3.2548 \(\int x^{-1-5 n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=50 \[ \frac{b x^{-4 n} \left (a+b x^n\right )^4}{20 a^2 n}-\frac{x^{-5 n} \left (a+b x^n\right )^4}{5 a n} \]

[Out]

-(a + b*x^n)^4/(5*a*n*x^(5*n)) + (b*(a + b*x^n)^4)/(20*a^2*n*x^(4*n))

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Rubi [A]  time = 0.0162479, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {266, 45, 37} \[ \frac{b x^{-4 n} \left (a+b x^n\right )^4}{20 a^2 n}-\frac{x^{-5 n} \left (a+b x^n\right )^4}{5 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 5*n)*(a + b*x^n)^3,x]

[Out]

-(a + b*x^n)^4/(5*a*n*x^(5*n)) + (b*(a + b*x^n)^4)/(20*a^2*n*x^(4*n))

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int x^{-1-5 n} \left (a+b x^n\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^6} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-5 n} \left (a+b x^n\right )^4}{5 a n}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^5} \, dx,x,x^n\right )}{5 a n}\\ &=-\frac{x^{-5 n} \left (a+b x^n\right )^4}{5 a n}+\frac{b x^{-4 n} \left (a+b x^n\right )^4}{20 a^2 n}\\ \end{align*}

Mathematica [A]  time = 0.0202982, size = 48, normalized size = 0.96 \[ -\frac{x^{-5 n} \left (15 a^2 b x^n+4 a^3+20 a b^2 x^{2 n}+10 b^3 x^{3 n}\right )}{20 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 5*n)*(a + b*x^n)^3,x]

[Out]

-(4*a^3 + 15*a^2*b*x^n + 20*a*b^2*x^(2*n) + 10*b^3*x^(3*n))/(20*n*x^(5*n))

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Maple [A]  time = 0.017, size = 63, normalized size = 1.3 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}} \left ( -{\frac{{a}^{3}}{5\,n}}-{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,n}}-{\frac{3\,b{a}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{4\,n}}-{\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^(-1-5*n)*(a+b*x^n)^3,x)

[Out]

(-1/5*a^3/n-1/2*b^3/n*exp(n*ln(x))^3-3/4*b*a^2/n*exp(n*ln(x))-b^2*a/n*exp(n*ln(x))^2)/exp(n*ln(x))^5

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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^(-1-5*n)*(a+b*x^n)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.38117, size = 105, normalized size = 2.1 \begin{align*} -\frac{10 \, b^{3} x^{3 \, n} + 20 \, a b^{2} x^{2 \, n} + 15 \, a^{2} b x^{n} + 4 \, a^{3}}{20 \, n x^{5 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-5*n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

-1/20*(10*b^3*x^(3*n) + 20*a*b^2*x^(2*n) + 15*a^2*b*x^n + 4*a^3)/(n*x^(5*n))

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Sympy [A]  time = 27.4295, size = 60, normalized size = 1.2 \begin{align*} \begin{cases} - \frac{a^{3} x^{- 5 n}}{5 n} - \frac{3 a^{2} b x^{- 4 n}}{4 n} - \frac{a b^{2} x^{- 3 n}}{n} - \frac{b^{3} x^{- 2 n}}{2 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**(-1-5*n)*(a+b*x**n)**3,x)

[Out]

Piecewise((-a**3*x**(-5*n)/(5*n) - 3*a**2*b*x**(-4*n)/(4*n) - a*b**2*x**(-3*n)/n - b**3*x**(-2*n)/(2*n), Ne(n,
0)), ((a + b)**3*log(x), True))

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Giac [A]  time = 1.21494, size = 65, normalized size = 1.3 \begin{align*} -\frac{10 \, b^{3} x^{3 \, n} + 20 \, a b^{2} x^{2 \, n} + 15 \, a^{2} b x^{n} + 4 \, a^{3}}{20 \, n x^{5 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-5*n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

-1/20*(10*b^3*x^(3*n) + 20*a*b^2*x^(2*n) + 15*a^2*b*x^n + 4*a^3)/(n*x^(5*n))