3.58 \(\int x (a+b x^2)^5 \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^6}{12 b} \]

[Out]

(a + b*x^2)^6/(12*b)

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Rubi [A]  time = 0.0024507, antiderivative size = 16, normalized size of antiderivative = 1., 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} \[ \frac{\left (a+b x^2\right )^6}{12 b} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^5,x]

[Out]

(a + b*x^2)^6/(12*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 )^5 \, dx &=\frac{\left (a+b x^2\right )^6}{12 b}\\ \end{align*}

Mathematica [A]  time = 0.0023003, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^6}{12 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^2)^5,x]

[Out]

(a + b*x^2)^6/(12*b)

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Maple [B]  time = 0.002, size = 58, normalized size = 3.6 \begin{align*}{\frac{{b}^{5}{x}^{12}}{12}}+{\frac{a{b}^{4}{x}^{10}}{2}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{8}}{4}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{6}}{3}}+{\frac{5\,{a}^{4}b{x}^{4}}{4}}+{\frac{{a}^{5}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x^2+a)^5,x)

[Out]

1/12*b^5*x^12+1/2*a*b^4*x^10+5/4*a^2*b^3*x^8+5/3*a^3*b^2*x^6+5/4*a^4*b*x^4+1/2*a^5*x^2

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Maxima [A]  time = 2.05634, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{6}}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^5,x, algorithm="maxima")

[Out]

1/12*(b*x^2 + a)^6/b

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Fricas [B]  time = 1.09957, size = 132, normalized size = 8.25 \begin{align*} \frac{1}{12} x^{12} b^{5} + \frac{1}{2} x^{10} b^{4} a + \frac{5}{4} x^{8} b^{3} a^{2} + \frac{5}{3} x^{6} b^{2} a^{3} + \frac{5}{4} x^{4} b a^{4} + \frac{1}{2} x^{2} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^5,x, algorithm="fricas")

[Out]

1/12*x^12*b^5 + 1/2*x^10*b^4*a + 5/4*x^8*b^3*a^2 + 5/3*x^6*b^2*a^3 + 5/4*x^4*b*a^4 + 1/2*x^2*a^5

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Sympy [B]  time = 0.067715, size = 65, normalized size = 4.06 \begin{align*} \frac{a^{5} x^{2}}{2} + \frac{5 a^{4} b x^{4}}{4} + \frac{5 a^{3} b^{2} x^{6}}{3} + \frac{5 a^{2} b^{3} x^{8}}{4} + \frac{a b^{4} x^{10}}{2} + \frac{b^{5} x^{12}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x**2+a)**5,x)

[Out]

a**5*x**2/2 + 5*a**4*b*x**4/4 + 5*a**3*b**2*x**6/3 + 5*a**2*b**3*x**8/4 + a*b**4*x**10/2 + b**5*x**12/12

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Giac [A]  time = 3.02084, size = 19, normalized size = 1.19 \begin{align*} \frac{{\left (b x^{2} + a\right )}^{6}}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(b*x^2+a)^5,x, algorithm="giac")

[Out]

1/12*(b*x^2 + a)^6/b