3.2624 \(\int \frac{x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=94 \[ \frac{4 b^3 \log \left (a+b x^n\right )}{a^5 n}-\frac{4 b^3 \log (x)}{a^5}-\frac{b^3}{a^4 n \left (a+b x^n\right )}-\frac{3 b^2 x^{-n}}{a^4 n}+\frac{b x^{-2 n}}{a^3 n}-\frac{x^{-3 n}}{3 a^2 n} \]

[Out]

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

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Rubi [A]  time = 0.137872, antiderivative size = 94, 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 \[ \frac{4 b^3 \log \left (a+b x^n\right )}{a^5 n}-\frac{4 b^3 \log (x)}{a^5}-\frac{b^3}{a^4 n \left (a+b x^n\right )}-\frac{3 b^2 x^{-n}}{a^4 n}+\frac{b x^{-2 n}}{a^3 n}-\frac{x^{-3 n}}{3 a^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.2003, size = 87, normalized size = 0.93 \[ - \frac{x^{- 3 n}}{3 a^{2} n} + \frac{b x^{- 2 n}}{a^{3} n} - \frac{b^{3}}{a^{4} n \left (a + b x^{n}\right )} - \frac{3 b^{2} x^{- n}}{a^{4} n} - \frac{4 b^{3} \log{\left (x^{n} \right )}}{a^{5} n} + \frac{4 b^{3} \log{\left (a + b x^{n} \right )}}{a^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.137936, size = 77, normalized size = 0.82 \[ \frac{x^{-3 n} \left (-a^3+3 a^2 b x^n+\frac{3 b^4 x^{4 n}}{a+b x^n}-9 a b^2 x^{2 n}\right )+12 b^3 \log \left (a x^{-n}+b\right )}{3 a^5 n} \]

Antiderivative was successfully verified.

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

[Out]

((-a^3 + 3*a^2*b*x^n - 9*a*b^2*x^(2*n) + (3*b^4*x^(4*n))/(a + b*x^n))/x^(3*n) +
12*b^3*Log[b + a/x^n])/(3*a^5*n)

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Maple [A]  time = 0.042, size = 135, normalized size = 1.4 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) } \left ( 4\,{\frac{{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{a}^{5}n}}-{\frac{1}{3\,an}}+{\frac{2\,b{{\rm e}^{n\ln \left ( x \right ) }}}{3\,{a}^{2}n}}-2\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-4\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}}-4\,{\frac{{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{a}^{5}}} \right ) }+4\,{\frac{{b}^{3}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{5}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-3*n)/(a+b*x^n)^2,x)

[Out]

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

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Maxima [A]  time = 1.45409, size = 127, normalized size = 1.35 \[ -\frac{12 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} + a^{3}}{3 \,{\left (a^{4} b n x^{4 \, n} + a^{5} n x^{3 \, n}\right )}} - \frac{4 \, b^{3} \log \left (x\right )}{a^{5}} + \frac{4 \, b^{3} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{5} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23224, size = 157, normalized size = 1.67 \[ -\frac{12 \, b^{4} n x^{4 \, n} \log \left (x\right ) + 6 \, a^{2} b^{2} x^{2 \, n} - 2 \, a^{3} b x^{n} + a^{4} + 12 \,{\left (a b^{3} n \log \left (x\right ) + a b^{3}\right )} x^{3 \, n} - 12 \,{\left (b^{4} x^{4 \, n} + a b^{3} x^{3 \, n}\right )} \log \left (b x^{n} + a\right )}{3 \,{\left (a^{5} b n x^{4 \, n} + a^{6} n x^{3 \, n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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