3.2631 \(\int \frac{x^{-1-2 n}}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=101 \[ -\frac{6 b^2 \log \left (a+b x^n\right )}{a^5 n}+\frac{6 b^2 \log (x)}{a^5}+\frac{3 b^2}{a^4 n \left (a+b x^n\right )}+\frac{3 b x^{-n}}{a^4 n}+\frac{b^2}{2 a^3 n \left (a+b x^n\right )^2}-\frac{x^{-2 n}}{2 a^3 n} \]

[Out]

-1/(2*a^3*n*x^(2*n)) + (3*b)/(a^4*n*x^n) + b^2/(2*a^3*n*(a + b*x^n)^2) + (3*b^2)
/(a^4*n*(a + b*x^n)) + (6*b^2*Log[x])/a^5 - (6*b^2*Log[a + b*x^n])/(a^5*n)

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Rubi [A]  time = 0.141831, antiderivative size = 101, 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 \[ -\frac{6 b^2 \log \left (a+b x^n\right )}{a^5 n}+\frac{6 b^2 \log (x)}{a^5}+\frac{3 b^2}{a^4 n \left (a+b x^n\right )}+\frac{3 b x^{-n}}{a^4 n}+\frac{b^2}{2 a^3 n \left (a+b x^n\right )^2}-\frac{x^{-2 n}}{2 a^3 n} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a^3*n*x^(2*n)) + (3*b)/(a^4*n*x^n) + b^2/(2*a^3*n*(a + b*x^n)^2) + (3*b^2)
/(a^4*n*(a + b*x^n)) + (6*b^2*Log[x])/a^5 - (6*b^2*Log[a + b*x^n])/(a^5*n)

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Rubi in Sympy [A]  time = 22.6642, size = 94, normalized size = 0.93 \[ \frac{b^{2}}{2 a^{3} n \left (a + b x^{n}\right )^{2}} - \frac{x^{- 2 n}}{2 a^{3} n} + \frac{3 b^{2}}{a^{4} n \left (a + b x^{n}\right )} + \frac{3 b x^{- n}}{a^{4} n} + \frac{6 b^{2} \log{\left (x^{n} \right )}}{a^{5} n} - \frac{6 b^{2} \log{\left (a + b x^{n} \right )}}{a^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-2*n)/(a+b*x**n)**3,x)

[Out]

b**2/(2*a**3*n*(a + b*x**n)**2) - x**(-2*n)/(2*a**3*n) + 3*b**2/(a**4*n*(a + b*x
**n)) + 3*b*x**(-n)/(a**4*n) + 6*b**2*log(x**n)/(a**5*n) - 6*b**2*log(a + b*x**n
)/(a**5*n)

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Mathematica [A]  time = 0.0822267, size = 97, normalized size = 0.96 \[ \frac{b^4}{2 a^5 n \left (a x^{-n}+b\right )^2}-\frac{4 b^3}{a^5 n \left (a x^{-n}+b\right )}-\frac{6 b^2 \log \left (a x^{-n}+b\right )}{a^5 n}+\frac{3 b x^{-n}}{a^4 n}-\frac{x^{-2 n}}{2 a^3 n} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a^3*n*x^(2*n)) + (3*b)/(a^4*n*x^n) + b^4/(2*a^5*n*(b + a/x^n)^2) - (4*b^3)
/(a^5*n*(b + a/x^n)) - (6*b^2*Log[b + a/x^n])/(a^5*n)

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Maple [A]  time = 0.058, size = 152, normalized size = 1.5 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( 9\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-{\frac{1}{2\,an}}+6\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}}+2\,{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}n}}+12\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}}+6\,{\frac{{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{a}^{5}}}+6\,{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}n}} \right ) }-6\,{\frac{{b}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{5}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2*n)/(a+b*x^n)^3,x)

[Out]

(9*b^2/a^3/n*exp(n*ln(x))^2-1/2/a/n+6*b^2/a^3*ln(x)*exp(n*ln(x))^2+2*b/a^2/n*exp
(n*ln(x))+12*b^3/a^4*ln(x)*exp(n*ln(x))^3+6*b^4/a^5*ln(x)*exp(n*ln(x))^4+6*b^3/a
^4/n*exp(n*ln(x))^3)/exp(n*ln(x))^2/(a+b*exp(n*ln(x)))^2-6*b^2/a^5/n*ln(a+b*exp(
n*ln(x)))

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Maxima [A]  time = 1.43712, size = 149, normalized size = 1.48 \[ \frac{12 \, b^{3} x^{3 \, n} + 18 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} - a^{3}}{2 \,{\left (a^{4} b^{2} n x^{4 \, n} + 2 \, a^{5} b n x^{3 \, n} + a^{6} n x^{2 \, n}\right )}} + \frac{6 \, b^{2} \log \left (x\right )}{a^{5}} - \frac{6 \, b^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n - 1)/(b*x^n + a)^3,x, algorithm="maxima")

[Out]

1/2*(12*b^3*x^(3*n) + 18*a*b^2*x^(2*n) + 4*a^2*b*x^n - a^3)/(a^4*b^2*n*x^(4*n) +
2*a^5*b*n*x^(3*n) + a^6*n*x^(2*n)) + 6*b^2*log(x)/a^5 - 6*b^2*log((b*x^n + a)/b
)/(a^5*n)

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Fricas [A]  time = 0.232328, size = 216, normalized size = 2.14 \[ \frac{12 \, b^{4} n x^{4 \, n} \log \left (x\right ) + 4 \, a^{3} b x^{n} - a^{4} + 12 \,{\left (2 \, a b^{3} n \log \left (x\right ) + a b^{3}\right )} x^{3 \, n} + 6 \,{\left (2 \, a^{2} b^{2} n \log \left (x\right ) + 3 \, a^{2} b^{2}\right )} x^{2 \, n} - 12 \,{\left (b^{4} x^{4 \, n} + 2 \, a b^{3} x^{3 \, n} + a^{2} b^{2} x^{2 \, n}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{5} b^{2} n x^{4 \, n} + 2 \, a^{6} b n x^{3 \, n} + a^{7} n x^{2 \, n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n - 1)/(b*x^n + a)^3,x, algorithm="fricas")

[Out]

1/2*(12*b^4*n*x^(4*n)*log(x) + 4*a^3*b*x^n - a^4 + 12*(2*a*b^3*n*log(x) + a*b^3)
*x^(3*n) + 6*(2*a^2*b^2*n*log(x) + 3*a^2*b^2)*x^(2*n) - 12*(b^4*x^(4*n) + 2*a*b^
3*x^(3*n) + a^2*b^2*x^(2*n))*log(b*x^n + a))/(a^5*b^2*n*x^(4*n) + 2*a^6*b*n*x^(3
*n) + a^7*n*x^(2*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**(-1-2*n)/(a+b*x**n)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n - 1)/(b*x^n + a)^3,x, algorithm="giac")

[Out]

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