3.2545 \(\int x^{-1-2 n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=48 \[ -\frac{3 a^2 b x^{-n}}{n}-\frac{a^3 x^{-2 n}}{2 n}+3 a b^2 \log (x)+\frac{b^3 x^n}{n} \]

[Out]

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

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Rubi [A]  time = 0.0214762, antiderivative size = 48, 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} \[ -\frac{3 a^2 b x^{-n}}{n}-\frac{a^3 x^{-2 n}}{2 n}+3 a b^2 \log (x)+\frac{b^3 x^n}{n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)*(a + b*x^n)^3,x]

[Out]

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

Mathematica [A]  time = 0.0258998, size = 44, normalized size = 0.92 \[ \frac{-3 a^2 b x^{-n}-\frac{1}{2} a^3 x^{-2 n}+3 a b^2 n \log (x)+b^3 x^n}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 2*n)*(a + b*x^n)^3,x]

[Out]

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

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Maple [A]  time = 0.013, size = 61, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+3\,{b}^{2}a\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-{\frac{{a}^{3}}{2\,n}}-3\,{\frac{b{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-2*n)*(a+b*x^n)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33808, size = 108, normalized size = 2.25 \begin{align*} \frac{6 \, a b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, b^{3} x^{3 \, n} - 6 \, a^{2} b x^{n} - a^{3}}{2 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

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

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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**(-1-2*n)*(a+b*x**n)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.2356, size = 69, normalized size = 1.44 \begin{align*} \frac{6 \, a b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, b^{3} x^{3 \, n} - 6 \, a^{2} b x^{n} - a^{3}}{2 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

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