3.3012 \(\int \frac{1}{x^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx\)

Optimal. Leaf size=125 \[ \frac{3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}+\frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}-\frac{1}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.113569, antiderivative size = 125, 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 \[ \frac{3 b^2 \log (x) \left (c x^n\right )^{2/n}}{a^4 x^2}-\frac{3 b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^4 x^2}+\frac{b^2 \left (c x^n\right )^{2/n}}{a^3 x^2 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}+\frac{2 b \left (c x^n\right )^{\frac{1}{n}}}{a^3 x^2}-\frac{1}{2 a^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.5913, size = 122, normalized size = 0.98 \[ - \frac{1}{2 a^{2} x^{2}} + \frac{b^{2} \left (c x^{n}\right )^{\frac{2}{n}}}{a^{3} x^{2} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )} + \frac{2 b \left (c x^{n}\right )^{\frac{1}{n}}}{a^{3} x^{2}} + \frac{3 b^{2} \left (c x^{n}\right )^{\frac{2}{n}} \log{\left (\left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{4} x^{2}} - \frac{3 b^{2} \left (c x^{n}\right )^{\frac{2}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a+b*(c*x**n)**(1/n))**2,x)

[Out]

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

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Mathematica [A]  time = 4.36541, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.051, size = 556, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*(c*x^n)^(1/n))^2,x)

[Out]

1/a/x^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/a^3/x*c^(1/n)*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*n*ln(x)+2*ln(x^n))/n)-3/2/a^2/x^2+3/a^4*(c^(1/n))^2*b^2*ln(x)*exp((I*Pi*cs
gn(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)-3/a^4*(c^(1/n))^2
*b^2*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)*cs
gn(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)*cs
gn(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 = 22.1929, size = 126, normalized size = 1.01 \[ -\frac{3 \, b^{2} c^{\frac{2}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{a^{4}} + \frac{3 \, b^{2} c^{\frac{2}{n}} \log \left (x\right )}{a^{4}} + \frac{1}{a b c^{\left (\frac{1}{n}\right )} x^{2}{\left (x^{n}\right )}^{\left (\frac{1}{n}\right )} + a^{2} x^{2}} + \frac{3 \,{\left (2 \, b c^{\left (\frac{1}{n}\right )} x - a\right )}}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.237625, size = 177, normalized size = 1.42 \[ \frac{6 \, b^{3} c^{\frac{3}{n}} x^{3} \log \left (x\right ) + 3 \, a^{2} b c^{\left (\frac{1}{n}\right )} x - a^{3} + 6 \,{\left (a b^{2} x^{2} \log \left (x\right ) + a b^{2} x^{2}\right )} c^{\frac{2}{n}} - 6 \,{\left (b^{3} c^{\frac{3}{n}} x^{3} + a b^{2} c^{\frac{2}{n}} x^{2}\right )} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{2 \,{\left (a^{4} b c^{\left (\frac{1}{n}\right )} x^{3} + a^{5} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(a+b*(c*x**n)**(1/n))**2,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{2} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((c*x^n)^(1/n)*b + a)^2*x^3),x, algorithm="giac")

[Out]

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