Optimal. Leaf size=94 \[ -\frac{2 b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^3 x}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x}-\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{1}{a^2 x} \]
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Rubi [A] time = 0.0894864, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^3 x}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x}-\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{1}{a^2 x} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*(c*x^n)^n^(-1))^2),x]
[Out]
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Rubi in Sympy [A] time = 12.8394, size = 90, normalized size = 0.96 \[ - \frac{b \left (c x^{n}\right )^{\frac{1}{n}}}{a^{2} x \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} - \frac{1}{a^{2} x} - \frac{2 b \left (c x^{n}\right )^{\frac{1}{n}} \log{\left (\left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{3} x} + \frac{2 b \left (c x^{n}\right )^{\frac{1}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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Mathematica [A] time = 4.33256, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[1/(x^2*(a + b*(c*x^n)^n^(-1))^2),x]
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Maple [C] time = 0.048, size = 440, normalized size = 4.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a+b*(c*x^n)^(1/n))^2,x)
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Maxima [A] time = 21.9765, size = 78, normalized size = 0.83 \[ \frac{2 \, b c^{\left (\frac{1}{n}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} + \frac{a}{x}\right )}{a^{3}} + \frac{1}{a b c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x} - \frac{2}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.236604, size = 134, normalized size = 1.43 \[ -\frac{2 \, b^{2} c^{\frac{2}{n}} x^{2} \log \left (x\right ) + a^{2} + 2 \,{\left (a b x \log \left (x\right ) + a b x\right )} c^{\left (\frac{1}{n}\right )} - 2 \,{\left (b^{2} c^{\frac{2}{n}} x^{2} + a b c^{\left (\frac{1}{n}\right )} x\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{a^{3} b c^{\left (\frac{1}{n}\right )} x^{2} + a^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^2),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^2),x, algorithm="giac")
[Out]