3.3071 \(\int \frac{\left (c x^n\right )^{\frac{1}{n}}}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^5} \, dx\)

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]

(a*x)/(4*b^2*(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^4) - x/(3*b^2*(c*x^n)^n^(-1)*
(a + b*(c*x^n)^n^(-1))^3)

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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]

(a*x)/(4*b^2*(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^4) - x/(3*b^2*(c*x^n)^n^(-1)*
(a + b*(c*x^n)^n^(-1))^3)

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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]

a*x*(c*x**n)**(-1/n)/(4*b**2*(a + b*(c*x**n)**(1/n))**4) - x*(c*x**n)**(-1/n)/(3
*b**2*(a + b*(c*x**n)**(1/n))**3)

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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]

(x*(c*x^n)^n^(-1)*(6*a^2 + 4*a*b*(c*x^n)^n^(-1) + b^2*(c*x^n)^(2/n)))/(12*a^3*(a
+ b*(c*x^n)^n^(-1))^4)

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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]

1/12*x/(a+b*exp(1/2*(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*csgn(I*c)
*csgn(I*c*x^n)-I*Pi*csgn(I*c*x^n)^3+I*Pi*csgn(I*c)*csgn(I*c*x^n)^2+2*ln(c)+2*ln(
x^n))/n))^4/a^3*((c^(1/n))^3*((x^n)^(1/n))^3*b^2*exp(3/2*I*Pi*csgn(I*c*x^n)*(csg
n(I*c*x^n)-csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n)+4*(c^(1/n))^2*((x^n)^(1/n)
)^2*a*b*exp(I*Pi*csgn(I*c*x^n)*(csgn(I*c*x^n)-csgn(I*c))*(-csgn(I*c*x^n)+csgn(I*
x^n))/n)+6*c^(1/n)*(x^n)^(1/n)*a^2*exp(1/2*I*Pi*csgn(I*c*x^n)*(csgn(I*c*x^n)-csg
n(I*c))*(-csgn(I*c*x^n)+csgn(I*x^n))/n))

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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]

1/12*(b^2*c^(3/n)*x*(x^n)^(3/n) + 4*a*b*c^(2/n)*x*(x^n)^(2/n) + 6*a^2*c^(1/n)*x*
(x^n)^(1/n))/(a^3*b^4*c^(4/n)*(x^n)^(4/n) + 4*a^4*b^3*c^(3/n)*(x^n)^(3/n) + 6*a^
5*b^2*c^(2/n)*(x^n)^(2/n) + 4*a^6*b*c^(1/n)*(x^n)^(1/n) + a^7)

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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]

-1/12*(4*b*c^(1/n)*x + a)/(b^6*c^(5/n)*x^4 + 4*a*b^5*c^(4/n)*x^3 + 6*a^2*b^4*c^(
3/n)*x^2 + 4*a^3*b^3*c^(2/n)*x + a^4*b^2*c^(1/n))

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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))**5,x)

[Out]

Exception raised: RecursionError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{5}}\,{d x} \]

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]

integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^5, x)