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

Optimal. Leaf size=68 \[ -\frac{2 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{5/2} n}+\frac{2 c x^{-n/2}}{b^2 n}-\frac{2 x^{-3 n/2}}{3 b n} \]

[Out]

-2/(3*b*n*x^((3*n)/2)) + (2*c)/(b^2*n*x^(n/2)) - (2*c^(3/2)*ArcTan[Sqrt[b]/(Sqrt
[c]*x^(n/2))])/(b^(5/2)*n)

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Rubi [A]  time = 0.089816, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{2 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )}{b^{5/2} n}+\frac{2 c x^{-n/2}}{b^2 n}-\frac{2 x^{-3 n/2}}{3 b n} \]

Antiderivative was successfully verified.

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

[Out]

-2/(3*b*n*x^((3*n)/2)) + (2*c)/(b^2*n*x^(n/2)) - (2*c^(3/2)*ArcTan[Sqrt[b]/(Sqrt
[c]*x^(n/2))])/(b^(5/2)*n)

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Rubi in Sympy [A]  time = 17.8674, size = 56, normalized size = 0.82 \[ - \frac{2 x^{- \frac{3 n}{2}}}{3 b n} + \frac{2 c x^{- \frac{n}{2}}}{b^{2} n} - \frac{2 c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x^{- \frac{n}{2}}}{\sqrt{c}} \right )}}{b^{\frac{5}{2}} n} \]

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**(-3*n/2)/(3*b*n) + 2*c*x**(-n/2)/(b**2*n) - 2*c**(3/2)*atan(sqrt(b)*x**(-n
/2)/sqrt(c))/(b**(5/2)*n)

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Mathematica [A]  time = 0.0892205, size = 62, normalized size = 0.91 \[ \frac{2 \left (\sqrt{b} x^{-3 n/2} \left (3 c x^n-b\right )-3 c^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x^{-n/2}}{\sqrt{c}}\right )\right )}{3 b^{5/2} n} \]

Antiderivative was successfully verified.

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

[Out]

(2*((Sqrt[b]*(-b + 3*c*x^n))/x^((3*n)/2) - 3*c^(3/2)*ArcTan[Sqrt[b]/(Sqrt[c]*x^(
n/2))]))/(3*b^(5/2)*n)

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Maple [A]  time = 0.122, size = 97, normalized size = 1.4 \[ 2\,{\frac{c}{n{b}^{2}{x}^{n/2}}}-{\frac{2}{3\,bn} \left ({x}^{{\frac{n}{2}}} \right ) ^{-3}}+{\frac{c}{{b}^{3}n}\sqrt{-bc}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{c}\sqrt{-bc}} \right ) }-{\frac{c}{{b}^{3}n}\sqrt{-bc}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{c}\sqrt{-bc}} \right ) } \]

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*c/b^2/n/(x^(1/2*n))-2/3/b/n/(x^(1/2*n))^3+1/b^3*(-b*c)^(1/2)*c/n*ln(x^(1/2*n)+
1/c*(-b*c)^(1/2))-1/b^3*(-b*c)^(1/2)*c/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 = 0.300984, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, b x^{3} x^{-\frac{3}{2} \, n - 3} - 6 \, c x x^{-\frac{1}{2} \, n - 1} - 3 \, c \sqrt{-\frac{c}{b}} \log \left (\frac{b x^{2} x^{-n - 2} - 2 \, b x x^{-\frac{1}{2} \, n - 1} \sqrt{-\frac{c}{b}} - c}{b x^{2} x^{-n - 2} + c}\right )}{3 \, b^{2} n}, -\frac{2 \,{\left (b x^{3} x^{-\frac{3}{2} \, n - 3} - 3 \, c x x^{-\frac{1}{2} \, n - 1} - 3 \, c \sqrt{\frac{c}{b}} \arctan \left (\frac{\sqrt{\frac{c}{b}}}{x x^{-\frac{1}{2} \, n - 1}}\right )\right )}}{3 \, b^{2} n}\right ] \]

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]

[-1/3*(2*b*x^3*x^(-3/2*n - 3) - 6*c*x*x^(-1/2*n - 1) - 3*c*sqrt(-c/b)*log((b*x^2
*x^(-n - 2) - 2*b*x*x^(-1/2*n - 1)*sqrt(-c/b) - c)/(b*x^2*x^(-n - 2) + c)))/(b^2
*n), -2/3*(b*x^3*x^(-3/2*n - 3) - 3*c*x*x^(-1/2*n - 1) - 3*c*sqrt(c/b)*arctan(sq
rt(c/b)/(x*x^(-1/2*n - 1))))/(b^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**(-1-1/2*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)