3.2618 \(\int \frac{x^{-1+3 n}}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac{a^2}{b^3 n \left (a+b x^n\right )}-\frac{2 a \log \left (a+b x^n\right )}{b^3 n}+\frac{x^n}{b^2 n} \]

[Out]

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

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Rubi [A]  time = 0.0783434, antiderivative size = 48, 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 \[ -\frac{a^2}{b^3 n \left (a+b x^n\right )}-\frac{2 a \log \left (a+b x^n\right )}{b^3 n}+\frac{x^n}{b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{b^{3} n \left (a + b x^{n}\right )} - \frac{2 a \log{\left (a + b x^{n} \right )}}{b^{3} n} + \frac{\int ^{x^{n}} \frac{1}{b^{2}}\, dx}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0511669, size = 38, normalized size = 0.79 \[ \frac{-\frac{a^2}{a+b x^n}-2 a \log \left (a+b x^n\right )+b x^n}{b^3 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.031, size = 59, normalized size = 1.2 \[{\frac{1}{a+b{{\rm e}^{n\ln \left ( x \right ) }}} \left ({\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{bn}}-2\,{\frac{{a}^{2}}{{b}^{3}n}} \right ) }-2\,{\frac{a\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{3}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3*n)/(a+b*x^n)^2,x)

[Out]

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

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Maxima [A]  time = 1.4509, size = 82, normalized size = 1.71 \[ \frac{b^{2} x^{2 \, n} + a b x^{n} - a^{2}}{b^{4} n x^{n} + a b^{3} n} - \frac{2 \, a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{3} n} \]

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]

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

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Fricas [A]  time = 0.228412, size = 80, normalized size = 1.67 \[ \frac{b^{2} x^{2 \, n} + a b x^{n} - a^{2} - 2 \,{\left (a b x^{n} + a^{2}\right )} \log \left (b x^{n} + a\right )}{b^{4} n x^{n} + a b^{3} n} \]

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]

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