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

Optimal. Leaf size=16 \[ \frac {\left (a+b x^2\right )^4}{8 b} \]

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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 )^4}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^3,x]

[Out]

(a + b*x^2)^4/(8*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 )^3 \, dx &=\frac {\left (a+b x^2\right )^4}{8 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 )^4}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^3,x]

[Out]

(a + b*x^2)^4/(8*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 )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x*(a + b*x^2)^3,x]

[Out]

IntegrateAlgebraic[x*(a + b*x^2)^3, x]

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fricas [B]  time = 0.89, size = 35, normalized size = 2.19 \begin {gather*} \frac {1}{8} x^{8} b^{3} + \frac {1}{2} x^{6} b^{2} a + \frac {3}{4} x^{4} b a^{2} + \frac {1}{2} x^{2} a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.00, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{4}}{8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x^2 + a)^4/b

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maple [B]  time = 0.00, size = 36, normalized size = 2.25 \begin {gather*} \frac {1}{8} b^{3} x^{8}+\frac {1}{2} a \,b^{2} x^{6}+\frac {3}{4} a^{2} b \,x^{4}+\frac {1}{2} a^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{2} + a\right )}^{4}}{8 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x^2 + a)^4/b

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mupad [B]  time = 0.06, size = 35, normalized size = 2.19 \begin {gather*} \frac {a^3\,x^2}{2}+\frac {3\,a^2\,b\,x^4}{4}+\frac {a\,b^2\,x^6}{2}+\frac {b^3\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^2)^3,x)

[Out]

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

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sympy [B]  time = 0.07, size = 37, normalized size = 2.31 \begin {gather*} \frac {a^{3} x^{2}}{2} + \frac {3 a^{2} b x^{4}}{4} + \frac {a b^{2} x^{6}}{2} + \frac {b^{3} x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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