3.3070 \(\int \frac{\left (c x^n\right )^{\frac{1}{n}}}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^4} \, dx\)

Optimal. Leaf size=70 \[ \frac{a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^3}-\frac{x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \]

[Out]

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

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

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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]

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

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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]

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

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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]

1/6*x/a^2/(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))^3*((c^(1/n))^2*((x^n)^(1/n))^2*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)+3*c^(1/n)*(x^n)^(1/n)*a*exp(1/2*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))

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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]

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

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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]

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

[Out]

Exception raised: RecursionError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{4}}\,{d x} \]

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]

integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^4, x)