Optimal. Leaf size=75 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{a+b (c x)^n}}+\frac{2}{3 a n \left (a+b (c x)^n\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.146465, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{a^{5/2} n}+\frac{2}{a^2 n \sqrt{a+b (c x)^n}}+\frac{2}{3 a n \left (a+b (c x)^n\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a + b*(c*x)^n)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 6.5052, size = 63, normalized size = 0.84 \[ \frac{2}{3 a n \left (a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}} + \frac{2}{a^{2} n \sqrt{a + b \left (c x\right )^{n}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(a+b*(c*x)**n)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.153711, size = 66, normalized size = 0.88 \[ \frac{2 \left (\frac{\sqrt{a} \left (4 a+3 b (c x)^n\right )}{\left (a+b (c x)^n\right )^{3/2}}-3 \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )\right )}{3 a^{5/2} n} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a + b*(c*x)^n)^(5/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.011, size = 59, normalized size = 0.8 \[{\frac{1}{n} \left ( -2\,{\frac{1}{{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{\sqrt{a+b \left ( cx \right ) ^{n}}{a}^{2}}}+{\frac{2}{3\,a} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{-{\frac{3}{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(a+b*(c*x)^n)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x\right )^{n} b + a\right )}^{\frac{5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x)^n*b + a)^(5/2)*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.288377, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, \left (c x\right )^{n} \sqrt{a} b + 3 \,{\left (\left (c x\right )^{n} b + a\right )}^{\frac{3}{2}} \log \left (\frac{\left (c x\right )^{n} \sqrt{a} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} a + 2 \, a^{\frac{3}{2}}}{\left (c x\right )^{n}}\right ) + 8 \, a^{\frac{3}{2}}}{3 \,{\left (\left (c x\right )^{n} a^{\frac{5}{2}} b n + a^{\frac{7}{2}} n\right )} \sqrt{\left (c x\right )^{n} b + a}}, \frac{2 \,{\left (3 \, \left (c x\right )^{n} \sqrt{-a} b + 3 \,{\left (\left (c x\right )^{n} b + a\right )}^{\frac{3}{2}} \arctan \left (\frac{a}{\sqrt{\left (c x\right )^{n} b + a} \sqrt{-a}}\right ) + 4 \, \sqrt{-a} a\right )}}{3 \,{\left (\left (c x\right )^{n} \sqrt{-a} a^{2} b n + \sqrt{-a} a^{3} n\right )} \sqrt{\left (c x\right )^{n} b + a}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x)^n*b + a)^(5/2)*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a + b \left (c x\right )^{n}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(a+b*(c*x)**n)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\left (c x\right )^{n} b + a\right )}^{\frac{5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((c*x)^n*b + a)^(5/2)*x),x, algorithm="giac")
[Out]