3.54 \(\int x (a+b x)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]

[Out]

(a^2*x^2)/2 + (2*a*b*x^3)/3 + (b^2*x^4)/4

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Rubi [A]  time = 0.0094348, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^2,x]

[Out]

(a^2*x^2)/2 + (2*a*b*x^3)/3 + (b^2*x^4)/4

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x (a+b x)^2 \, dx &=\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx\\ &=\frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4}\\ \end{align*}

Mathematica [A]  time = 0.0013115, size = 30, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^2,x]

[Out]

(a^2*x^2)/2 + (2*a*b*x^3)/3 + (b^2*x^4)/4

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{2\,ab{x}^{3}}{3}}+{\frac{{b}^{2}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x+a)^2,x)

[Out]

1/2*a^2*x^2+2/3*a*b*x^3+1/4*b^2*x^4

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Maxima [A]  time = 1.02898, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4 + 2/3*a*b*x^3 + 1/2*a^2*x^2

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Fricas [A]  time = 1.28362, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{4} x^{4} b^{2} + \frac{2}{3} x^{3} b a + \frac{1}{2} x^{2} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*b^2 + 2/3*x^3*b*a + 1/2*x^2*a^2

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Sympy [A]  time = 0.087937, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x+a)**2,x)

[Out]

a**2*x**2/2 + 2*a*b*x**3/3 + b**2*x**4/4

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Giac [A]  time = 1.21765, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, a^{2} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b^2*x^4 + 2/3*a*b*x^3 + 1/2*a^2*x^2