3.2465 \(\int x^3 (a+b x^n)^3 \, dx\)

Optimal. Leaf size=65 \[ \frac{3 a^2 b x^{n+4}}{n+4}+\frac{a^3 x^4}{4}+\frac{3 a b^2 x^{2 (n+2)}}{2 (n+2)}+\frac{b^3 x^{3 n+4}}{3 n+4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0315563, antiderivative size = 65, 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} \[ \frac{3 a^2 b x^{n+4}}{n+4}+\frac{a^3 x^4}{4}+\frac{3 a b^2 x^{2 (n+2)}}{2 (n+2)}+\frac{b^3 x^{3 n+4}}{3 n+4} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^n)^3,x]

[Out]

(a^3*x^4)/4 + (3*a*b^2*x^(2*(2 + n)))/(2*(2 + n)) + (3*a^2*b*x^(4 + n))/(4 + n) + (b^3*x^(4 + 3*n))/(4 + 3*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 )^3 \, dx &=\int \left (a^3 x^3+b^3 x^{3 (1+n)}+3 a^2 b x^{3+n}+3 a b^2 x^{3+2 n}\right ) \, dx\\ &=\frac{a^3 x^4}{4}+\frac{3 a b^2 x^{2 (2+n)}}{2 (2+n)}+\frac{3 a^2 b x^{4+n}}{4+n}+\frac{b^3 x^{4+3 n}}{4+3 n}\\ \end{align*}

Mathematica [A]  time = 0.0402591, size = 58, normalized size = 0.89 \[ \frac{1}{4} x^4 \left (\frac{12 a^2 b x^n}{n+4}+a^3+\frac{6 a b^2 x^{2 n}}{n+2}+\frac{4 b^3 x^{3 n}}{3 n+4}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^n)^3,x]

[Out]

(x^4*(a^3 + (12*a^2*b*x^n)/(4 + n) + (6*a*b^2*x^(2*n))/(2 + n) + (4*b^3*x^(3*n))/(4 + 3*n)))/4

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 65, normalized size = 1. \begin{align*}{\frac{{a}^{3}{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{4} \left ({x}^{n} \right ) ^{3}}{4+3\,n}}+{\frac{3\,a{b}^{2}{x}^{4} \left ({x}^{n} \right ) ^{2}}{4+2\,n}}+3\,{\frac{{a}^{2}b{x}^{4}{x}^{n}}{4+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.0637, size = 315, normalized size = 4.85 \begin{align*} \frac{4 \,{\left (b^{3} n^{2} + 6 \, b^{3} n + 8 \, b^{3}\right )} x^{4} x^{3 \, n} + 6 \,{\left (3 \, a b^{2} n^{2} + 16 \, a b^{2} n + 16 \, a b^{2}\right )} x^{4} x^{2 \, n} + 12 \,{\left (3 \, a^{2} b n^{2} + 10 \, a^{2} b n + 8 \, a^{2} b\right )} x^{4} x^{n} +{\left (3 \, a^{3} n^{3} + 22 \, a^{3} n^{2} + 48 \, a^{3} n + 32 \, a^{3}\right )} x^{4}}{4 \,{\left (3 \, n^{3} + 22 \, n^{2} + 48 \, n + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*(b^3*n^2 + 6*b^3*n + 8*b^3)*x^4*x^(3*n) + 6*(3*a*b^2*n^2 + 16*a*b^2*n + 16*a*b^2)*x^4*x^(2*n) + 12*(3*a
^2*b*n^2 + 10*a^2*b*n + 8*a^2*b)*x^4*x^n + (3*a^3*n^3 + 22*a^3*n^2 + 48*a^3*n + 32*a^3)*x^4)/(3*n^3 + 22*n^2 +
48*n + 32)

