Optimal. Leaf size=34 \[ -\frac{x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
[Out]
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Rubi [A] time = 0.0212443, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c*x^n)^n^(-1))^(-3),x]
[Out]
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Rubi in Sympy [A] time = 2.28924, size = 27, normalized size = 0.79 \[ - \frac{x \left (c x^{n}\right )^{- \frac{1}{n}}}{2 b \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*(c*x**n)**(1/n))**3,x)
[Out]
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Mathematica [A] time = 0.0212443, size = 34, normalized size = 1. \[ -\frac{x \left (c x^n\right )^{-1/n}}{2 b \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(c*x^n)^n^(-1))^(-3),x]
[Out]
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Maple [C] time = 0.041, size = 209, normalized size = 6.2 \[{\frac{x}{2\,{a}^{2}} \left ( b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}+2\,a \right ) \left ( a+b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*(c*x^n)^(1/n))^3,x)
[Out]
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Maxima [A] time = 1.41803, size = 93, normalized size = 2.74 \[ \frac{b c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + 2 \, a x}{2 \,{\left (a^{2} b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 2 \, a^{3} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^(-3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232197, size = 58, normalized size = 1.71 \[ -\frac{1}{2 \,{\left (b^{3} c^{\frac{3}{n}} x^{2} + 2 \, a b^{2} c^{\frac{2}{n}} x + a^{2} b c^{\left (\frac{1}{n}\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^(-3),x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*(c*x**n)**(1/n))**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((c*x^n)^(1/n)*b + a)^(-3),x, algorithm="giac")
[Out]