3.502 \(\int \frac{x^{-1+\frac{n}{2}}}{b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=50 \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{3/2} n}-\frac{2 x^{-n/2}}{b n} \]

[Out]

-2/(b*n*x^(n/2)) + (2*Sqrt[c]*ArcTan[Sqrt[b]/(Sqrt[c]*x^(n/2))])/(b^(3/2)*n)

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Rubi [A]  time = 0.0684434, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{3/2} n}-\frac{2 x^{-n/2}}{b n} \]

Antiderivative was successfully verified.

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

[Out]

-2/(b*n*x^(n/2)) + (2*Sqrt[c]*ArcTan[Sqrt[b]/(Sqrt[c]*x^(n/2))])/(b^(3/2)*n)

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Rubi in Sympy [A]  time = 13.4537, size = 39, normalized size = 0.78 \[ - \frac{2 x^{- \frac{n}{2}}}{b n} + \frac{2 \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{b} x^{- \frac{n}{2}}}{\sqrt{c}} \right )}}{b^{\frac{3}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**(-n/2)/(b*n) + 2*sqrt(c)*atan(sqrt(b)*x**(-n/2)/sqrt(c))/(b**(3/2)*n)

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Mathematica [A]  time = 0.0406458, size = 50, normalized size = 1. \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{3/2} n}-\frac{2 x^{-n/2}}{b n} \]

Antiderivative was successfully verified.

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

[Out]

-2/(b*n*x^(n/2)) + (2*Sqrt[c]*ArcTan[Sqrt[b]/(Sqrt[c]*x^(n/2))])/(b^(3/2)*n)

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Maple [A]  time = 0.093, size = 79, normalized size = 1.6 \[ -2\,{\frac{1}{bn{x}^{n/2}}}+{\frac{1}{{b}^{2}n}\sqrt{-bc}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{c}\sqrt{-bc}} \right ) }-{\frac{1}{{b}^{2}n}\sqrt{-bc}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{c}\sqrt{-bc}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+1/2*n)/(b*x^n+c*x^(2*n)),x)

[Out]

-2/b/n/(x^(1/2*n))+1/b^2*(-b*c)^(1/2)/n*ln(x^(1/2*n)-1/c*(-b*c)^(1/2))-1/b^2*(-b
*c)^(1/2)/n*ln(x^(1/2*n)+1/c*(-b*c)^(1/2))

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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(x^(1/2*n - 1)/(c*x^(2*n) + b*x^n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298401, size = 1, normalized size = 0.02 \[ \left [\frac{x x^{\frac{1}{2} \, n - 1} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} x^{n - 2} - 2 \, b x x^{\frac{1}{2} \, n - 1} \sqrt{-\frac{c}{b}} - b}{c x^{2} x^{n - 2} + b}\right ) - 2}{b n x x^{\frac{1}{2} \, n - 1}}, \frac{2 \,{\left (x x^{\frac{1}{2} \, n - 1} \sqrt{\frac{c}{b}} \arctan \left (\frac{b \sqrt{\frac{c}{b}}}{c x x^{\frac{1}{2} \, n - 1}}\right ) - 1\right )}}{b n x x^{\frac{1}{2} \, n - 1}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[(x*x^(1/2*n - 1)*sqrt(-c/b)*log((c*x^2*x^(n - 2) - 2*b*x*x^(1/2*n - 1)*sqrt(-c/
b) - b)/(c*x^2*x^(n - 2) + b)) - 2)/(b*n*x*x^(1/2*n - 1)), 2*(x*x^(1/2*n - 1)*sq
rt(c/b)*arctan(b*sqrt(c/b)/(c*x*x^(1/2*n - 1))) - 1)/(b*n*x*x^(1/2*n - 1))]

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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+1/2*n)/(b*x**n+c*x**(2*n)),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{1}{2} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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