3.2477 \(\int \frac{1}{x \left (a+b x^n\right )^3} \, dx\)

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

[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 = 0.0841658, 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 \[ -\frac{\log \left (a+b x^n\right )}{a^3 n}+\frac{\log (x)}{a^3}+\frac{1}{a^2 n \left (a+b x^n\right )}+\frac{1}{2 a n \left (a+b x^n\right )^2} \]

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)

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Rubi in Sympy [A]  time = 12.9604, size = 51, normalized size = 0.88 \[ \frac{1}{2 a n \left (a + b x^{n}\right )^{2}} + \frac{1}{a^{2} n \left (a + b x^{n}\right )} + \frac{\log{\left (x^{n} \right )}}{a^{3} n} - \frac{\log{\left (a + b x^{n} \right )}}{a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(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**n)/(a**3*n) - log(a
+ b*x**n)/(a**3*n)

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Mathematica [A]  time = 0.171271, size = 48, normalized size = 0.83 \[ \frac{\frac{\frac{a \left (3 a+2 b x^n\right )}{\left (a+b x^n\right )^2}-2 \log \left (a+b x^n\right )}{n}+2 \log (x)}{2 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^n)^3),x]

[Out]

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

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Maple [A]  time = 0.004, size = 62, normalized size = 1.1 \[{\frac{\ln \left ({x}^{n} \right ) }{n{a}^{3}}}-{\frac{\ln \left ( a+b{x}^{n} \right ) }{n{a}^{3}}}+{\frac{1}{{a}^{2}n \left ( a+b{x}^{n} \right ) }}+{\frac{1}{2\,an \left ( a+b{x}^{n} \right ) ^{2}}} \]

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 = 1.44153, size = 96, normalized size = 1.66 \[ \frac{2 \, b x^{n} + 3 \, a}{2 \,{\left (a^{2} b^{2} n x^{2 \, n} + 2 \, a^{3} b n x^{n} + a^{4} n\right )}} - \frac{\log \left (b x^{n} + a\right )}{a^{3} n} + \frac{\log \left (x^{n}\right )}{a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^3*x),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 = 0.227049, size = 143, normalized size = 2.47 \[ \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, a^{2} n \log \left (x\right ) + 3 \, a^{2} + 2 \,{\left (2 \, a b n \log \left (x\right ) + a b\right )} x^{n} - 2 \,{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{3} b^{2} n x^{2 \, n} + 2 \, a^{4} b n x^{n} + a^{5} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^3*x),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 = 5.56056, size = 406, normalized size = 7. \[ \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a^{3}} & \text{for}\: b = 0 \\- \frac{x^{- 3 n}}{3 b^{3} n} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{3}} & \text{for}\: n = 0 \\\frac{2 a^{2} n \log{\left (x \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac{2 a^{2} \log{\left (\frac{a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac{3 a^{2}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac{4 a b n x^{n} \log{\left (x \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac{4 a b x^{n} \log{\left (\frac{a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac{2 a b x^{n}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} + \frac{2 b^{2} n x^{2 n} \log{\left (x \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} - \frac{2 b^{2} x^{2 n} \log{\left (\frac{a}{b} + x^{n} \right )}}{2 a^{5} n + 4 a^{4} b n x^{n} + 2 a^{3} b^{2} n x^{2 n}} & \text{otherwise} \end{cases} \]

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)), (log(x)/a**3, Eq(b, 0)),
(-x**(-3*n)/(3*b**3*n), Eq(a, 0)), (log(x)/(a + b)**3, Eq(n, 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)) + 3*a**2/(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)) + 2*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)), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{3} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^n + a)^3*x), x)