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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]