3.2547 \(\int x^{-1-4 n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=24 \[ -\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n} \]

[Out]

-(a + b*x^n)^4/(4*a*n*x^(4*n))

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Rubi [A]  time = 0.0049612, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 4*n)*(a + b*x^n)^3,x]

[Out]

-(a + b*x^n)^4/(4*a*n*x^(4*n))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0066993, size = 24, normalized size = 1. \[ -\frac{x^{-4 n} \left (a+b x^n\right )^4}{4 a n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 4*n)*(a + b*x^n)^3,x]

[Out]

-(a + b*x^n)^4/(4*a*n*x^(4*n))

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Maple [B]  time = 0.014, size = 63, normalized size = 2.6 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}} \left ( -{\frac{{a}^{3}}{4\,n}}-{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-{\frac{b{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{n}}-{\frac{3\,{b}^{2}a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-4*n)*(a+b*x^n)^3,x)

[Out]

(-1/4*a^3/n-b^3/n*exp(n*ln(x))^3-b*a^2/n*exp(n*ln(x))-3/2*b^2*a/n*exp(n*ln(x))^2)/exp(n*ln(x))^4

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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^(-1-4*n)*(a+b*x^n)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.33464, size = 97, normalized size = 4.04 \begin{align*} -\frac{4 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} + a^{3}}{4 \, n x^{4 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

-1/4*(4*b^3*x^(3*n) + 6*a*b^2*x^(2*n) + 4*a^2*b*x^n + a^3)/(n*x^(4*n))

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Sympy [A]  time = 29.6429, size = 56, normalized size = 2.33 \begin{align*} \begin{cases} - \frac{a^{3} x^{- 4 n}}{4 n} - \frac{a^{2} b x^{- 3 n}}{n} - \frac{3 a b^{2} x^{- 2 n}}{2 n} - \frac{b^{3} x^{- n}}{n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{3} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-4*n)*(a+b*x**n)**3,x)

[Out]

Piecewise((-a**3*x**(-4*n)/(4*n) - a**2*b*x**(-3*n)/n - 3*a*b**2*x**(-2*n)/(2*n) - b**3*x**(-n)/n, Ne(n, 0)),
((a + b)**3*log(x), True))

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Giac [A]  time = 1.14749, size = 62, normalized size = 2.58 \begin{align*} -\frac{4 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} + a^{3}}{4 \, n x^{4 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

-1/4*(4*b^3*x^(3*n) + 6*a*b^2*x^(2*n) + 4*a^2*b*x^n + a^3)/(n*x^(4*n))