Optimal. Leaf size=67 \[ \frac{a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]
[Out]
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Rubi [A] time = 0.0601779, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*(c*x^n)^n^(-1))^2,x]
[Out]
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Rubi in Sympy [A] time = 8.60182, size = 56, normalized size = 0.84 \[ \frac{a x^{2} \left (c x^{n}\right )^{- \frac{2}{n}}}{b^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} + \frac{x^{2} \left (c x^{n}\right )^{- \frac{2}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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Mathematica [A] time = 4.19184, size = 0, normalized size = 0. \[ \int \frac{x}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x/(a + b*(c*x^n)^n^(-1))^2,x]
[Out]
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Maple [C] time = 0.052, size = 435, normalized size = 6.5 \[{\frac{{x}^{2}}{a} \left ( a+b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-1}}-{\frac{x}{a\sqrt [n]{c}b}{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}}+{\frac{1}{ \left ( \sqrt [n]{c} \right ) ^{2}{b}^{2}}\ln \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ){{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{n}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*(c*x^n)^(1/n))^2,x)
[Out]
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Maxima [A] time = 22.4892, size = 86, normalized size = 1.28 \[ \frac{x^{2}}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} - \frac{c^{-\frac{1}{n}} x}{a b} + \frac{c^{-\frac{2}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((c*x^n)^(1/n)*b + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22947, size = 70, normalized size = 1.04 \[ \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) + a}{b^{3} c^{\frac{3}{n}} x + a b^{2} c^{\frac{2}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((c*x^n)^(1/n)*b + a)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*(c*x**n)**(1/n))**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((c*x^n)^(1/n)*b + a)^2,x, algorithm="giac")
[Out]