________________________________________________________________________________________

Sympy [A]  time = 14.8165, size = 507, normalized size = 7.8 \begin{align*} \begin{cases} \frac{a^{3} x^{4}}{4} + 3 a^{2} b \log{\left (x \right )} - \frac{3 a b^{2}}{4 x^{4}} - \frac{b^{3}}{8 x^{8}} & \text{for}\: n = -4 \\\frac{a^{3} x^{4}}{4} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} \log{\left (x \right )} - \frac{b^{3}}{2 x^{2}} & \text{for}\: n = -2 \\\frac{a^{3} x^{4}}{4} + \frac{9 a^{2} b x^{\frac{8}{3}}}{8} + \frac{9 a b^{2} x^{\frac{4}{3}}}{4} + b^{3} \log{\left (x \right )} & \text{for}\: n = - \frac{4}{3} \\\frac{3 a^{3} n^{3} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{22 a^{3} n^{2} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{48 a^{3} n x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{32 a^{3} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{36 a^{2} b n^{2} x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{120 a^{2} b n x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{96 a^{2} b x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{18 a b^{2} n^{2} x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{96 a b^{2} n x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{96 a b^{2} x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{4 b^{3} n^{2} x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{24 b^{3} n x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{32 b^{3} x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x**4/4 + 3*a**2*b*log(x) - 3*a*b**2/(4*x**4) - b**3/(8*x**8), Eq(n, -4)), (a**3*x**4/4 + 3*a**
2*b*x**2/2 + 3*a*b**2*log(x) - b**3/(2*x**2), Eq(n, -2)), (a**3*x**4/4 + 9*a**2*b*x**(8/3)/8 + 9*a*b**2*x**(4/
3)/4 + b**3*log(x), Eq(n, -4/3)), (3*a**3*n**3*x**4/(12*n**3 + 88*n**2 + 192*n + 128) + 22*a**3*n**2*x**4/(12*
n**3 + 88*n**2 + 192*n + 128) + 48*a**3*n*x**4/(12*n**3 + 88*n**2 + 192*n + 128) + 32*a**3*x**4/(12*n**3 + 88*
n**2 + 192*n + 128) + 36*a**2*b*n**2*x**4*x**n/(12*n**3 + 88*n**2 + 192*n + 128) + 120*a**2*b*n*x**4*x**n/(12*
n**3 + 88*n**2 + 192*n + 128) + 96*a**2*b*x**4*x**n/(12*n**3 + 88*n**2 + 192*n + 128) + 18*a*b**2*n**2*x**4*x*
*(2*n)/(12*n**3 + 88*n**2 + 192*n + 128) + 96*a*b**2*n*x**4*x**(2*n)/(12*n**3 + 88*n**2 + 192*n + 128) + 96*a*
b**2*x**4*x**(2*n)/(12*n**3 + 88*n**2 + 192*n + 128) + 4*b**3*n**2*x**4*x**(3*n)/(12*n**3 + 88*n**2 + 192*n +
128) + 24*b**3*n*x**4*x**(3*n)/(12*n**3 + 88*n**2 + 192*n + 128) + 32*b**3*x**4*x**(3*n)/(12*n**3 + 88*n**2 +
192*n + 128), True))

________________________________________________________________________________________

Giac [B]  time = 1.19445, size = 254, normalized size = 3.91 \begin{align*} \frac{4 \, b^{3} n^{2} x^{4} x^{3 \, n} + 18 \, a b^{2} n^{2} x^{4} x^{2 \, n} + 36 \, a^{2} b n^{2} x^{4} x^{n} + 3 \, a^{3} n^{3} x^{4} + 24 \, b^{3} n x^{4} x^{3 \, n} + 96 \, a b^{2} n x^{4} x^{2 \, n} + 120 \, a^{2} b n x^{4} x^{n} + 22 \, a^{3} n^{2} x^{4} + 32 \, b^{3} x^{4} x^{3 \, n} + 96 \, a b^{2} x^{4} x^{2 \, n} + 96 \, a^{2} b x^{4} x^{n} + 48 \, a^{3} n x^{4} + 32 \, a^{3} x^{4}}{4 \,{\left (3 \, n^{3} + 22 \, n^{2} + 48 \, n + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*b^3*n^2*x^4*x^(3*n) + 18*a*b^2*n^2*x^4*x^(2*n) + 36*a^2*b*n^2*x^4*x^n + 3*a^3*n^3*x^4 + 24*b^3*n*x^4*x^
(3*n) + 96*a*b^2*n*x^4*x^(2*n) + 120*a^2*b*n*x^4*x^n + 22*a^3*n^2*x^4 + 32*b^3*x^4*x^(3*n) + 96*a*b^2*x^4*x^(2
*n) + 96*a^2*b*x^4*x^n + 48*a^3*n*x^4 + 32*a^3*x^4)/(3*n^3 + 22*n^2 + 48*n + 32)