3.2755 \(\int \frac{(c x)^{-1-\frac{n}{2}}}{a+b x^n} \, dx\)

Optimal. Leaf size=74 \[ \frac{2 \sqrt{b} x^{n/2} (c x)^{-n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} c n}-\frac{2 (c x)^{-n/2}}{a c n} \]

[Out]

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

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Rubi [A]  time = 0.103681, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{2 \sqrt{b} x^{n/2} (c x)^{-n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} c n}-\frac{2 (c x)^{-n/2}}{a c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.828, size = 56, normalized size = 0.76 \[ - \frac{2 \left (c x\right )^{- \frac{n}{2}}}{a c n} + \frac{2 \sqrt{b} x^{\frac{n}{2}} \left (c x\right )^{- \frac{n}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0407082, size = 61, normalized size = 0.82 \[ -\frac{2 (c x)^{-n/2} \left (\sqrt{a}-\sqrt{b} x^{n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )\right )}{a^{3/2} c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{n}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249168, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, x e^{\left (-\frac{1}{2} \,{\left (n + 2\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n + 2\right )} \log \left (x\right )\right )} - \sqrt{-\frac{b c^{-n - 2}}{a}} \log \left (\frac{a x^{2} e^{\left (-{\left (n + 2\right )} \log \left (c\right ) -{\left (n + 2\right )} \log \left (x\right )\right )} + 2 \, a \sqrt{-\frac{b c^{-n - 2}}{a}} x e^{\left (-\frac{1}{2} \,{\left (n + 2\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n + 2\right )} \log \left (x\right )\right )} - b c^{-n - 2}}{a x^{2} e^{\left (-{\left (n + 2\right )} \log \left (c\right ) -{\left (n + 2\right )} \log \left (x\right )\right )} + b c^{-n - 2}}\right )}{a n}, -\frac{2 \,{\left (x e^{\left (-\frac{1}{2} \,{\left (n + 2\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (n + 2\right )} \log \left (x\right )\right )} + \sqrt{\frac{b c^{-n - 2}}{a}} \arctan \left (\frac{\sqrt{\frac{b c^{-n - 2}}{a}} e^{\left (\frac{1}{2} \,{\left (n + 2\right )} \log \left (c\right ) + \frac{1}{2} \,{\left (n + 2\right )} \log \left (x\right )\right )}}{x}\right )\right )}}{a n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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