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

Optimal. Leaf size=30 \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8

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Rubi [A]  time = 0.0177522, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^2,x]

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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^3 \left (a+b x^2\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8}\\ \end{align*}

Mathematica [A]  time = 0.0006755, size = 30, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^2)^2,x]

[Out]

(a^2*x^4)/4 + (a*b*x^6)/3 + (b^2*x^8)/8

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{ab{x}^{6}}{3}}+{\frac{{b}^{2}{x}^{8}}{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*x^4+1/3*a*b*x^6+1/8*b^2*x^8

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Maxima [A]  time = 2.32416, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*x^8 + 1/3*a*b*x^6 + 1/4*a^2*x^4

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Fricas [A]  time = 1.26203, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{8} x^{8} b^{2} + \frac{1}{3} x^{6} b a + \frac{1}{4} x^{4} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8*b^2 + 1/3*x^6*b*a + 1/4*x^4*a^2

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Sympy [A]  time = 0.060431, size = 24, normalized size = 0.8 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**4/4 + a*b*x**6/3 + b**2*x**8/8

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Giac [A]  time = 1.34731, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^2*x^8 + 1/3*a*b*x^6 + 1/4*a^2*x^4