3.2681 \(\int \frac{x^{-1+\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=98 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{4 b^{5/2} n}-\frac{3 a x^{n/2} \sqrt{a+b x^n}}{4 b^2 n}+\frac{x^{3 n/2} \sqrt{a+b x^n}}{2 b n} \]

[Out]

(-3*a*x^(n/2)*Sqrt[a + b*x^n])/(4*b^2*n) + (x^((3*n)/2)*Sqrt[a + b*x^n])/(2*b*n)
+ (3*a^2*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(4*b^(5/2)*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.108191, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{4 b^{5/2} n}-\frac{3 a x^{n/2} \sqrt{a+b x^n}}{4 b^2 n}+\frac{x^{3 n/2} \sqrt{a+b x^n}}{2 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + (5*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(-3*a*x^(n/2)*Sqrt[a + b*x^n])/(4*b^2*n) + (x^((3*n)/2)*Sqrt[a + b*x^n])/(2*b*n)
+ (3*a^2*ArcTanh[(Sqrt[b]*x^(n/2))/Sqrt[a + b*x^n]])/(4*b^(5/2)*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.3045, size = 116, normalized size = 1.18 \[ \frac{a^{2} x^{\frac{3 n}{2}}}{2 b n \left (a + b x^{n}\right )^{\frac{3}{2}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )^{2}} - \frac{3 a^{2} x^{\frac{n}{2}}}{4 b^{2} n \sqrt{a + b x^{n}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )} + \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{4 b^{\frac{5}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+5/2*n)/(a+b*x**n)**(1/2),x)

[Out]

a**2*x**(3*n/2)/(2*b*n*(a + b*x**n)**(3/2)*(-b*x**n/(a + b*x**n) + 1)**2) - 3*a*
*2*x**(n/2)/(4*b**2*n*sqrt(a + b*x**n)*(-b*x**n/(a + b*x**n) + 1)) + 3*a**2*atan
h(sqrt(b)*x**(n/2)/sqrt(a + b*x**n))/(4*b**(5/2)*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0847465, size = 80, normalized size = 0.82 \[ \frac{3 a^2 \log \left (\sqrt{b} \sqrt{a+b x^n}+b x^{n/2}\right )+\sqrt{b} x^{n/2} \sqrt{a+b x^n} \left (2 b x^n-3 a\right )}{4 b^{5/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + (5*n)/2)/Sqrt[a + b*x^n],x]

[Out]

(Sqrt[b]*x^(n/2)*Sqrt[a + b*x^n]*(-3*a + 2*b*x^n) + 3*a^2*Log[b*x^(n/2) + Sqrt[b
]*Sqrt[a + b*x^n]])/(4*b^(5/2)*n)

_______________________________________________________________________________________

Maple [A]  time = 0.051, size = 82, normalized size = 0.8 \[ -{\frac{1}{4\,{b}^{2}n}{{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \left ( -2\,b \left ({{\rm e}^{1/2\,n\ln \left ( x \right ) }} \right ) ^{2}+3\,a \right ) \sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}}}+{\frac{3\,{a}^{2}}{4\,n}\ln \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}}\sqrt{b}+\sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+5/2*n)/(a+b*x^n)^(1/2),x)

[Out]

-1/4*exp(1/2*n*ln(x))*(-2*b*exp(1/2*n*ln(x))^2+3*a)*(a+b*exp(1/2*n*ln(x))^2)^(1/
2)/b^2/n+3/4*a^2/b^(5/2)/n*ln(exp(1/2*n*ln(x))*b^(1/2)+(a+b*exp(1/2*n*ln(x))^2)^
(1/2))

_______________________________________________________________________________________

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(x^(5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.25389, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \log \left (-2 \, \sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} - 2 \, b^{\frac{3}{2}} x^{n} - a \sqrt{b}\right ) + 2 \,{\left (2 \, b^{\frac{3}{2}} x^{\frac{3}{2} \, n} - 3 \, a \sqrt{b} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{8 \, b^{\frac{5}{2}} n}, \frac{3 \, a^{2} \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right ) +{\left (2 \, \sqrt{-b} b x^{\frac{3}{2} \, n} - 3 \, a \sqrt{-b} x^{\frac{1}{2} \, n}\right )} \sqrt{b x^{n} + a}}{4 \, \sqrt{-b} b^{2} n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

[1/8*(3*a^2*log(-2*sqrt(b*x^n + a)*b*x^(1/2*n) - 2*b^(3/2)*x^n - a*sqrt(b)) + 2*
(2*b^(3/2)*x^(3/2*n) - 3*a*sqrt(b)*x^(1/2*n))*sqrt(b*x^n + a))/(b^(5/2)*n), 1/4*
(3*a^2*arctan(sqrt(-b)*x^(1/2*n)/sqrt(b*x^n + a)) + (2*sqrt(-b)*b*x^(3/2*n) - 3*
a*sqrt(-b)*x^(1/2*n))*sqrt(b*x^n + a))/(sqrt(-b)*b^2*n)]

_______________________________________________________________________________________

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(x**(-1+5/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(5/2*n - 1)/sqrt(b*x^n + a), x)