3.2550 \(\int x^{-1-7 n} (a+b x^n)^3 \, dx\)

Optimal. Leaf size=63 \[ -\frac{a^2 b x^{-6 n}}{2 n}-\frac{a^3 x^{-7 n}}{7 n}-\frac{3 a b^2 x^{-5 n}}{5 n}-\frac{b^3 x^{-4 n}}{4 n} \]

[Out]

-a^3/(7*n*x^(7*n)) - (a^2*b)/(2*n*x^(6*n)) - (3*a*b^2)/(5*n*x^(5*n)) - b^3/(4*n*x^(4*n))

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Rubi [A]  time = 0.0235806, antiderivative size = 63, 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} \[ -\frac{a^2 b x^{-6 n}}{2 n}-\frac{a^3 x^{-7 n}}{7 n}-\frac{3 a b^2 x^{-5 n}}{5 n}-\frac{b^3 x^{-4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 7*n)*(a + b*x^n)^3,x]

[Out]

-a^3/(7*n*x^(7*n)) - (a^2*b)/(2*n*x^(6*n)) - (3*a*b^2)/(5*n*x^(5*n)) - b^3/(4*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 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^{-1-7 n} \left (a+b x^n\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^8} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^3}{x^8}+\frac{3 a^2 b}{x^7}+\frac{3 a b^2}{x^6}+\frac{b^3}{x^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 x^{-7 n}}{7 n}-\frac{a^2 b x^{-6 n}}{2 n}-\frac{3 a b^2 x^{-5 n}}{5 n}-\frac{b^3 x^{-4 n}}{4 n}\\ \end{align*}

Mathematica [A]  time = 0.0204867, size = 48, normalized size = 0.76 \[ -\frac{x^{-7 n} \left (70 a^2 b x^n+20 a^3+84 a b^2 x^{2 n}+35 b^3 x^{3 n}\right )}{140 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 7*n)*(a + b*x^n)^3,x]

[Out]

-(20*a^3 + 70*a^2*b*x^n + 84*a*b^2*x^(2*n) + 35*b^3*x^(3*n))/(140*n*x^(7*n))

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Maple [A]  time = 0.018, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{3\,{b}^{2}a}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{b{a}^{2}}{2\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{{a}^{3}}{7\,n \left ({x}^{n} \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-7*n)*(a+b*x^n)^3,x)

[Out]

-1/4*b^3/n/(x^n)^4-3/5*b^2*a/n/(x^n)^5-1/2*b*a^2/n/(x^n)^6-1/7*a^3/n/(x^n)^7

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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^(-1-7*n)*(a+b*x^n)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.32103, size = 108, normalized size = 1.71 \begin{align*} -\frac{35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-7*n)*(a+b*x^n)^3,x, algorithm="fricas")

[Out]

-1/140*(35*b^3*x^(3*n) + 84*a*b^2*x^(2*n) + 70*a^2*b*x^n + 20*a^3)/(n*x^(7*n))

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Sympy [A]  time = 31.7118, size = 61, normalized size = 0.97 \begin{align*} \begin{cases} - \frac{a^{3} x^{- 7 n}}{7 n} - \frac{a^{2} b x^{- 6 n}}{2 n} - \frac{3 a b^{2} x^{- 5 n}}{5 n} - \frac{b^{3} x^{- 4 n}}{4 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**(-1-7*n)*(a+b*x**n)**3,x)

[Out]

Piecewise((-a**3*x**(-7*n)/(7*n) - a**2*b*x**(-6*n)/(2*n) - 3*a*b**2*x**(-5*n)/(5*n) - b**3*x**(-4*n)/(4*n), N
e(n, 0)), ((a + b)**3*log(x), True))

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Giac [A]  time = 1.19891, size = 65, normalized size = 1.03 \begin{align*} -\frac{35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-7*n)*(a+b*x^n)^3,x, algorithm="giac")

[Out]

-1/140*(35*b^3*x^(3*n) + 84*a*b^2*x^(2*n) + 70*a^2*b*x^n + 20*a^3)/(n*x^(7*n))