3.2460 \(\int x (a+b x^n)^2 \, dx\)

Optimal. Leaf size=44 \[ \frac{a^2 x^2}{2}+\frac{2 a b x^{n+2}}{n+2}+\frac{b^2 x^{2 (n+1)}}{2 (n+1)} \]

[Out]

(a^2*x^2)/2 + (b^2*x^(2*(1 + n)))/(2*(1 + n)) + (2*a*b*x^(2 + n))/(2 + n)

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Rubi [A]  time = 0.0162416, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {270} \[ \frac{a^2 x^2}{2}+\frac{2 a b x^{n+2}}{n+2}+\frac{b^2 x^{2 (n+1)}}{2 (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^n)^2,x]

[Out]

(a^2*x^2)/2 + (b^2*x^(2*(1 + n)))/(2*(1 + n)) + (2*a*b*x^(2 + n))/(2 + n)

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 \left (a+b x^n\right )^2 \, dx &=\int \left (a^2 x+2 a b x^{1+n}+b^2 x^{1+2 n}\right ) \, dx\\ &=\frac{a^2 x^2}{2}+\frac{b^2 x^{2 (1+n)}}{2 (1+n)}+\frac{2 a b x^{2+n}}{2+n}\\ \end{align*}

Mathematica [A]  time = 0.0323536, size = 37, normalized size = 0.84 \[ \frac{1}{2} x^2 \left (a^2+\frac{4 a b x^n}{n+2}+\frac{b^2 x^{2 n}}{n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^n)^2,x]

[Out]

(x^2*(a^2 + (4*a*b*x^n)/(2 + n) + (b^2*x^(2*n))/(1 + n)))/2

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Maple [A]  time = 0.008, size = 47, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{{b}^{2}{x}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2+2\,n}}+2\,{\frac{ab{x}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{2+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*x^2+1/2*b^2/(1+n)*x^2*exp(n*ln(x))^2+2*a*b/(2+n)*x^2*exp(n*ln(x))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.06011, size = 150, normalized size = 3.41 \begin{align*} \frac{{\left (b^{2} n + 2 \, b^{2}\right )} x^{2} x^{2 \, n} + 4 \,{\left (a b n + a b\right )} x^{2} x^{n} +{\left (a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}\right )} x^{2}}{2 \,{\left (n^{2} + 3 \, n + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((b^2*n + 2*b^2)*x^2*x^(2*n) + 4*(a*b*n + a*b)*x^2*x^n + (a^2*n^2 + 3*a^2*n + 2*a^2)*x^2)/(n^2 + 3*n + 2)

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Sympy [A]  time = 0.455159, size = 201, normalized size = 4.57 \begin{align*} \begin{cases} \frac{a^{2} x^{2}}{2} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{2 x^{2}} & \text{for}\: n = -2 \\\frac{a^{2} x^{2}}{2} + 2 a b x + b^{2} \log{\left (x \right )} & \text{for}\: n = -1 \\\frac{a^{2} n^{2} x^{2}}{2 n^{2} + 6 n + 4} + \frac{3 a^{2} n x^{2}}{2 n^{2} + 6 n + 4} + \frac{2 a^{2} x^{2}}{2 n^{2} + 6 n + 4} + \frac{4 a b n x^{2} x^{n}}{2 n^{2} + 6 n + 4} + \frac{4 a b x^{2} x^{n}}{2 n^{2} + 6 n + 4} + \frac{b^{2} n x^{2} x^{2 n}}{2 n^{2} + 6 n + 4} + \frac{2 b^{2} x^{2} x^{2 n}}{2 n^{2} + 6 n + 4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x**2/2 + 2*a*b*log(x) - b**2/(2*x**2), Eq(n, -2)), (a**2*x**2/2 + 2*a*b*x + b**2*log(x), Eq(n,
-1)), (a**2*n**2*x**2/(2*n**2 + 6*n + 4) + 3*a**2*n*x**2/(2*n**2 + 6*n + 4) + 2*a**2*x**2/(2*n**2 + 6*n + 4)
+ 4*a*b*n*x**2*x**n/(2*n**2 + 6*n + 4) + 4*a*b*x**2*x**n/(2*n**2 + 6*n + 4) + b**2*n*x**2*x**(2*n)/(2*n**2 + 6
*n + 4) + 2*b**2*x**2*x**(2*n)/(2*n**2 + 6*n + 4), True))

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Giac [B]  time = 1.17757, size = 117, normalized size = 2.66 \begin{align*} \frac{b^{2} n x^{2} x^{2 \, n} + 4 \, a b n x^{2} x^{n} + a^{2} n^{2} x^{2} + 2 \, b^{2} x^{2} x^{2 \, n} + 4 \, a b x^{2} x^{n} + 3 \, a^{2} n x^{2} + 2 \, a^{2} x^{2}}{2 \,{\left (n^{2} + 3 \, n + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*n*x^2*x^(2*n) + 4*a*b*n*x^2*x^n + a^2*n^2*x^2 + 2*b^2*x^2*x^(2*n) + 4*a*b*x^2*x^n + 3*a^2*n*x^2 + 2*a
^2*x^2)/(n^2 + 3*n + 2)