3.2777 \(\int \frac{(c x)^{-1+\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=54 \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]

[Out]

(2*(c*x)^(n/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(Sqrt[b]*c*n*x^(n/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0781306, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]

Antiderivative was successfully verified.

[In]  Int[(c*x)^(-1 + n/2)/Sqrt[a + b*x^n],x]

[Out]

(2*(c*x)^(n/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(Sqrt[b]*c*n*x^(n/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.64597, size = 42, normalized size = 0.78 \[ \frac{2 x^{- \frac{n}{2}} \left (c x\right )^{\frac{n}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{\sqrt{b} c n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x)**(-1+1/2*n)/(a+b*x**n)**(1/2),x)

[Out]

2*x**(-n/2)*(c*x)**(n/2)*atanh(sqrt(b)*x**(n/2)/sqrt(a + b*x**n))/(sqrt(b)*c*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0759691, size = 54, normalized size = 1. \[ \frac{2 x^{-n/2} (c x)^{n/2} \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{\sqrt{b} c n} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x)^(-1 + n/2)/Sqrt[a + b*x^n],x]

[Out]

(2*(c*x)^(n/2)*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(Sqrt[b]*c*n*x^(n/2))

_______________________________________________________________________________________

Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-1+{\frac{n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x)^(-1+1/2*n)/(a+b*x^n)^(1/2),x)

[Out]

int((c*x)^(-1+1/2*n)/(a+b*x^n)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(1/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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((c*x)^(1/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

Sympy [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((c*x)**(-1+1/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x\right )^{\frac{1}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x)^(1/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate((c*x)^(1/2*n - 1)/sqrt(b*x^n + a), x)