3.3068 \(\int \frac{\left (c x^n\right )^{\frac{1}{n}}}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{a x \left (c x^n\right )^{-1/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0704411, antiderivative size = 63, 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 \[ \frac{a x \left (c x^n\right )^{-1/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{x \left (c x^n\right )^{-1/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.004, size = 53, normalized size = 0.84 \[ \frac{a x \left (c x^{n}\right )^{- \frac{1}{n}}}{b^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} + \frac{x \left (c x^{n}\right )^{- \frac{1}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{2}} \]

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))**2,x)

[Out]

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

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Mathematica [A]  time = 4.19963, size = 0, normalized size = 0. \[ \int \frac{\left (c x^n\right )^{\frac{1}{n}}}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^2,x]

[Out]

Integrate[(c*x^n)^n^(-1)/(a + b*(c*x^n)^n^(-1))^2, x]

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Maple [C]  time = 0.047, size = 322, normalized size = 5.1 \[ -{\frac{x}{b} \left ( a+b{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}} \right ) ^{-1}}+{\frac{1}{{b}^{2}\sqrt [n]{c}}\ln \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ){{\rm e}^{-{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^n)^(1/n)/(a+b*(c*x^n)^(1/n))^2,x)

[Out]

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

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Maxima [A]  time = 23.7521, size = 70, normalized size = 1.11 \[ -\frac{x}{b^{2} c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a b} + \frac{c^{-\frac{1}{n}} \log \left (b^{2} c^{\left (\frac{1}{n}\right )} x + a b\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219076, size = 68, normalized size = 1.08 \[ \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) + a}{b^{3} c^{\frac{2}{n}} x + a b^{2} c^{\left (\frac{1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{n}\right )^{\frac{1}{n}}}{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**n)**(1/n)/(a+b*(c*x**n)**(1/n))**2,x)

[Out]

Integral((c*x**n)**(1/n)/(a + b*(c*x**n)**(1/n))**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{n}\right )^{\left (\frac{1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^2,x, algorithm="giac")

[Out]

integrate((c*x^n)^(1/n)/((c*x^n)^(1/n)*b + a)^2, x)