3.3005 \(\int \frac{1}{x^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \, dx\)

Optimal. Leaf size=87 \[ \frac{b^2 \log (x) \left (c x^n\right )^{2/n}}{a^3 x^2}-\frac{b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x^2}+\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x^2}-\frac{1}{2 a x^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0769668, antiderivative size = 87, 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 \[ \frac{b^2 \log (x) \left (c x^n\right )^{2/n}}{a^3 x^2}-\frac{b^2 \left (c x^n\right )^{2/n} \log \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )}{a^3 x^2}+\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a^2 x^2}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.5294, size = 85, normalized size = 0.98 \[ - \frac{1}{2 a x^{2}} + \frac{b \left (c x^{n}\right )^{\frac{1}{n}}}{a^{2} x^{2}} + \frac{b^{2} \left (c x^{n}\right )^{\frac{2}{n}} \log{\left (\left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{3} x^{2}} - \frac{b^{2} \left (c x^{n}\right )^{\frac{2}{n}} \log{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}} \right )}}{a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 4.7998, size = 0, normalized size = 0. \[ \int \frac{1}{x^3 \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.1, size = 446, normalized size = 5.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 22.0393, size = 86, normalized size = 0.99 \[ -\frac{b^{2} c^{\frac{2}{n}} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right )}{a^{3}} + \frac{b^{2} c^{\frac{2}{n}} \log \left (x\right )}{a^{3}} + \frac{2 \, b c^{\left (\frac{1}{n}\right )} x - a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.237146, size = 88, normalized size = 1.01 \[ -\frac{2 \, b^{2} c^{\frac{2}{n}} x^{2} \log \left (b c^{\left (\frac{1}{n}\right )} x + a\right ) - 2 \, b^{2} c^{\frac{2}{n}} x^{2} \log \left (x\right ) - 2 \, a b c^{\left (\frac{1}{n}\right )} x + a^{2}}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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