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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 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])

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]]

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 = 0.04, size = 48, normalized size = 0.76 \[ -\frac {x^{-7 n} \left (20 a^3+70 a^2 b x^n+84 a b^2 x^{2 n}+35 b^3 x^{3 n}\right )}{140 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 48, normalized size = 0.76 \[ -\frac {35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \]

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))

________________________________________________________________________________________

giac [A]  time = 0.21, size = 48, normalized size = 0.76 \[ -\frac {35 \, b^{3} x^{3 \, n} + 84 \, a b^{2} x^{2 \, n} + 70 \, a^{2} b x^{n} + 20 \, a^{3}}{140 \, n x^{7 \, n}} \]

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))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 56, normalized size = 0.89 \[ -\frac {a^{3} x^{-7 n}}{7 n}-\frac {a^{2} b \,x^{-6 n}}{2 n}-\frac {3 a \,b^{2} x^{-5 n}}{5 n}-\frac {b^{3} x^{-4 n}}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 63, normalized size = 1.00 \[ -\frac {a^{3}}{7 \, n x^{7 \, n}} - \frac {a^{2} b}{2 \, n x^{6 \, n}} - \frac {3 \, a b^{2}}{5 \, n x^{5 \, n}} - \frac {b^{3}}{4 \, n x^{4 \, n}} \]

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]

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

________________________________________________________________________________________

mupad [B]  time = 1.37, size = 63, normalized size = 1.00 \[ -\frac {a^3}{7\,n\,x^{7\,n}}-\frac {b^3}{4\,n\,x^{4\,n}}-\frac {3\,a\,b^2}{5\,n\,x^{5\,n}}-\frac {a^2\,b}{2\,n\,x^{6\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^3/x^(7*n + 1),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 27.13, size = 61, normalized size = 0.97 \[ \begin {cases} - \frac {a^{3} x^{- 7 n}}{7 n} - \frac {a^{2} b x^{- 6 n}}{2 n} - \frac {3 a b^{2} x^{- 5 n}}{5 n} - \frac {b^{3} x^{- 4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{3} \log {\relax (x )} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________