3.2527 \(\int x^{-1-2 n} \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2 x^{-2 n}}{2 n}-\frac{2 a b x^{-n}}{n}+b^2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0477683, antiderivative size = 34, 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 \[ -\frac{a^2 x^{-2 n}}{2 n}-\frac{2 a b x^{-n}}{n}+b^2 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.65974, size = 31, normalized size = 0.91 \[ - \frac{a^{2} x^{- 2 n}}{2 n} - \frac{2 a b x^{- n}}{n} + \frac{b^{2} \log{\left (x^{n} \right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0587054, size = 28, normalized size = 0.82 \[ b^2 \log (x)-\frac{a x^{-2 n} \left (a+4 b x^n\right )}{2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 43, normalized size = 1.3 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ({b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-{\frac{{a}^{2}}{2\,n}}-2\,{\frac{a{{\rm e}^{n\ln \left ( x \right ) }}b}{n}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2*n)*(a+b*x^n)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224981, size = 51, normalized size = 1.5 \[ \frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) - 4 \, a b x^{n} - a^{2}}{2 \, n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219392, size = 54, normalized size = 1.59 \[ \frac{{\left (2 \, b^{2} n e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) - 4 \, a b e^{\left (n{\rm ln}\left (x\right )\right )} - a^{2}\right )} e^{\left (-2 \, n{\rm ln}\left (x\right )\right )}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b^2*n*e^(2*n*ln(x))*ln(x) - 4*a*b*e^(n*ln(x)) - a^2)*e^(-2*n*ln(x))/n