Optimal. Leaf size=53 \[ \frac{x^2 \left (c x^n\right )^{-1/n}}{b}-\frac{a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]
[Out]
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Rubi [A] time = 0.0493574, antiderivative size = 53, 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 \[ \frac{x^2 \left (c x^n\right )^{-1/n}}{b}-\frac{a x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]
Antiderivative was successfully verified.
[In] Int[x/(a + b*(c*x^n)^n^(-1)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a x^{2} \left (c x^{n}\right )^{- \frac{2}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{2}} + x^{2} \left (c x^{n}\right )^{- \frac{2}{n}} \int ^{\left (c x^{n}\right )^{\frac{1}{n}}} \frac{1}{b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*(c*x**n)**(1/n)),x)
[Out]
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Mathematica [A] time = 4.64948, size = 0, normalized size = 0. \[ \int \frac{x}{a+b \left (c x^n\right )^{\frac{1}{n}}} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[x/(a + b*(c*x^n)^n^(-1)),x]
[Out]
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Maple [C] time = 0.084, size = 325, normalized size = 6.1 \[{\frac{x}{\sqrt [n]{c}b}{{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) \pi \,{\it csgn} \left ( ic \right ) -i \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\pi +i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}}-{\frac{a}{ \left ( \sqrt [n]{c} \right ) ^{2}{b}^{2}}\ln \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\pi +2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ){{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) \pi \,{\it csgn} \left ( ic \right ) -i \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\pi +i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{n}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*(c*x^n)^(1/n)),x)
[Out]
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Maxima [A] time = 22.6675, size = 50, normalized size = 0.94 \[ \frac{c^{-\frac{1}{n}} x}{b} - \frac{a c^{-\frac{2}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((c*x^n)^(1/n)*b + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.253974, size = 49, normalized size = 0.92 \[ \frac{b c^{\left (\frac{1}{n}\right )} x - a \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{2} c^{\frac{2}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((c*x^n)^(1/n)*b + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{a + b \left (c x^{n}\right )^{\frac{1}{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*(c*x**n)**(1/n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((c*x^n)^(1/n)*b + a),x, algorithm="giac")
[Out]