3.2636 \(\int \frac{1}{x (a+b x^n)^3} \, dx\)

Optimal. Leaf size=58 \[ \frac{1}{a^2 n \left (a+b x^n\right )}-\frac{\log \left (a+b x^n\right )}{a^3 n}+\frac{\log (x)}{a^3}+\frac{1}{2 a n \left (a+b x^n\right )^2} \]

[Out]

1/(2*a*n*(a + b*x^n)^2) + 1/(a^2*n*(a + b*x^n)) + Log[x]/a^3 - Log[a + b*x^n]/(a^3*n)

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Rubi [A]  time = 0.0309098, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac{1}{a^2 n \left (a+b x^n\right )}-\frac{\log \left (a+b x^n\right )}{a^3 n}+\frac{\log (x)}{a^3}+\frac{1}{2 a n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^n)^3),x]

[Out]

1/(2*a*n*(a + b*x^n)^2) + 1/(a^2*n*(a + b*x^n)) + Log[x]/a^3 - Log[a + b*x^n]/(a^3*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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0615218, size = 47, normalized size = 0.81 \[ \frac{\frac{a \left (3 a+2 b x^n\right )}{\left (a+b x^n\right )^2}-2 \log \left (a+b x^n\right )+2 n \log (x)}{2 a^3 n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^n)^3),x]

[Out]

((a*(3*a + 2*b*x^n))/(a + b*x^n)^2 + 2*n*Log[x] - 2*Log[a + b*x^n])/(2*a^3*n)

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Maple [A]  time = 0., size = 62, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({x}^{n} \right ) }{n{a}^{3}}}-{\frac{\ln \left ( a+b{x}^{n} \right ) }{n{a}^{3}}}+{\frac{1}{{a}^{2}n \left ( a+b{x}^{n} \right ) }}+{\frac{1}{2\,an \left ( a+b{x}^{n} \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.970463, size = 96, normalized size = 1.66 \begin{align*} \frac{2 \, b x^{n} + 3 \, a}{2 \,{\left (a^{2} b^{2} n x^{2 \, n} + 2 \, a^{3} b n x^{n} + a^{4} n\right )}} - \frac{\log \left (b x^{n} + a\right )}{a^{3} n} + \frac{\log \left (x^{n}\right )}{a^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.999372, size = 244, normalized size = 4.21 \begin{align*} \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, a^{2} n \log \left (x\right ) + 3 \, a^{2} + 2 \,{\left (2 \, a b n \log \left (x\right ) + a b\right )} x^{n} - 2 \,{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.90427, size = 411, normalized size = 7.09 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac{x^{- 3 n}}{3 b^{3} n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{3}} & \text{for}\: n = 0 \\\frac{\log{\left (x \right )}}{a^{3}} & \text{for}\: b = 0 \\\frac{2 a^{2} n \log{\left (x \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac{2 a^{2} \log{\left (\frac{a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac{4 a b n x^{n} \log{\left (x \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac{4 a b x^{n} \log{\left (\frac{a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac{4 a b x^{n}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac{2 b^{2} n x^{2 n} \log{\left (x \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac{2 b^{2} x^{2 n} \log{\left (\frac{a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac{3 b^{2} x^{2 n}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^n + a)^3*x), x)