3.2762 \(\int \frac{(c x)^{-1-\frac{3 n}{2}}}{a+b x^n} \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 b^{3/2} x^{3 n/2} (c x)^{-3 n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{5/2} c n}+\frac{2 b x^n (c x)^{-3 n/2}}{a^2 c n}-\frac{2 (c x)^{-3 n/2}}{3 a c n} \]

[Out]

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

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Rubi [A]  time = 0.140504, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 b^{3/2} x^{3 n/2} (c x)^{-3 n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{5/2} c n}+\frac{2 b x^n (c x)^{-3 n/2}}{a^2 c n}-\frac{2 (c x)^{-3 n/2}}{3 a c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.175, size = 85, normalized size = 0.85 \[ - \frac{2 \left (c x\right )^{- \frac{3 n}{2}}}{3 a c n} + \frac{2 b x^{n} \left (c x\right )^{- \frac{3 n}{2}}}{a^{2} c n} - \frac{2 b^{\frac{3}{2}} x^{\frac{3 n}{2}} \left (c x\right )^{- \frac{3 n}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0584139, size = 72, normalized size = 0.72 \[ -\frac{2 (c x)^{-3 n/2} \left (3 b^{3/2} x^{3 n/2} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )+\sqrt{a} \left (a-3 b x^n\right )\right )}{3 a^{5/2} c n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{\frac{1}{a+b{x}^{n}} \left ( cx \right ) ^{-1-{\frac{3\,n}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((c*x)^(-1-3/2*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)^(-3/2*n - 1)/(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312595, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b c^{-n - \frac{2}{3}} \sqrt{-\frac{b c^{-n - \frac{2}{3}}}{a}} \log \left (-\frac{2 \, a \sqrt{-\frac{b c^{-n - \frac{2}{3}}}{a}} x^{\frac{1}{3}} e^{\left (-\frac{1}{6} \,{\left (3 \, n + 2\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (3 \, n + 2\right )} \log \left (x\right )\right )} - a x^{\frac{2}{3}} e^{\left (-\frac{1}{3} \,{\left (3 \, n + 2\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (3 \, n + 2\right )} \log \left (x\right )\right )} + b c^{-n - \frac{2}{3}}}{a x^{\frac{2}{3}} e^{\left (-\frac{1}{3} \,{\left (3 \, n + 2\right )} \log \left (c\right ) - \frac{1}{3} \,{\left (3 \, n + 2\right )} \log \left (x\right )\right )} + b c^{-n - \frac{2}{3}}}\right ) + 6 \, b c^{-n - \frac{2}{3}} x^{\frac{1}{3}} e^{\left (-\frac{1}{6} \,{\left (3 \, n + 2\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (3 \, n + 2\right )} \log \left (x\right )\right )} - 2 \, a x e^{\left (-\frac{1}{2} \,{\left (3 \, n + 2\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (3 \, n + 2\right )} \log \left (x\right )\right )}}{3 \, a^{2} n}, \frac{2 \,{\left (3 \, b c^{-n - \frac{2}{3}} \sqrt{\frac{b c^{-n - \frac{2}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{b c^{-n - \frac{2}{3}}}{a}} e^{\left (\frac{1}{6} \,{\left (3 \, n + 2\right )} \log \left (c\right ) + \frac{1}{6} \,{\left (3 \, n + 2\right )} \log \left (x\right )\right )}}{x^{\frac{1}{3}}}\right ) + 3 \, b c^{-n - \frac{2}{3}} x^{\frac{1}{3}} e^{\left (-\frac{1}{6} \,{\left (3 \, n + 2\right )} \log \left (c\right ) - \frac{1}{6} \,{\left (3 \, n + 2\right )} \log \left (x\right )\right )} - a x e^{\left (-\frac{1}{2} \,{\left (3 \, n + 2\right )} \log \left (c\right ) - \frac{1}{2} \,{\left (3 \, n + 2\right )} \log \left (x\right )\right )}\right )}}{3 \, a^{2} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(-3/2*n - 1)/(b*x^n + a),x, algorithm="fricas")

[Out]

[1/3*(3*b*c^(-n - 2/3)*sqrt(-b*c^(-n - 2/3)/a)*log(-(2*a*sqrt(-b*c^(-n - 2/3)/a)
*x^(1/3)*e^(-1/6*(3*n + 2)*log(c) - 1/6*(3*n + 2)*log(x)) - a*x^(2/3)*e^(-1/3*(3
*n + 2)*log(c) - 1/3*(3*n + 2)*log(x)) + b*c^(-n - 2/3))/(a*x^(2/3)*e^(-1/3*(3*n
+ 2)*log(c) - 1/3*(3*n + 2)*log(x)) + b*c^(-n - 2/3))) + 6*b*c^(-n - 2/3)*x^(1/
3)*e^(-1/6*(3*n + 2)*log(c) - 1/6*(3*n + 2)*log(x)) - 2*a*x*e^(-1/2*(3*n + 2)*lo
g(c) - 1/2*(3*n + 2)*log(x)))/(a^2*n), 2/3*(3*b*c^(-n - 2/3)*sqrt(b*c^(-n - 2/3)
/a)*arctan(sqrt(b*c^(-n - 2/3)/a)*e^(1/6*(3*n + 2)*log(c) + 1/6*(3*n + 2)*log(x)
)/x^(1/3)) + 3*b*c^(-n - 2/3)*x^(1/3)*e^(-1/6*(3*n + 2)*log(c) - 1/6*(3*n + 2)*l
og(x)) - a*x*e^(-1/2*(3*n + 2)*log(c) - 1/2*(3*n + 2)*log(x)))/(a^2*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)**(-1-3/2*n)/(a+b*x**n),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{-\frac{3}{2} \, n - 1}}{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(-3/2*n - 1)/(b*x^n + a),x, algorithm="giac")

[Out]

integrate((c*x)^(-3/2*n - 1)/(b*x^n + a), x)