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

Optimal. Leaf size=30 \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

[Out]

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

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Rubi [A]  time = 0.0099083, antiderivative size = 30, 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} \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x)^2,x]

[Out]

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

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)^2 \, dx &=\int \left (a^2 x^2+2 a b x^3+b^2 x^4\right ) \, dx\\ &=\frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0029569, size = 30, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{3}}{3}}+{\frac{ab{x}^{4}}{2}}+{\frac{{b}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05648, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.35776, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{5} x^{5} b^{2} + \frac{1}{2} x^{4} b a + \frac{1}{3} x^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.075105, size = 24, normalized size = 0.8 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1918, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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