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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 48, 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} \[ -\frac {3 a^2 b x^{-n}}{n}-\frac {a^3 x^{-2 n}}{2 n}+3 a b^2 \log (x)+\frac {b^3 x^n}{n} \]

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 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^{-1-2 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*}

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Mathematica [A]  time = 0.05, size = 44, normalized size = 0.92 \[ \frac {-\frac {1}{2} a^3 x^{-2 n}-3 a^2 b x^{-n}+3 a b^2 n \log (x)+b^3 x^n}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/2*a^3/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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fricas [A]  time = 0.78, size = 51, normalized size = 1.06 \[ \frac {6 \, a b^{2} n x^{2 \, n} \log \relax (x) + 2 \, b^{3} x^{3 \, n} - 6 \, a^{2} b x^{n} - a^{3}}{2 \, n x^{2 \, n}} \]

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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giac [A]  time = 0.21, size = 51, normalized size = 1.06 \[ \frac {6 \, a b^{2} n x^{2 \, n} \log \relax (x) + 2 \, b^{3} x^{3 \, n} - 6 \, a^{2} b x^{n} - a^{3}}{2 \, n x^{2 \, n}} \]

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))

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maple [A]  time = 0.02, size = 61, normalized size = 1.27 \[ \left (3 a \,b^{2} {\mathrm e}^{2 n \ln \relax (x )} \ln \relax (x )-\frac {3 a^{2} b \,{\mathrm e}^{n \ln \relax (x )}}{n}+\frac {b^{3} {\mathrm e}^{3 n \ln \relax (x )}}{n}-\frac {a^{3}}{2 n}\right ) {\mathrm e}^{-2 n \ln \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.65, size = 48, normalized size = 1.00 \[ 3 \, a b^{2} \log \relax (x) + \frac {b^{3} x^{n}}{n} - \frac {a^{3}}{2 \, n x^{2 \, n}} - \frac {3 \, a^{2} b}{n x^{n}} \]

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]

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

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mupad [B]  time = 1.36, size = 48, normalized size = 1.00 \[ \frac {b^3\,x^n}{n}+3\,a\,b^2\,\ln \relax (x)-\frac {a^3}{2\,n\,x^{2\,n}}-\frac {3\,a^2\,b}{n\,x^n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^3/x^(2*n + 1),x)

[Out]

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

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sympy [A]  time = 117.45, size = 318, normalized size = 6.62 \[ \begin {cases} a^{3} x + 6 a^{2} b \sqrt {x} + 3 a b^{2} \log {\relax (x )} - \frac {2 b^{3}}{\sqrt {x}} & \text {for}\: n = - \frac {1}{2} \\\left (a + b\right )^{3} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {2 a^{3} n}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac {a^{3}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac {12 a^{2} b n x^{n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} - \frac {6 a^{2} b x^{n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {12 a b^{2} n^{2} x^{2 n} \log {\relax (x )}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {6 a b^{2} n x^{2 n} \log {\relax (x )}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {6 a b^{2} n x^{2 n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {4 b^{3} n x^{3 n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} + \frac {2 b^{3} x^{3 n}}{4 n^{2} x^{2 n} + 2 n x^{2 n}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x + 6*a**2*b*sqrt(x) + 3*a*b**2*log(x) - 2*b**3/sqrt(x), Eq(n, -1/2)), ((a + b)**3*log(x), Eq(
n, 0)), (-2*a**3*n/(4*n**2*x**(2*n) + 2*n*x**(2*n)) - a**3/(4*n**2*x**(2*n) + 2*n*x**(2*n)) - 12*a**2*b*n*x**n
/(4*n**2*x**(2*n) + 2*n*x**(2*n)) - 6*a**2*b*x**n/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 12*a*b**2*n**2*x**(2*n)*l
og(x)/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 6*a*b**2*n*x**(2*n)*log(x)/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 6*a*b**
2*n*x**(2*n)/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 4*b**3*n*x**(3*n)/(4*n**2*x**(2*n) + 2*n*x**(2*n)) + 2*b**3*x*
*(3*n)/(4*n**2*x**(2*n) + 2*n*x**(2*n)), True))

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