3.2458 \(\int x^3 (a+b x^n)^2 \, dx\)

Optimal. Leaf size=44 \[ \frac{a^2 x^4}{4}+\frac{2 a b x^{n+4}}{n+4}+\frac{b^2 x^{2 (n+2)}}{2 (n+2)} \]

[Out]

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

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Rubi [A]  time = 0.0178456, antiderivative size = 44, 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} \[ \frac{a^2 x^4}{4}+\frac{2 a b x^{n+4}}{n+4}+\frac{b^2 x^{2 (n+2)}}{2 (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^n)^2,x]

[Out]

(a^2*x^4)/4 + (b^2*x^(2*(2 + n)))/(2*(2 + n)) + (2*a*b*x^(4 + n))/(4 + 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^3 \left (a+b x^n\right )^2 \, dx &=\int \left (a^2 x^3+2 a b x^{3+n}+b^2 x^{3+2 n}\right ) \, dx\\ &=\frac{a^2 x^4}{4}+\frac{b^2 x^{2 (2+n)}}{2 (2+n)}+\frac{2 a b x^{4+n}}{4+n}\\ \end{align*}

Mathematica [A]  time = 0.0343121, size = 38, normalized size = 0.86 \[ \frac{1}{4} x^4 \left (a^2+\frac{8 a b x^n}{n+4}+\frac{2 b^2 x^{2 n}}{n+2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^n)^2,x]

[Out]

(x^4*(a^2 + (8*a*b*x^n)/(4 + n) + (2*b^2*x^(2*n))/(2 + n)))/4

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Maple [A]  time = 0.01, size = 47, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{{b}^{2}{x}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{4+2\,n}}+2\,{\frac{ab{x}^{4}{{\rm e}^{n\ln \left ( x \right ) }}}{4+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^2*x^4+1/2*b^2/(2+n)*x^4*exp(n*ln(x))^2+2*a*b/(4+n)*x^4*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^3*(a+b*x^n)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01665, size = 155, normalized size = 3.52 \begin{align*} \frac{2 \,{\left (b^{2} n + 4 \, b^{2}\right )} x^{4} x^{2 \, n} + 8 \,{\left (a b n + 2 \, a b\right )} x^{4} x^{n} +{\left (a^{2} n^{2} + 6 \, a^{2} n + 8 \, a^{2}\right )} x^{4}}{4 \,{\left (n^{2} + 6 \, n + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*(b^2*n + 4*b^2)*x^4*x^(2*n) + 8*(a*b*n + 2*a*b)*x^4*x^n + (a^2*n^2 + 6*a^2*n + 8*a^2)*x^4)/(n^2 + 6*n +
8)

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Sympy [A]  time = 0.824018, size = 202, normalized size = 4.59 \begin{align*} \begin{cases} \frac{a^{2} x^{4}}{4} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{4 x^{4}} & \text{for}\: n = -4 \\\frac{a^{2} x^{4}}{4} + a b x^{2} + b^{2} \log{\left (x \right )} & \text{for}\: n = -2 \\\frac{a^{2} n^{2} x^{4}}{4 n^{2} + 24 n + 32} + \frac{6 a^{2} n x^{4}}{4 n^{2} + 24 n + 32} + \frac{8 a^{2} x^{4}}{4 n^{2} + 24 n + 32} + \frac{8 a b n x^{4} x^{n}}{4 n^{2} + 24 n + 32} + \frac{16 a b x^{4} x^{n}}{4 n^{2} + 24 n + 32} + \frac{2 b^{2} n x^{4} x^{2 n}}{4 n^{2} + 24 n + 32} + \frac{8 b^{2} x^{4} x^{2 n}}{4 n^{2} + 24 n + 32} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*x**n)**2,x)

[Out]

Piecewise((a**2*x**4/4 + 2*a*b*log(x) - b**2/(4*x**4), Eq(n, -4)), (a**2*x**4/4 + a*b*x**2 + b**2*log(x), Eq(n
, -2)), (a**2*n**2*x**4/(4*n**2 + 24*n + 32) + 6*a**2*n*x**4/(4*n**2 + 24*n + 32) + 8*a**2*x**4/(4*n**2 + 24*n
+ 32) + 8*a*b*n*x**4*x**n/(4*n**2 + 24*n + 32) + 16*a*b*x**4*x**n/(4*n**2 + 24*n + 32) + 2*b**2*n*x**4*x**(2*
n)/(4*n**2 + 24*n + 32) + 8*b**2*x**4*x**(2*n)/(4*n**2 + 24*n + 32), True))

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Giac [B]  time = 1.23935, size = 119, normalized size = 2.7 \begin{align*} \frac{2 \, b^{2} n x^{4} x^{2 \, n} + 8 \, a b n x^{4} x^{n} + a^{2} n^{2} x^{4} + 8 \, b^{2} x^{4} x^{2 \, n} + 16 \, a b x^{4} x^{n} + 6 \, a^{2} n x^{4} + 8 \, a^{2} x^{4}}{4 \,{\left (n^{2} + 6 \, n + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*b^2*n*x^4*x^(2*n) + 8*a*b*n*x^4*x^n + a^2*n^2*x^4 + 8*b^2*x^4*x^(2*n) + 16*a*b*x^4*x^n + 6*a^2*n*x^4 +
8*a^2*x^4)/(n^2 + 6*n + 8)