Optimal. Leaf size=70 \[ \frac{a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}-\frac{x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
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Rubi [A] time = 0.0759947, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}-\frac{x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^4,x]
[Out]
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Rubi in Sympy [A] time = 18.0886, size = 58, normalized size = 0.83 \[ \frac{a x \left (c x^{n}\right )^{- \frac{1}{n}}}{3 b^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{3}} - \frac{x \left (c x^{n}\right )^{- \frac{1}{n}}}{2 b^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**n)**(1/n)/(a+b*(c*x**n)**(1/n))**4,x)
[Out]
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Mathematica [A] time = 0.144542, size = 47, normalized size = 0.67 \[ \frac{x \left (c x^n\right )^{\frac{1}{n}} \left (3 a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{6 a^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^4,x]
[Out]
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Maple [C] time = 99.751, size = 242, normalized size = 3.5 \[{\frac{x}{6\,{a}^{2}} \left ( \left ( \sqrt [n]{c} \right ) ^{2} \left ( \sqrt [n]{{x}^{n}} \right ) ^{2}b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) -{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) }{n}}}}+3\,\sqrt [n]{c}\sqrt [n]{{x}^{n}}a{{\rm e}^{{\frac{i/2\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) -{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) }{n}}}} \right ) \left ( a+b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n))^4,x)
[Out]
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Maxima [A] time = 1.39182, size = 147, normalized size = 2.1 \[ \frac{b c^{\frac{2}{n}} x{\left (x^{n}\right )}^{\frac{2}{n}} + 3 \, a c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{6 \,{\left (a^{2} b^{3} c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + 3 \, a^{3} b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 3 \, a^{4} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217647, size = 100, normalized size = 1.43 \[ -\frac{3 \, b c^{\left (\frac{1}{n}\right )} x + a}{6 \,{\left (b^{5} c^{\frac{4}{n}} x^{3} + 3 \, a b^{4} c^{\frac{3}{n}} x^{2} + 3 \, a^{2} b^{3} c^{\frac{2}{n}} x + a^{3} b^{2} c^{\left (\frac{1}{n}\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^4,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((c*x**n)**(1/n)/(a+b*(c*x**n)**(1/n))**4,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^4,x, algorithm="giac")
[Out]