Optimal. Leaf size=70 \[ \frac{a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}-\frac{x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.076499, 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 \[ \frac{a x \left (c x^n\right )^{-1/n}}{4 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4}-\frac{x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3} \]
Antiderivative was successfully verified.
[In] Int[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^5,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.9434, size = 58, normalized size = 0.83 \[ \frac{a x \left (c x^{n}\right )^{- \frac{1}{n}}}{4 b^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{4}} - \frac{x \left (c x^{n}\right )^{- \frac{1}{n}}}{3 b^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{3}} \]
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))**5,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.227877, size = 66, normalized size = 0.94 \[ \frac{x \left (c x^n\right )^{\frac{1}{n}} \left (6 a^2+4 a b \left (c x^n\right )^{\frac{1}{n}}+b^2 \left (c x^n\right )^{2/n}\right )}{12 a^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^5,x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.079, size = 316, normalized size = 4.5 \[{\frac{x}{12\,{a}^{3}} \left ( \left ( \sqrt [n]{c} \right ) ^{3} \left ( \sqrt [n]{{x}^{n}} \right ) ^{3}{b}^{2}{{\rm e}^{{\frac{{\frac{3\,i}{2}}\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) -{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) }{n}}}}+4\, \left ( \sqrt [n]{c} \right ) ^{2} \left ( \sqrt [n]{{x}^{n}} \right ) ^{2}ab{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) -{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) }{n}}}}+6\,\sqrt [n]{c}\sqrt [n]{{x}^{n}}{a}^{2}{{\rm e}^{{\frac{i/2\pi \,{\it csgn} \left ( ic{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) -{\it csgn} \left ( ic \right ) \right ) \left ( -{\it csgn} \left ( ic{x}^{n} \right ) +{\it csgn} \left ( i{x}^{n} \right ) \right ) }{n}}}} \right ) \left ( a+b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n))^5,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.39939, size = 213, normalized size = 3.04 \[ \frac{b^{2} c^{\frac{3}{n}} x{\left (x^{n}\right )}^{\frac{3}{n}} + 4 \, a b c^{\frac{2}{n}} x{\left (x^{n}\right )}^{\frac{2}{n}} + 6 \, a^{2} c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )}}{12 \,{\left (a^{3} b^{4} c^{\frac{4}{n}}{\left (x^{n}\right )}^{\frac{4}{n}} + 4 \, a^{4} b^{3} c^{\frac{3}{n}}{\left (x^{n}\right )}^{\frac{3}{n}} + 6 \, a^{5} b^{2} c^{\frac{2}{n}}{\left (x^{n}\right )}^{\frac{2}{n}} + 4 \, a^{6} b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^5,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.218151, size = 124, normalized size = 1.77 \[ -\frac{4 \, b c^{\left (\frac{1}{n}\right )} x + a}{12 \,{\left (b^{6} c^{\frac{5}{n}} x^{4} + 4 \, a b^{5} c^{\frac{4}{n}} x^{3} + 6 \, a^{2} b^{4} c^{\frac{3}{n}} x^{2} + 4 \, a^{3} b^{3} c^{\frac{2}{n}} x + a^{4} b^{2} c^{\left (\frac{1}{n}\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^5,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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))**5,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^5,x, algorithm="giac")
[Out]