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

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

[Out]

(a + c*x^2 + d*x^3)^(1 + n)/(1 + n)

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Rubi [A]  time = 0.0177172, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1588} \[ \frac{\left (a+c x^2+d x^3\right )^{n+1}}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[(2*c*x + 3*d*x^2)*(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 \left (2 c x+3 d x^2\right ) \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*}

Mathematica [A]  time = 0.011625, 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[(2*c*x + 3*d*x^2)*(a + c*x^2 + d*x^3)^n,x]

[Out]

(a + x^2*(c + d*x))^(1 + n)/(1 + n)

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Maple [A]  time = 0.003, size = 23, normalized size = 1.1 \begin{align*}{\frac{ \left ( d{x}^{3}+c{x}^{2}+a \right ) ^{1+n}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62482, size = 69, normalized size = 3.14 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + a\right )}{\left (d x^{3} + c x^{2} + a\right )}^{n}}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*d*x^2+2*c*x)*(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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.17653, size = 82, normalized size = 3.73 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + a\right )}^{n} d x^{3} +{\left (d x^{3} + c x^{2} + a\right )}^{n} c x^{2} +{\left (d x^{3} + c x^{2} + a\right )}^{n} a}{n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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