3.2459 \(\int x^2 (a+b x^n)^2 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^2 x^3}{3}+\frac{2 a b x^{n+3}}{n+3}+\frac{b^2 x^{2 n+3}}{2 n+3} \]

[Out]

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

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Rubi [A]  time = 0.0199481, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{a^2 x^3}{3}+\frac{2 a b x^{n+3}}{n+3}+\frac{b^2 x^{2 n+3}}{2 n+3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^n)^2,x]

[Out]

(a^2*x^3)/3 + (2*a*b*x^(3 + n))/(3 + n) + (b^2*x^(3 + 2*n))/(3 + 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^2 \left (a+b x^n\right )^2 \, dx &=\int \left (a^2 x^2+b^2 x^{2 (1+n)}+2 a b x^{2+n}\right ) \, dx\\ &=\frac{a^2 x^3}{3}+\frac{2 a b x^{3+n}}{3+n}+\frac{b^2 x^{3+2 n}}{3+2 n}\\ \end{align*}

Mathematica [A]  time = 0.0350172, size = 40, normalized size = 0.93 \[ \frac{1}{3} x^3 \left (a^2+\frac{6 a b x^n}{n+3}+\frac{3 b^2 x^{2 n}}{2 n+3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^n)^2,x]

[Out]

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

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Maple [A]  time = 0.01, size = 48, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{x}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{3+2\,n}}+{\frac{{x}^{3}{a}^{2}}{3}}+2\,{\frac{{x}^{3}ab{{\rm e}^{n\ln \left ( x \right ) }}}{3+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2/(3+2*n)*x^3*exp(n*ln(x))^2+1/3*x^3*a^2+2*a*b/(3+n)*x^3*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^2*(a+b*x^n)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.08461, size = 163, normalized size = 3.79 \begin{align*} \frac{3 \,{\left (b^{2} n + 3 \, b^{2}\right )} x^{3} x^{2 \, n} + 6 \,{\left (2 \, a b n + 3 \, a b\right )} x^{3} x^{n} +{\left (2 \, a^{2} n^{2} + 9 \, a^{2} n + 9 \, a^{2}\right )} x^{3}}{3 \,{\left (2 \, n^{2} + 9 \, n + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(b^2*n + 3*b^2)*x^3*x^(2*n) + 6*(2*a*b*n + 3*a*b)*x^3*x^n + (2*a^2*n^2 + 9*a^2*n + 9*a^2)*x^3)/(2*n^2 +
 9*n + 9)

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Sympy [A]  time = 1.72169, size = 211, normalized size = 4.91 \begin{align*} \begin{cases} \frac{a^{2} x^{3}}{3} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{3 x^{3}} & \text{for}\: n = -3 \\\frac{a^{2} x^{3}}{3} + \frac{4 a b x^{\frac{3}{2}}}{3} + b^{2} \log{\left (x \right )} & \text{for}\: n = - \frac{3}{2} \\\frac{2 a^{2} n^{2} x^{3}}{6 n^{2} + 27 n + 27} + \frac{9 a^{2} n x^{3}}{6 n^{2} + 27 n + 27} + \frac{9 a^{2} x^{3}}{6 n^{2} + 27 n + 27} + \frac{12 a b n x^{3} x^{n}}{6 n^{2} + 27 n + 27} + \frac{18 a b x^{3} x^{n}}{6 n^{2} + 27 n + 27} + \frac{3 b^{2} n x^{3} x^{2 n}}{6 n^{2} + 27 n + 27} + \frac{9 b^{2} x^{3} x^{2 n}}{6 n^{2} + 27 n + 27} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*x**3/3 + 2*a*b*log(x) - b**2/(3*x**3), Eq(n, -3)), (a**2*x**3/3 + 4*a*b*x**(3/2)/3 + b**2*log(
x), Eq(n, -3/2)), (2*a**2*n**2*x**3/(6*n**2 + 27*n + 27) + 9*a**2*n*x**3/(6*n**2 + 27*n + 27) + 9*a**2*x**3/(6
*n**2 + 27*n + 27) + 12*a*b*n*x**3*x**n/(6*n**2 + 27*n + 27) + 18*a*b*x**3*x**n/(6*n**2 + 27*n + 27) + 3*b**2*
n*x**3*x**(2*n)/(6*n**2 + 27*n + 27) + 9*b**2*x**3*x**(2*n)/(6*n**2 + 27*n + 27), True))

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Giac [B]  time = 1.21544, size = 123, normalized size = 2.86 \begin{align*} \frac{3 \, b^{2} n x^{3} x^{2 \, n} + 12 \, a b n x^{3} x^{n} + 2 \, a^{2} n^{2} x^{3} + 9 \, b^{2} x^{3} x^{2 \, n} + 18 \, a b x^{3} x^{n} + 9 \, a^{2} n x^{3} + 9 \, a^{2} x^{3}}{3 \,{\left (2 \, n^{2} + 9 \, n + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b^2*n*x^3*x^(2*n) + 12*a*b*n*x^3*x^n + 2*a^2*n^2*x^3 + 9*b^2*x^3*x^(2*n) + 18*a*b*x^3*x^n + 9*a^2*n*x^3
 + 9*a^2*x^3)/(2*n^2 + 9*n + 9)