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

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

[Out]

(c*x^2 + d*x^3)^(1 + n)/(1 + n)

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Rubi [A]  time = 0.012103, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1588} \[ \frac{\left (c x^2+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0046984, size = 19, normalized size = 0.9 \[ \frac{\left (x^2 (c+d x)\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(c + d*x))^(1 + n)/(1 + n)

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Maple [A]  time = 0.002, size = 28, normalized size = 1.3 \begin{align*}{\frac{ \left ( d{x}^{3}+c{x}^{2} \right ) ^{n}{x}^{2} \left ( dx+c \right ) }{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17194, size = 43, normalized size = 2.05 \begin{align*} \frac{{\left (d x^{3} + c x^{2}\right )} e^{\left (n \log \left (d x + c\right ) + 2 \, n \log \left (x\right )\right )}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.41657, size = 58, normalized size = 2.76 \begin{align*} \frac{{\left (d x^{3} + c x^{2}\right )}{\left (d x^{3} + c x^{2}\right )}^{n}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.773654, size = 53, normalized size = 2.52 \begin{align*} \begin{cases} \frac{c x^{2} \left (c x^{2} + d x^{3}\right )^{n}}{n + 1} + \frac{d x^{3} \left (c x^{2} + d x^{3}\right )^{n}}{n + 1} & \text{for}\: n \neq -1 \\2 \log{\left (x \right )} + \log{\left (\frac{c}{d} + x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c*x**2*(c*x**2 + d*x**3)**n/(n + 1) + d*x**3*(c*x**2 + d*x**3)**n/(n + 1), Ne(n, -1)), (2*log(x) +
log(c/d + x), True))

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Giac [B]  time = 1.29148, size = 58, normalized size = 2.76 \begin{align*} \frac{{\left (d x^{3} + c x^{2}\right )}^{n} d x^{3} +{\left (d x^{3} + c x^{2}\right )}^{n} c x^{2}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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