3.2467 \(\int x (a+b x^n)^3 \, dx\)

Optimal. Leaf size=65 \[ \frac{3 a^2 b x^{n+2}}{n+2}+\frac{a^3 x^2}{2}+\frac{3 a b^2 x^{2 (n+1)}}{2 (n+1)}+\frac{b^3 x^{3 n+2}}{3 n+2} \]

[Out]

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

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Rubi [A]  time = 0.0256181, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {270} \[ \frac{3 a^2 b x^{n+2}}{n+2}+\frac{a^3 x^2}{2}+\frac{3 a b^2 x^{2 (n+1)}}{2 (n+1)}+\frac{b^3 x^{3 n+2}}{3 n+2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^n)^3,x]

[Out]

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

Mathematica [A]  time = 0.0355971, size = 58, normalized size = 0.89 \[ \frac{1}{2} x^2 \left (\frac{6 a^2 b x^n}{n+2}+a^3+\frac{3 a b^2 x^{2 n}}{n+1}+\frac{2 b^3 x^{3 n}}{3 n+2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^n)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 71, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}{x}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2+3\,n}}+{\frac{{x}^{2}{a}^{3}}{2}}+3\,{\frac{{a}^{2}b{x}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{2+n}}+{\frac{3\,a{b}^{2}{x}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2+2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.07138, size = 306, normalized size = 4.71 \begin{align*} \frac{2 \,{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{2} x^{3 \, n} + 3 \,{\left (3 \, a b^{2} n^{2} + 8 \, a b^{2} n + 4 \, a b^{2}\right )} x^{2} x^{2 \, n} + 6 \,{\left (3 \, a^{2} b n^{2} + 5 \, a^{2} b n + 2 \, a^{2} b\right )} x^{2} x^{n} +{\left (3 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 12 \, a^{3} n + 4 \, a^{3}\right )} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*(b^3*n^2 + 3*b^3*n + 2*b^3)*x^2*x^(3*n) + 3*(3*a*b^2*n^2 + 8*a*b^2*n + 4*a*b^2)*x^2*x^(2*n) + 6*(3*a^2*
b*n^2 + 5*a^2*b*n + 2*a^2*b)*x^2*x^n + (3*a^3*n^3 + 11*a^3*n^2 + 12*a^3*n + 4*a^3)*x^2)/(3*n^3 + 11*n^2 + 12*n
+ 4)

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Sympy [A]  time = 2.17456, size = 500, normalized size = 7.69 \begin{align*} \begin{cases} \frac{a^{3} x^{2}}{2} + 3 a^{2} b \log{\left (x \right )} - \frac{3 a b^{2}}{2 x^{2}} - \frac{b^{3}}{4 x^{4}} & \text{for}\: n = -2 \\\frac{a^{3} x^{2}}{2} + 3 a^{2} b x + 3 a b^{2} \log{\left (x \right )} - \frac{b^{3}}{x} & \text{for}\: n = -1 \\\frac{a^{3} x^{2}}{2} + \frac{9 a^{2} b x^{\frac{4}{3}}}{4} + \frac{9 a b^{2} x^{\frac{2}{3}}}{2} + b^{3} \log{\left (x \right )} & \text{for}\: n = - \frac{2}{3} \\\frac{3 a^{3} n^{3} x^{2}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{11 a^{3} n^{2} x^{2}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{12 a^{3} n x^{2}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{4 a^{3} x^{2}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{18 a^{2} b n^{2} x^{2} x^{n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{30 a^{2} b n x^{2} x^{n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{12 a^{2} b x^{2} x^{n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{9 a b^{2} n^{2} x^{2} x^{2 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{24 a b^{2} n x^{2} x^{2 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{12 a b^{2} x^{2} x^{2 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{2 b^{3} n^{2} x^{2} x^{3 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{6 b^{3} n x^{2} x^{3 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} + \frac{4 b^{3} x^{2} x^{3 n}}{6 n^{3} + 22 n^{2} + 24 n + 8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x**2/2 + 3*a**2*b*log(x) - 3*a*b**2/(2*x**2) - b**3/(4*x**4), Eq(n, -2)), (a**3*x**2/2 + 3*a**
2*b*x + 3*a*b**2*log(x) - b**3/x, Eq(n, -1)), (a**3*x**2/2 + 9*a**2*b*x**(4/3)/4 + 9*a*b**2*x**(2/3)/2 + b**3*
log(x), Eq(n, -2/3)), (3*a**3*n**3*x**2/(6*n**3 + 22*n**2 + 24*n + 8) + 11*a**3*n**2*x**2/(6*n**3 + 22*n**2 +
24*n + 8) + 12*a**3*n*x**2/(6*n**3 + 22*n**2 + 24*n + 8) + 4*a**3*x**2/(6*n**3 + 22*n**2 + 24*n + 8) + 18*a**2
*b*n**2*x**2*x**n/(6*n**3 + 22*n**2 + 24*n + 8) + 30*a**2*b*n*x**2*x**n/(6*n**3 + 22*n**2 + 24*n + 8) + 12*a**
2*b*x**2*x**n/(6*n**3 + 22*n**2 + 24*n + 8) + 9*a*b**2*n**2*x**2*x**(2*n)/(6*n**3 + 22*n**2 + 24*n + 8) + 24*a
*b**2*n*x**2*x**(2*n)/(6*n**3 + 22*n**2 + 24*n + 8) + 12*a*b**2*x**2*x**(2*n)/(6*n**3 + 22*n**2 + 24*n + 8) +
2*b**3*n**2*x**2*x**(3*n)/(6*n**3 + 22*n**2 + 24*n + 8) + 6*b**3*n*x**2*x**(3*n)/(6*n**3 + 22*n**2 + 24*n + 8)
+ 4*b**3*x**2*x**(3*n)/(6*n**3 + 22*n**2 + 24*n + 8), True))

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Giac [B]  time = 1.20892, size = 254, normalized size = 3.91 \begin{align*} \frac{2 \, b^{3} n^{2} x^{2} x^{3 \, n} + 9 \, a b^{2} n^{2} x^{2} x^{2 \, n} + 18 \, a^{2} b n^{2} x^{2} x^{n} + 3 \, a^{3} n^{3} x^{2} + 6 \, b^{3} n x^{2} x^{3 \, n} + 24 \, a b^{2} n x^{2} x^{2 \, n} + 30 \, a^{2} b n x^{2} x^{n} + 11 \, a^{3} n^{2} x^{2} + 4 \, b^{3} x^{2} x^{3 \, n} + 12 \, a b^{2} x^{2} x^{2 \, n} + 12 \, a^{2} b x^{2} x^{n} + 12 \, a^{3} n x^{2} + 4 \, a^{3} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b^3*n^2*x^2*x^(3*n) + 9*a*b^2*n^2*x^2*x^(2*n) + 18*a^2*b*n^2*x^2*x^n + 3*a^3*n^3*x^2 + 6*b^3*n*x^2*x^(3
*n) + 24*a*b^2*n*x^2*x^(2*n) + 30*a^2*b*n*x^2*x^n + 11*a^3*n^2*x^2 + 4*b^3*x^2*x^(3*n) + 12*a*b^2*x^2*x^(2*n)
+ 12*a^2*b*x^2*x^n + 12*a^3*n*x^2 + 4*a^3*x^2)/(3*n^3 + 11*n^2 + 12*n + 4)