3.1308 \(\int \frac{x^{3/2}}{\sqrt{a+b x^5}} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]

[Out]

(2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*Sqrt[b])

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Rubi [A]  time = 0.0651626, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)/Sqrt[a + b*x^5],x]

[Out]

(2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*Sqrt[b])

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Rubi in Sympy [A]  time = 6.80623, size = 29, normalized size = 0.91 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{b} x^{\frac{5}{2}}}{\sqrt{a + b x^{5}}} \right )}}{5 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)/(b*x**5+a)**(1/2),x)

[Out]

2*atanh(sqrt(b)*x**(5/2)/sqrt(a + b*x**5))/(5*sqrt(b))

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Mathematica [A]  time = 0.0604771, size = 32, normalized size = 1. \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} x^{5/2}}{\sqrt{a+b x^5}}\right )}{5 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)/Sqrt[a + b*x^5],x]

[Out]

(2*ArcTanh[(Sqrt[b]*x^(5/2))/Sqrt[a + b*x^5]])/(5*Sqrt[b])

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{1{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{b{x}^{5}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)/(b*x^5+a)^(1/2),x)

[Out]

int(x^(3/2)/(b*x^5+a)^(1/2),x)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.690606, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-4 \,{\left (2 \, b^{2} x^{7} + a b x^{2}\right )} \sqrt{b x^{5} + a} \sqrt{x} -{\left (8 \, b^{2} x^{10} + 8 \, a b x^{5} + a^{2}\right )} \sqrt{b}\right )}{10 \, \sqrt{b}}, \frac{\arctan \left (\frac{2 \, \sqrt{b x^{5} + a} \sqrt{-b} x^{\frac{5}{2}}}{2 \, b x^{5} + a}\right )}{5 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(b*x^5 + a),x, algorithm="fricas")

[Out]

[1/10*log(-4*(2*b^2*x^7 + a*b*x^2)*sqrt(b*x^5 + a)*sqrt(x) - (8*b^2*x^10 + 8*a*b
*x^5 + a^2)*sqrt(b))/sqrt(b), 1/5*arctan(2*sqrt(b*x^5 + a)*sqrt(-b)*x^(5/2)/(2*b
*x^5 + a))/sqrt(-b)]

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Sympy [A]  time = 6.80554, size = 24, normalized size = 0.75 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{5}{2}}}{\sqrt{a}} \right )}}{5 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)/(b*x**5+a)**(1/2),x)

[Out]

2*asinh(sqrt(b)*x**(5/2)/sqrt(a))/(5*sqrt(b))

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GIAC/XCAS [A]  time = 0.23552, size = 55, normalized size = 1.72 \[ -\frac{2 \, \arctan \left (\frac{\sqrt{b + \frac{a}{x^{5}}}}{\sqrt{-b}}\right )}{5 \, \sqrt{-b}} + \frac{2 \, \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right )}{5 \, \sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(3/2)/sqrt(b*x^5 + a),x, algorithm="giac")

[Out]

-2/5*arctan(sqrt(b + a/x^5)/sqrt(-b))/sqrt(-b) + 2/5*arctan(sqrt(b)/sqrt(-b))/sq
rt(-b)