x^{2} + a\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.32, size = 22, normalized size = 1.00 \[ \frac {{\left (d x^{3} + c x^{2} + a\right )}^{n + 1}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x^3 + c*x^2 + a)^(n + 1)/(n + 1)

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maple [A]  time = 0.00, size = 23, normalized size = 1.05 \[ \frac {\left (d \,x^{3}+c \,x^{2}+a \right )^{n +1}}{n +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.75, size = 32, normalized size = 1.45 \[ \frac {{\left (d x^{3} + c x^{2} + a\right )} {\left (d x^{3} + c x^{2} + a\right )}^{n}}{n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.14, size = 43, normalized size = 1.95 \[ \left (\frac {a}{n+1}+\frac {c\,x^2}{n+1}+\frac {d\,x^3}{n+1}\right )\,{\left (d\,x^3+c\,x^2+a\right )}^n \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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