3.2617 \(\int \frac{x^{-1+4 n}}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{a^3}{b^4 n \left (a+b x^n\right )}+\frac{3 a^2 \log \left (a+b x^n\right )}{b^4 n}-\frac{2 a x^n}{b^3 n}+\frac{x^{2 n}}{2 b^2 n} \]

[Out]

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

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Rubi [A]  time = 0.10375, antiderivative size = 66, 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 \[ \frac{a^3}{b^4 n \left (a+b x^n\right )}+\frac{3 a^2 \log \left (a+b x^n\right )}{b^4 n}-\frac{2 a x^n}{b^3 n}+\frac{x^{2 n}}{2 b^2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 4*n)/(a + b*x^n)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3}}{b^{4} n \left (a + b x^{n}\right )} + \frac{3 a^{2} \log{\left (a + b x^{n} \right )}}{b^{4} n} - \frac{2 a x^{n}}{b^{3} n} + \frac{\int ^{x^{n}} x\, dx}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+4*n)/(a+b*x**n)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0585582, size = 54, normalized size = 0.82 \[ \frac{\frac{2 a^3}{a+b x^n}+6 a^2 \log \left (a+b x^n\right )-4 a b x^n+b^2 x^{2 n}}{2 b^4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 4*n)/(a + b*x^n)^2,x]

[Out]

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

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Maple [A]  time = 0.035, size = 78, normalized size = 1.2 \[{\frac{1}{a+b{{\rm e}^{n\ln \left ( x \right ) }}} \left ( 3\,{\frac{{a}^{3}}{{b}^{4}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,bn}}-{\frac{3\,a \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,{b}^{2}n}} \right ) }+3\,{\frac{{a}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{4}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+4*n)/(a+b*x^n)^2,x)

[Out]

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

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Maxima [A]  time = 1.4517, size = 105, normalized size = 1.59 \[ \frac{b^{3} x^{3 \, n} - 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 2 \, a^{3}}{2 \,{\left (b^{5} n x^{n} + a b^{4} n\right )}} + \frac{3 \, a^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{b^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.228132, size = 103, normalized size = 1.56 \[ \frac{b^{3} x^{3 \, n} - 3 \, a b^{2} x^{2 \, n} - 4 \, a^{2} b x^{n} + 2 \, a^{3} + 6 \,{\left (a^{2} b x^{n} + a^{3}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (b^{5} n x^{n} + a b^{4} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + a)^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(4*n - 1)/(b*x^n + a)^2,x, algorithm="giac")

[Out]

integrate(x^(4*n - 1)/(b*x^n + a)^2, x)