3.2598 \(\int \frac{x^{-1-3 n}}{a+b x^n} \, dx\)

Optimal. Leaf size=76 \[ \frac{b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac{b^3 \log (x)}{a^4}-\frac{b^2 x^{-n}}{a^3 n}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{x^{-3 n}}{3 a n} \]

[Out]

-1/(3*a*n*x^(3*n)) + b/(2*a^2*n*x^(2*n)) - b^2/(a^3*n*x^n) - (b^3*Log[x])/a^4 +
(b^3*Log[a + b*x^n])/(a^4*n)

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Rubi [A]  time = 0.0945419, antiderivative size = 76, 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 \[ \frac{b^3 \log \left (a+b x^n\right )}{a^4 n}-\frac{b^3 \log (x)}{a^4}-\frac{b^2 x^{-n}}{a^3 n}+\frac{b x^{-2 n}}{2 a^2 n}-\frac{x^{-3 n}}{3 a n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 3*n)/(a + b*x^n),x]

[Out]

-1/(3*a*n*x^(3*n)) + b/(2*a^2*n*x^(2*n)) - b^2/(a^3*n*x^n) - (b^3*Log[x])/a^4 +
(b^3*Log[a + b*x^n])/(a^4*n)

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Rubi in Sympy [A]  time = 15.5498, size = 66, normalized size = 0.87 \[ - \frac{x^{- 3 n}}{3 a n} + \frac{b x^{- 2 n}}{2 a^{2} n} - \frac{b^{2} x^{- n}}{a^{3} n} - \frac{b^{3} \log{\left (x^{n} \right )}}{a^{4} n} + \frac{b^{3} \log{\left (a + b x^{n} \right )}}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-3*n)/(a+b*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.0403614, size = 61, normalized size = 0.8 \[ \frac{x^{-3 n} \left (a \left (-2 a^2+3 a b x^n-6 b^2 x^{2 n}\right )+6 b^3 x^{3 n} \log \left (a x^{-n}+b\right )\right )}{6 a^4 n} \]

Antiderivative was successfully verified.

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

[Out]

(a*(-2*a^2 + 3*a*b*x^n - 6*b^2*x^(2*n)) + 6*b^3*x^(3*n)*Log[b + a/x^n])/(6*a^4*n
*x^(3*n))

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Maple [A]  time = 0.038, size = 88, normalized size = 1.2 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}} \left ( -{\frac{1}{3\,an}}+{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,{a}^{2}n}}-{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}} \right ) }+{\frac{{b}^{3}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{4}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3*n)/(a+b*x^n),x)

[Out]

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

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Maxima [A]  time = 1.42239, size = 93, normalized size = 1.22 \[ -\frac{b^{3} \log \left (x\right )}{a^{4}} + \frac{b^{3} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{4} n} - \frac{{\left (6 \, b^{2} x^{2 \, n} - 3 \, a b x^{n} + 2 \, a^{2}\right )} x^{-3 \, n}}{6 \, a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b^3*log(x)/a^4 + b^3*log((b*x^n + a)/b)/(a^4*n) - 1/6*(6*b^2*x^(2*n) - 3*a*b*x^
n + 2*a^2)*x^(-3*n)/(a^3*n)

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Fricas [A]  time = 0.228453, size = 97, normalized size = 1.28 \[ -\frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 6 \, b^{3} x^{3 \, n} \log \left (b x^{n} + a\right ) + 6 \, a b^{2} x^{2 \, n} - 3 \, a^{2} b x^{n} + 2 \, a^{3}}{6 \, a^{4} n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(6*b^3*n*x^(3*n)*log(x) - 6*b^3*x^(3*n)*log(b*x^n + a) + 6*a*b^2*x^(2*n) -
3*a^2*b*x^n + 2*a^3)/(a^4*n*x^(3*n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-3*n)/(a+b*x**n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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