3.66 \(\int x^2 (a+b x)^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^3}{3}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^6}{6} \]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^4)/4 + (3*a*b^2*x^5)/5 + (b^3*x^6)/6

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Rubi [A]  time = 0.0146646, antiderivative size = 43, 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 = {43} \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^3}{3}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^4)/4 + (3*a*b^2*x^5)/5 + (b^3*x^6)/6

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^2 (a+b x)^3 \, dx &=\int \left (a^3 x^2+3 a^2 b x^3+3 a b^2 x^4+b^3 x^5\right ) \, dx\\ &=\frac{a^3 x^3}{3}+\frac{3}{4} a^2 b x^4+\frac{3}{5} a b^2 x^5+\frac{b^3 x^6}{6}\\ \end{align*}

Mathematica [A]  time = 0.0017854, size = 43, normalized size = 1. \[ \frac{3}{4} a^2 b x^4+\frac{a^3 x^3}{3}+\frac{3}{5} a b^2 x^5+\frac{b^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^4)/4 + (3*a*b^2*x^5)/5 + (b^3*x^6)/6

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{3}}{3}}+{\frac{3\,{a}^{2}b{x}^{4}}{4}}+{\frac{3\,a{b}^{2}{x}^{5}}{5}}+{\frac{{b}^{3}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3+3/4*a^2*b*x^4+3/5*a*b^2*x^5+1/6*b^3*x^6

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Maxima [A]  time = 1.03505, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{5} \, a b^{2} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/5*a*b^2*x^5 + 3/4*a^2*b*x^4 + 1/3*a^3*x^3

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Fricas [A]  time = 1.36393, size = 80, normalized size = 1.86 \begin{align*} \frac{1}{6} x^{6} b^{3} + \frac{3}{5} x^{5} b^{2} a + \frac{3}{4} x^{4} b a^{2} + \frac{1}{3} x^{3} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*b^3 + 3/5*x^5*b^2*a + 3/4*x^4*b*a^2 + 1/3*x^3*a^3

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Sympy [A]  time = 0.067495, size = 39, normalized size = 0.91 \begin{align*} \frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{4}}{4} + \frac{3 a b^{2} x^{5}}{5} + \frac{b^{3} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + 3*a**2*b*x**4/4 + 3*a*b**2*x**5/5 + b**3*x**6/6

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Giac [A]  time = 1.16336, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{5} \, a b^{2} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/5*a*b^2*x^5 + 3/4*a^2*b*x^4 + 1/3*a^3*x^3