3.2466 \(\int x^2 (a+b x^n)^3 \, dx\)

Optimal. Leaf size=66 \[ \frac{3 a^2 b x^{n+3}}{n+3}+\frac{a^3 x^3}{3}+\frac{3 a b^2 x^{2 n+3}}{2 n+3}+\frac{b^3 x^{3 (n+1)}}{3 (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0265586, antiderivative size = 66, 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} \[ \frac{3 a^2 b x^{n+3}}{n+3}+\frac{a^3 x^3}{3}+\frac{3 a b^2 x^{2 n+3}}{2 n+3}+\frac{b^3 x^{3 (n+1)}}{3 (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^n)^3,x]

[Out]

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

Mathematica [A]  time = 0.0353533, size = 57, normalized size = 0.86 \[ \frac{1}{3} x^3 \left (\frac{9 a^2 b x^n}{n+3}+a^3+\frac{9 a b^2 x^{2 n}}{2 n+3}+\frac{b^3 x^{3 n}}{n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^n)^3,x]

[Out]

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

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Maple [A]  time = 0.008, size = 72, normalized size = 1.1 \begin{align*}{\frac{{a}^{3}{x}^{3}}{3}}+{\frac{{b}^{3}{x}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3+3\,n}}+3\,{\frac{{a}^{2}b{x}^{3}{{\rm e}^{n\ln \left ( x \right ) }}}{3+n}}+3\,{\frac{{x}^{3}a{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{3+2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.08527, size = 304, normalized size = 4.61 \begin{align*} \frac{{\left (2 \, b^{3} n^{2} + 9 \, b^{3} n + 9 \, b^{3}\right )} x^{3} x^{3 \, n} + 9 \,{\left (a b^{2} n^{2} + 4 \, a b^{2} n + 3 \, a b^{2}\right )} x^{3} x^{2 \, n} + 9 \,{\left (2 \, a^{2} b n^{2} + 5 \, a^{2} b n + 3 \, a^{2} b\right )} x^{3} x^{n} +{\left (2 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 18 \, a^{3} n + 9 \, a^{3}\right )} x^{3}}{3 \,{\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((2*b^3*n^2 + 9*b^3*n + 9*b^3)*x^3*x^(3*n) + 9*(a*b^2*n^2 + 4*a*b^2*n + 3*a*b^2)*x^3*x^(2*n) + 9*(2*a^2*b*
n^2 + 5*a^2*b*n + 3*a^2*b)*x^3*x^n + (2*a^3*n^3 + 11*a^3*n^2 + 18*a^3*n + 9*a^3)*x^3)/(2*n^3 + 11*n^2 + 18*n +
9)

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Sympy [A]  time = 4.45146, size = 500, normalized size = 7.58 \begin{align*} \begin{cases} \frac{a^{3} x^{3}}{3} + 3 a^{2} b \log{\left (x \right )} - \frac{a b^{2}}{x^{3}} - \frac{b^{3}}{6 x^{6}} & \text{for}\: n = -3 \\\frac{a^{3} x^{3}}{3} + 2 a^{2} b x^{\frac{3}{2}} + 3 a b^{2} \log{\left (x \right )} - \frac{2 b^{3}}{3 x^{\frac{3}{2}}} & \text{for}\: n = - \frac{3}{2} \\\frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} x + b^{3} \log{\left (x \right )} & \text{for}\: n = -1 \\\frac{2 a^{3} n^{3} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{11 a^{3} n^{2} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{18 a^{3} n x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{9 a^{3} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{18 a^{2} b n^{2} x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{45 a^{2} b n x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{27 a^{2} b x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{9 a b^{2} n^{2} x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{36 a b^{2} n x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{27 a b^{2} x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{2 b^{3} n^{2} x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{9 b^{3} n x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{9 b^{3} x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 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)**3,x)

[Out]

Piecewise((a**3*x**3/3 + 3*a**2*b*log(x) - a*b**2/x**3 - b**3/(6*x**6), Eq(n, -3)), (a**3*x**3/3 + 2*a**2*b*x*
*(3/2) + 3*a*b**2*log(x) - 2*b**3/(3*x**(3/2)), Eq(n, -3/2)), (a**3*x**3/3 + 3*a**2*b*x**2/2 + 3*a*b**2*x + b*
*3*log(x), Eq(n, -1)), (2*a**3*n**3*x**3/(6*n**3 + 33*n**2 + 54*n + 27) + 11*a**3*n**2*x**3/(6*n**3 + 33*n**2
+ 54*n + 27) + 18*a**3*n*x**3/(6*n**3 + 33*n**2 + 54*n + 27) + 9*a**3*x**3/(6*n**3 + 33*n**2 + 54*n + 27) + 18
*a**2*b*n**2*x**3*x**n/(6*n**3 + 33*n**2 + 54*n + 27) + 45*a**2*b*n*x**3*x**n/(6*n**3 + 33*n**2 + 54*n + 27) +
27*a**2*b*x**3*x**n/(6*n**3 + 33*n**2 + 54*n + 27) + 9*a*b**2*n**2*x**3*x**(2*n)/(6*n**3 + 33*n**2 + 54*n + 2
7) + 36*a*b**2*n*x**3*x**(2*n)/(6*n**3 + 33*n**2 + 54*n + 27) + 27*a*b**2*x**3*x**(2*n)/(6*n**3 + 33*n**2 + 54
*n + 27) + 2*b**3*n**2*x**3*x**(3*n)/(6*n**3 + 33*n**2 + 54*n + 27) + 9*b**3*n*x**3*x**(3*n)/(6*n**3 + 33*n**2
+ 54*n + 27) + 9*b**3*x**3*x**(3*n)/(6*n**3 + 33*n**2 + 54*n + 27), True))

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Giac [B]  time = 1.27289, size = 254, normalized size = 3.85 \begin{align*} \frac{2 \, b^{3} n^{2} x^{3} x^{3 \, n} + 9 \, a b^{2} n^{2} x^{3} x^{2 \, n} + 18 \, a^{2} b n^{2} x^{3} x^{n} + 2 \, a^{3} n^{3} x^{3} + 9 \, b^{3} n x^{3} x^{3 \, n} + 36 \, a b^{2} n x^{3} x^{2 \, n} + 45 \, a^{2} b n x^{3} x^{n} + 11 \, a^{3} n^{2} x^{3} + 9 \, b^{3} x^{3} x^{3 \, n} + 27 \, a b^{2} x^{3} x^{2 \, n} + 27 \, a^{2} b x^{3} x^{n} + 18 \, a^{3} n x^{3} + 9 \, a^{3} x^{3}}{3 \,{\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(2*b^3*n^2*x^3*x^(3*n) + 9*a*b^2*n^2*x^3*x^(2*n) + 18*a^2*b*n^2*x^3*x^n + 2*a^3*n^3*x^3 + 9*b^3*n*x^3*x^(3
*n) + 36*a*b^2*n*x^3*x^(2*n) + 45*a^2*b*n*x^3*x^n + 11*a^3*n^2*x^3 + 9*b^3*x^3*x^(3*n) + 27*a*b^2*x^3*x^(2*n)
+ 27*a^2*b*x^3*x^n + 18*a^3*n*x^3 + 9*a^3*x^3)/(2*n^3 + 11*n^2 + 18*n + 9)