3.2544 \(\int x^{-1-n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=49 \[ 3 a^2 b \log (x)-\frac{a^3 x^{-n}}{n}+\frac{3 a b^2 x^n}{n}+\frac{b^3 x^{2 n}}{2 n} \]

[Out]

-(a^3/(n*x^n)) + (3*a*b^2*x^n)/n + (b^3*x^(2*n))/(2*n) + 3*a^2*b*Log[x]

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Rubi [A]  time = 0.0222159, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ 3 a^2 b \log (x)-\frac{a^3 x^{-n}}{n}+\frac{3 a b^2 x^n}{n}+\frac{b^3 x^{2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*(a + b*x^n)^3,x]

[Out]

-(a^3/(n*x^n)) + (3*a*b^2*x^n)/n + (b^3*x^(2*n))/(2*n) + 3*a^2*b*Log[x]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0250209, size = 45, normalized size = 0.92 \[ \frac{3 a^2 b n \log (x)-a^3 x^{-n}+3 a b^2 x^n+\frac{1}{2} b^3 x^{2 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)*(a + b*x^n)^3,x]

[Out]

(-(a^3/x^n) + 3*a*b^2*x^n + (b^3*x^(2*n))/2 + 3*a^2*b*n*Log[x])/n

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Maple [A]  time = 0.013, size = 62, normalized size = 1.3 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 3\,b{a}^{2}\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}-{\frac{{a}^{3}}{n}}+{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,n}}+3\,{\frac{{b}^{2}a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-n)*(a+b*x^n)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27646, size = 103, normalized size = 2.1 \begin{align*} \frac{6 \, a^{2} b n x^{n} \log \left (x\right ) + b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{3}}{2 \, n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 79.7773, size = 277, normalized size = 5.65 \begin{align*} \begin{cases} a^{3} x + 3 a^{2} b \log{\left (x \right )} - \frac{3 a b^{2}}{x} - \frac{b^{3}}{2 x^{2}} & \text{for}\: n = -1 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{for}\: n = 0 \\- \frac{2 a^{3} n}{2 n^{2} x^{n} + 2 n x^{n}} - \frac{2 a^{3}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a^{2} b n^{2} x^{n} \log{\left (x \right )}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a^{2} b n x^{n} \log{\left (x \right )}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a^{2} b n x^{n}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a b^{2} n x^{2 n}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{6 a b^{2} x^{2 n}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{b^{3} n x^{3 n}}{2 n^{2} x^{n} + 2 n x^{n}} + \frac{b^{3} x^{3 n}}{2 n^{2} x^{n} + 2 n x^{n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-n)*(a+b*x**n)**3,x)

[Out]

Piecewise((a**3*x + 3*a**2*b*log(x) - 3*a*b**2/x - b**3/(2*x**2), Eq(n, -1)), ((a + b)**3*log(x), Eq(n, 0)), (
-2*a**3*n/(2*n**2*x**n + 2*n*x**n) - 2*a**3/(2*n**2*x**n + 2*n*x**n) + 6*a**2*b*n**2*x**n*log(x)/(2*n**2*x**n
+ 2*n*x**n) + 6*a**2*b*n*x**n*log(x)/(2*n**2*x**n + 2*n*x**n) + 6*a**2*b*n*x**n/(2*n**2*x**n + 2*n*x**n) + 6*a
*b**2*n*x**(2*n)/(2*n**2*x**n + 2*n*x**n) + 6*a*b**2*x**(2*n)/(2*n**2*x**n + 2*n*x**n) + b**3*n*x**(3*n)/(2*n*
*2*x**n + 2*n*x**n) + b**3*x**(3*n)/(2*n**2*x**n + 2*n*x**n), True))

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Giac [A]  time = 1.20743, size = 65, normalized size = 1.33 \begin{align*} \frac{6 \, a^{2} b n x^{n} \log \left (x\right ) + b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{3}}{2 \, n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

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