Optimal. Leaf size=125 \[ \frac{3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}+\frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}-\frac{1}{2 a^2 x^2} \]
[Out]
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Rubi [A] time = 0.113569, antiderivative size = 125, 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 \[ \frac{3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}+\frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}-\frac{1}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*(c*x^n)^n^(-1))^2),x]
[Out]
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Rubi in Sympy [A] time = 16.5913, size = 122, normalized size = 0.98 \[ - \frac{1}{2 a^{2} x^{2}} + \frac{b^{2} \left (c x^{n}\right )^{\frac{2}{n}}}{a^{3} x^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} + \frac{2 b \left (c x^{n}\right )^{\frac{1}{n}}}{a^{3} x^{2}} + \frac{3 b^{2} \left (c x^{n}\right )^{\frac{2}{n}} \log{\left (\left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{4} x^{2}} - \frac{3 b^{2} \left (c x^{n}\right )^{\frac{2}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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Mathematica [A] time = 4.36541, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[1/(x^3*(a + b*(c*x^n)^n^(-1))^2),x]
[Out]
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Maple [C] time = 0.051, size = 556, normalized size = 4.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(a+b*(c*x^n)^(1/n))^2,x)
[Out]
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Maxima [A] time = 22.1929, size = 126, normalized size = 1.01 \[ -\frac{3 \, b^{2} c^{\frac{2}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{a^{4}} + \frac{3 \, b^{2} c^{\frac{2}{n}} \log \left (x\right )}{a^{4}} + \frac{1}{a b c^{\left (\frac{1}{n}\right )} x^{2}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x^{2}} + \frac{3 \,{\left (2 \, b c^{\left (\frac{1}{n}\right )} x - a\right )}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237625, size = 177, normalized size = 1.42 \[ \frac{6 \, b^{3} c^{\frac{3}{n}} x^{3} \log \left (x\right ) + 3 \, a^{2} b c^{\left (\frac{1}{n}\right )} x - a^{3} + 6 \,{\left (a b^{2} x^{2} \log \left (x\right ) + a b^{2} x^{2}\right )} c^{\frac{2}{n}} - 6 \,{\left (b^{3} c^{\frac{3}{n}} x^{3} + a b^{2} c^{\frac{2}{n}} x^{2}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \,{\left (a^{4} b c^{\left (\frac{1}{n}\right )} x^{3} + a^{5} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^3),x, algorithm="giac")
[Out]