3.87 \(\int (2 x+3 x^2)^3 \, dx\)

Optimal. Leaf size=25 \[ \frac{27 x^7}{7}+9 x^6+\frac{36 x^5}{5}+2 x^4 \]

[Out]

2*x^4 + (36*x^5)/5 + 9*x^6 + (27*x^7)/7

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Rubi [A]  time = 0.0071173, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {611} \[ \frac{27 x^7}{7}+9 x^6+\frac{36 x^5}{5}+2 x^4 \]

Antiderivative was successfully verified.

[In]

Int[(2*x + 3*x^2)^3,x]

[Out]

2*x^4 + (36*x^5)/5 + 9*x^6 + (27*x^7)/7

Rule 611

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] ||  !PerfectSquareQ[b^2 - 4*a*c])

Rubi steps

\begin{align*} \int \left (2 x+3 x^2\right )^3 \, dx &=\int \left (8 x^3+36 x^4+54 x^5+27 x^6\right ) \, dx\\ &=2 x^4+\frac{36 x^5}{5}+9 x^6+\frac{27 x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0016558, size = 25, normalized size = 1. \[ \frac{27 x^7}{7}+9 x^6+\frac{36 x^5}{5}+2 x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[(2*x + 3*x^2)^3,x]

[Out]

2*x^4 + (36*x^5)/5 + 9*x^6 + (27*x^7)/7

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Maple [A]  time = 0., size = 22, normalized size = 0.9 \begin{align*} 2\,{x}^{4}+{\frac{36\,{x}^{5}}{5}}+9\,{x}^{6}+{\frac{27\,{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+2*x)^3,x)

[Out]

2*x^4+36/5*x^5+9*x^6+27/7*x^7

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Maxima [A]  time = 0.929313, size = 28, normalized size = 1.12 \begin{align*} \frac{27}{7} \, x^{7} + 9 \, x^{6} + \frac{36}{5} \, x^{5} + 2 \, x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/7*x^7 + 9*x^6 + 36/5*x^5 + 2*x^4

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Fricas [A]  time = 1.57105, size = 50, normalized size = 2. \begin{align*} \frac{27}{7} x^{7} + 9 x^{6} + \frac{36}{5} x^{5} + 2 x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/7*x^7 + 9*x^6 + 36/5*x^5 + 2*x^4

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Sympy [A]  time = 0.054514, size = 22, normalized size = 0.88 \begin{align*} \frac{27 x^{7}}{7} + 9 x^{6} + \frac{36 x^{5}}{5} + 2 x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+2*x)**3,x)

[Out]

27*x**7/7 + 9*x**6 + 36*x**5/5 + 2*x**4

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Giac [A]  time = 1.04843, size = 28, normalized size = 1.12 \begin{align*} \frac{27}{7} \, x^{7} + 9 \, x^{6} + \frac{36}{5} \, x^{5} + 2 \, x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/7*x^7 + 9*x^6 + 36/5*x^5 + 2*x^4