Optimal. Leaf size=52 \[ \frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.112795, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*(c*x)^n)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 4.9284, size = 42, normalized size = 0.81 \[ \frac{2}{a n \sqrt{a + b \left (c x\right )^{n}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(a+b*(c*x)**n)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0604115, size = 52, normalized size = 1. \[ \frac{2}{a n \sqrt{a+b (c x)^n}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{3/2} n} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a + b*(c*x)^n)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 43, normalized size = 0.8 \[{\frac{1}{n} \left ( -2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{\sqrt{a+b \left ( cx \right ) ^{n}}a}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(a+b*(c*x)^n)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x\right )^{n} b + a\right )}^{\frac{3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x)^n*b + a)^(3/2)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.285816, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\left (c x\right )^{n} b + a} \log \left (\frac{\left (c x\right )^{n} \sqrt{a} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} a + 2 \, a^{\frac{3}{2}}}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt{a}}{\sqrt{\left (c x\right )^{n} b + a} a^{\frac{3}{2}} n}, \frac{2 \,{\left (\sqrt{\left (c x\right )^{n} b + a} \arctan \left (\frac{a}{\sqrt{\left (c x\right )^{n} b + a} \sqrt{-a}}\right ) + \sqrt{-a}\right )}}{\sqrt{\left (c x\right )^{n} b + a} \sqrt{-a} a n}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x)^n*b + a)^(3/2)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(a+b*(c*x)**n)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x\right )^{n} b + a\right )}^{\frac{3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x)^n*b + a)^(3/2)*x),x, algorithm="giac")
[Out]