3.2546 \(\int x^{-1-3 n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=52 \[ -\frac{3 a^2 b x^{-2 n}}{2 n}-\frac{a^3 x^{-3 n}}{3 n}-\frac{3 a b^2 x^{-n}}{n}+b^3 \log (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0217421, antiderivative size = 52, 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} \[ -\frac{3 a^2 b x^{-2 n}}{2 n}-\frac{a^3 x^{-3 n}}{3 n}-\frac{3 a b^2 x^{-n}}{n}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)*(a + b*x^n)^3,x]

[Out]

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

Mathematica [A]  time = 0.0398875, size = 43, normalized size = 0.83 \[ b^3 \log (x)-\frac{a x^{-3 n} \left (2 a^2+9 a b x^n+18 b^2 x^{2 n}\right )}{6 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 3*n)*(a + b*x^n)^3,x]

[Out]

-(a*(2*a^2 + 9*a*b*x^n + 18*b^2*x^(2*n)))/(6*n*x^(3*n)) + b^3*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 61, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ({b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}-{\frac{{a}^{3}}{3\,n}}-{\frac{3\,b{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{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^(-1-3*n)*(a+b*x^n)^3,x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.48915, size = 112, normalized size = 2.15 \begin{align*} \frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 18 \, a b^{2} x^{2 \, n} - 9 \, a^{2} b x^{n} - 2 \, a^{3}}{6 \, n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*b^3*n*x^(3*n)*log(x) - 18*a*b^2*x^(2*n) - 9*a^2*b*x^n - 2*a^3)/(n*x^(3*n))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.21462, size = 69, normalized size = 1.33 \begin{align*} \frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 18 \, a b^{2} x^{2 \, n} - 9 \, a^{2} b x^{n} - 2 \, a^{3}}{6 \, n x^{3 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*b^3*n*x^(3*n)*log(x) - 18*a*b^2*x^(2*n) - 9*a^2*b*x^n - 2*a^3)/(n*x^(3*n))