3.2682 \(\int \frac{x^{-1+\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=62 \[ \frac{x^{n/2} \sqrt{a+b x^n}}{b n}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} n} \]

[Out]

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

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Rubi [A]  time = 0.071625, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x^{n/2} \sqrt{a+b x^n}}{b n}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{n/2}}{\sqrt{a+b x^n}}\right )}{b^{3/2} n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + (3*n)/2)/Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.1099, size = 63, normalized size = 1.02 \[ \frac{a x^{\frac{n}{2}}}{b n \sqrt{a + b x^{n}} \left (- \frac{b x^{n}}{a + b x^{n}} + 1\right )} - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{n}{2}}}{\sqrt{a + b x^{n}}} \right )}}{b^{\frac{3}{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+3/2*n)/(a+b*x**n)**(1/2),x)

[Out]

a*x**(n/2)/(b*n*sqrt(a + b*x**n)*(-b*x**n/(a + b*x**n) + 1)) - a*atanh(sqrt(b)*x
**(n/2)/sqrt(a + b*x**n))/(b**(3/2)*n)

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Mathematica [A]  time = 0.0503704, size = 65, normalized size = 1.05 \[ \frac{x^{n/2} \sqrt{a+b x^n}}{b n}-\frac{a \log \left (\sqrt{b} \sqrt{a+b x^n}+b x^{n/2}\right )}{b^{3/2} n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + (3*n)/2)/Sqrt[a + b*x^n],x]

[Out]

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

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Maple [A]  time = 0.049, size = 64, normalized size = 1. \[{\frac{1}{bn}{{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}}\sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}}}-{\frac{a}{n}\ln \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}}\sqrt{b}+\sqrt{a+b \left ({{\rm e}^{{\frac{n\ln \left ( x \right ) }{2}}}} \right ) ^{2}} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+3/2*n)/(a+b*x^n)^(1/2),x)

[Out]

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

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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^(3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248847, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{b x^{n} + a} \sqrt{b} x^{\frac{1}{2} \, n} + a \log \left (2 \, \sqrt{b x^{n} + a} b x^{\frac{1}{2} \, n} - 2 \, b^{\frac{3}{2}} x^{n} - a \sqrt{b}\right )}{2 \, b^{\frac{3}{2}} n}, \frac{\sqrt{b x^{n} + a} \sqrt{-b} x^{\frac{1}{2} \, n} - a \arctan \left (\frac{\sqrt{-b} x^{\frac{1}{2} \, n}}{\sqrt{b x^{n} + a}}\right )}{\sqrt{-b} b n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

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

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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+3/2*n)/(a+b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")

[Out]

integrate(x^(3/2*n - 1)/sqrt(b*x^n + a), x)