3.3004 \(\int \frac{1}{x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \, dx\)

Optimal. Leaf size=60 \[ -\frac{b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^2 x}+\frac{b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2 x}-\frac{1}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0615283, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^2 x}+\frac{b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^2 x}-\frac{1}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.70763, size = 58, normalized size = 0.97 \[ - \frac{1}{a x} - \frac{b \left (c x^{n}\right )^{\frac{1}{n}} \log{\left (\left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{2} x} + \frac{b \left (c x^{n}\right )^{\frac{1}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.7746, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.096, size = 331, normalized size = 5.5 \[ -{\frac{1}{ax}}+{\frac{\sqrt [n]{c}b}{{a}^{2}}\ln \left ( b{{\rm e}^{{\frac{-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) +i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+2\,\ln \left ( c \right ) +2\,\ln \left ({x}^{n} \right ) -2\,n\ln \left ( x \right ) }{2\,n}}}}x+a \right ){{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}}-{\frac{\sqrt [n]{c}b\ln \left ( x \right ) }{{a}^{2}}{{\rm e}^{{\frac{i\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic \right ){\it csgn} \left ( ic{x}^{n} \right ) -i\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+i\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-2\,n\ln \left ( x \right ) +2\,\ln \left ({x}^{n} \right ) }{2\,n}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 21.9054, size = 45, normalized size = 0.75 \[ \frac{b c^{\left (\frac{1}{n}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} + \frac{a}{x}\right )}{a^{2}} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237059, size = 55, normalized size = 0.92 \[ \frac{b c^{\left (\frac{1}{n}\right )} x \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) - b c^{\left (\frac{1}{n}\right )} x \log \left (x\right ) - a}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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