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

Optimal. Leaf size=16 \[ \frac {\left (a+b x^2\right )^3}{6 b} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {261} \begin {gather*} \frac {\left (a+b x^2\right )^3}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2,x]

[Out]

(a + b*x^2)^3/(6*b)

Rule 261

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

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^3}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^2,x]

[Out]

(a + b*x^2)^3/(6*b)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a+b x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(a + b*x^2)^2,x]

[Out]

IntegrateAlgebraic[x*(a + b*x^2)^2, x]

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fricas [A]  time = 0.49, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{6} x^{6} b^{2} + \frac {1}{2} x^{4} b a + \frac {1}{2} x^{2} a^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.05, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{3}}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^2 + a)^3/b

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maple [A]  time = 0.00, size = 25, normalized size = 1.56 \begin {gather*} \frac {1}{6} b^{2} x^{6}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{3}}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x^2 + a)^3/b

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mupad [B]  time = 0.03, size = 24, normalized size = 1.50 \begin {gather*} \frac {a^2\,x^2}{2}+\frac {a\,b\,x^4}{2}+\frac {b^2\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^2,x)

[Out]

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

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sympy [B]  time = 0.06, size = 24, normalized size = 1.50 \begin {gather*} \frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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