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]

1/3*a^2*x^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 = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.90, size = 78, normalized size = 1.81 \[ \frac {3 \, {\left (b^{2} n + 3 \, b^{2}\right )} x^{3} x^{2 \, n} + 6 \, {\left (2 \, a b n + 3 \, a b\right )} x^{3} x^{n} + {\left (2 \, a^{2} n^{2} + 9 \, a^{2} n + 9 \, a^{2}\right )} x^{3}}{3 \, {\left (2 \, n^{2} + 9 \, n + 9\right )}} \]

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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giac [B]  time = 0.17, size = 91, normalized size = 2.12 \[ \frac {3 \, b^{2} n x^{3} x^{2 \, n} + 12 \, a b n x^{3} x^{n} + 2 \, a^{2} n^{2} x^{3} + 9 \, b^{2} x^{3} x^{2 \, n} + 18 \, a b x^{3} x^{n} + 9 \, a^{2} n x^{3} + 9 \, a^{2} x^{3}}{3 \, {\left (2 \, n^{2} + 9 \, n + 9\right )}} \]

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)

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maple [A]  time = 0.01, size = 48, normalized size = 1.12 \[ \frac {2 a b \,x^{3} {\mathrm e}^{n \ln \relax (x )}}{n +3}+\frac {b^{2} x^{3} {\mathrm e}^{2 n \ln \relax (x )}}{2 n +3}+\frac {a^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 41, normalized size = 0.95 \[ \frac {1}{3} \, a^{2} x^{3} + \frac {b^{2} x^{2 \, n + 3}}{2 \, n + 3} + \frac {2 \, a b x^{n + 3}}{n + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.31, size = 43, normalized size = 1.00 \[ \frac {a^2\,x^3}{3}+\frac {b^2\,x^{2\,n}\,x^3}{2\,n+3}+\frac {2\,a\,b\,x^n\,x^3}{n+3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^n)^2,x)

[Out]

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

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sympy [A]  time = 1.92, size = 211, normalized size = 4.91 \[ \begin {cases} \frac {a^{2} x^{3}}{3} + 2 a b \log {\relax (x )} - \frac {b^{2}}{3 x^{3}} & \text {for}\: n = -3 \\\frac {a^{2} x^{3}}{3} + \frac {4 a b x^{\frac {3}{2}}}{3} + b^{2} \log {\relax (x )} & \text {for}\: n = - \frac {3}{2} \\\frac {2 a^{2} n^{2} x^{3}}{6 n^{2} + 27 n + 27} + \frac {9 a^{2} n x^{3}}{6 n^{2} + 27 n + 27} + \frac {9 a^{2} x^{3}}{6 n^{2} + 27 n + 27} + \frac {12 a b n x^{3} x^{n}}{6 n^{2} + 27 n + 27} + \frac {18 a b x^{3} x^{n}}{6 n^{2} + 27 n + 27} + \frac {3 b^{2} n x^{3} x^{2 n}}{6 n^{2} + 27 n + 27} + \frac {9 b^{2} x^{3} x^{2 n}}{6 n^{2} + 27 n + 27} & \text {otherwise} \end {cases} \]

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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