3.3011 \(\int \frac{1}{x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2 b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^3 x}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x}-\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{1}{a^2 x} \]

[Out]

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

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Rubi [A]  time = 0.0894864, antiderivative size = 94, 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 \[ -\frac{2 b \log (x) \left (c x^n\right )^{\frac{1}{n}}}{a^3 x}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x}-\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}-\frac{1}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.8394, size = 90, normalized size = 0.96 \[ - \frac{b \left (c x^{n}\right )^{\frac{1}{n}}}{a^{2} x \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} - \frac{1}{a^{2} x} - \frac{2 b \left (c x^{n}\right )^{\frac{1}{n}} \log{\left (\left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{3} x} + \frac{2 b \left (c x^{n}\right )^{\frac{1}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 4.33256, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.048, size = 440, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a/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))-2/a^2/x+2/a^3*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*cs
gn(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)*csg
n(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)-2/a^3*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 = 21.9765, size = 78, normalized size = 0.83 \[ \frac{2 \, b c^{\left (\frac{1}{n}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} + \frac{a}{x}\right )}{a^{3}} + \frac{1}{a b c^{\left (\frac{1}{n}\right )} x{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x} - \frac{2}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236604, size = 134, normalized size = 1.43 \[ -\frac{2 \, b^{2} c^{\frac{2}{n}} x^{2} \log \left (x\right ) + a^{2} + 2 \,{\left (a b x \log \left (x\right ) + a b x\right )} c^{\left (\frac{1}{n}\right )} - 2 \,{\left (b^{2} c^{\frac{2}{n}} x^{2} + a b c^{\left (\frac{1}{n}\right )} x\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{a^{3} b c^{\left (\frac{1}{n}\right )} x^{2} + a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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