3.3008 \(\int \frac{x}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

Optimal. Leaf size=67 \[ \frac{a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0601779, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a x^2 \left (c x^n\right )^{-2/n}}{b^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{x^2 \left (c x^n\right )^{-2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.60182, size = 56, normalized size = 0.84 \[ \frac{a x^{2} \left (c x^{n}\right )^{- \frac{2}{n}}}{b^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} + \frac{x^{2} \left (c x^{n}\right )^{- \frac{2}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.19184, size = 0, normalized size = 0. \[ \int \frac{x}{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.052, size = 435, normalized size = 6.5 \[{\frac{{x}^{2}}{a} \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{x}{a\sqrt [n]{c}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\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}}+{\frac{1}{ \left ( \sqrt [n]{c} \right ) ^{2}{b}^{2}}\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 ) }{n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*(c*x^n)^(1/n))^2,x)

[Out]

x^2/a/(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))-1/a/(c^(1/n))/b*x*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)+1/(c^(1/n))^2/b^2*ln(b*exp(1/2*(-I*Pi*csgn(I*x^n)*csg
n(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(-(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 = 22.4892, size = 86, normalized size = 1.28 \[ \frac{x^{2}}{a b c^{\left (\frac{1}{n}\right )}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2}} - \frac{c^{-\frac{1}{n}} x}{a b} + \frac{c^{-\frac{2}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22947, size = 70, normalized size = 1.04 \[ \frac{{\left (b c^{\left (\frac{1}{n}\right )} x + a\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) + a}{b^{3} c^{\frac{3}{n}} x + a b^{2} c^{\frac{2}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((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^(3/n)*x + a*b^2*c^(2/n))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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