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

Optimal. Leaf size=30 \[ \frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{b^2 x^9}{9} \]

[Out]

(a^2*x^5)/5 + (2*a*b*x^7)/7 + (b^2*x^9)/9

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Rubi [A]  time = 0.0150718, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1584, 270} \[ \frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{b^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a*x + b*x^3)^2,x]

[Out]

(a^2*x^5)/5 + (2*a*b*x^7)/7 + (b^2*x^9)/9

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0007609, size = 30, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{2}{7} a b x^7+\frac{b^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a*x + b*x^3)^2,x]

[Out]

(a^2*x^5)/5 + (2*a*b*x^7)/7 + (b^2*x^9)/9

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{5}}{5}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{{b}^{2}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*x^5+2/7*a*b*x^7+1/9*b^2*x^9

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Maxima [A]  time = 1.0588, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{9} \, b^{2} x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{5} \, a^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^2*x^9 + 2/7*a*b*x^7 + 1/5*a^2*x^5

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Fricas [A]  time = 1.20442, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{9} x^{9} b^{2} + \frac{2}{7} x^{7} b a + \frac{1}{5} x^{5} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*b^2 + 2/7*x^7*b*a + 1/5*x^5*a^2

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Sympy [A]  time = 0.060723, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{2 a b x^{7}}{7} + \frac{b^{2} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**5/5 + 2*a*b*x**7/7 + b**2*x**9/9

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Giac [A]  time = 1.13094, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{9} \, b^{2} x^{9} + \frac{2}{7} \, a b x^{7} + \frac{1}{5} \, a^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^2*x^9 + 2/7*a*b*x^7 + 1/5*a^2*x^5