Optimal. Leaf size=85 \[ \frac{2 c \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{a^{3/2} (1-n) \sqrt{c x}}-\frac{2 \sqrt{c x}}{a (1-n) \sqrt{a x+b x^n}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.235133, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{2 c \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b x^n}}\right )}{a^{3/2} (1-n) \sqrt{c x}}-\frac{2 \sqrt{c x}}{a (1-n) \sqrt{a x+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*x]/(a*x + b*x^n)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 23.5995, size = 70, normalized size = 0.82 \[ - \frac{2 \sqrt{c x}}{a \left (- n + 1\right ) \sqrt{a x + b x^{n}}} + \frac{2 \sqrt{c x} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b x^{n}}} \right )}}{a^{\frac{3}{2}} \sqrt{x} \left (- n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(1/2)/(a*x+b*x**n)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.20224, size = 104, normalized size = 1.22 \[ \frac{2 \sqrt{c x} \left (\sqrt{a} \sqrt{x}-\sqrt{b} x^{n/2} \sqrt{\frac{a x^{1-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{1}{2}-\frac{n}{2}}}{\sqrt{b}}\right )\right )}{a^{3/2} (n-1) \sqrt{x} \sqrt{a x+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*x]/(a*x + b*x^n)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.05, size = 0, normalized size = 0. \[ \int{1\sqrt{cx} \left ( ax+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x)^(1/2)/(a*x+b*x^n)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{{\left (a x + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(a*x + b*x^n)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(a*x + b*x^n)^(3/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{\left (a x + b x^{n}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x)**(1/2)/(a*x+b*x**n)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x}}{{\left (a x + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x)/(a*x + b*x^n)^(3/2),x, algorithm="giac")
[Out]