3.852 \(\int \frac{1}{x \left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{2 a \sqrt{a+b x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}} \]

[Out]

1/(2*a*Sqrt[a + b*x^4]) - ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]/(2*a^(3/2))

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Rubi [A]  time = 0.0715207, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{2 a \sqrt{a+b x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^4)^(3/2)),x]

[Out]

1/(2*a*Sqrt[a + b*x^4]) - ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]/(2*a^(3/2))

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Rubi in Sympy [A]  time = 7.19185, size = 37, normalized size = 0.8 \[ \frac{1}{2 a \sqrt{a + b x^{4}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x^{4}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**4+a)**(3/2),x)

[Out]

1/(2*a*sqrt(a + b*x**4)) - atanh(sqrt(a + b*x**4)/sqrt(a))/(2*a**(3/2))

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Mathematica [A]  time = 0.101797, size = 46, normalized size = 1. \[ \frac{1}{2 a \sqrt{a+b x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b x^4}}{\sqrt{a}}\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^4)^(3/2)),x]

[Out]

1/(2*a*Sqrt[a + b*x^4]) - ArcTanh[Sqrt[a + b*x^4]/Sqrt[a]]/(2*a^(3/2))

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Maple [A]  time = 0.019, size = 44, normalized size = 1. \[{\frac{1}{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}}-{\frac{1}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^4+a)^(3/2),x)

[Out]

1/2/a/(b*x^4+a)^(1/2)-1/2/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^4+a)^(1/2))/x^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(1/((b*x^4 + a)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292922, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b x^{4} + a} \log \left (\frac{{\left (b x^{4} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{4} + a} a}{x^{4}}\right ) + 2 \, \sqrt{a}}{4 \, \sqrt{b x^{4} + a} a^{\frac{3}{2}}}, \frac{\sqrt{b x^{4} + a} \arctan \left (\frac{a}{\sqrt{b x^{4} + a} \sqrt{-a}}\right ) + \sqrt{-a}}{2 \, \sqrt{b x^{4} + a} \sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.84632, size = 184, normalized size = 4. \[ \frac{2 a^{3} \sqrt{1 + \frac{b x^{4}}{a}}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} + \frac{a^{3} \log{\left (\frac{b x^{4}}{a} \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{4}}{a}} + 1 \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} + \frac{a^{2} b x^{4} \log{\left (\frac{b x^{4}}{a} \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} - \frac{2 a^{2} b x^{4} \log{\left (\sqrt{1 + \frac{b x^{4}}{a}} + 1 \right )}}{4 a^{\frac{9}{2}} + 4 a^{\frac{7}{2}} b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**4+a)**(3/2),x)

[Out]

2*a**3*sqrt(1 + b*x**4/a)/(4*a**(9/2) + 4*a**(7/2)*b*x**4) + a**3*log(b*x**4/a)/
(4*a**(9/2) + 4*a**(7/2)*b*x**4) - 2*a**3*log(sqrt(1 + b*x**4/a) + 1)/(4*a**(9/2
) + 4*a**(7/2)*b*x**4) + a**2*b*x**4*log(b*x**4/a)/(4*a**(9/2) + 4*a**(7/2)*b*x*
*4) - 2*a**2*b*x**4*log(sqrt(1 + b*x**4/a) + 1)/(4*a**(9/2) + 4*a**(7/2)*b*x**4)

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GIAC/XCAS [A]  time = 0.215509, size = 55, normalized size = 1.2 \[ \frac{\arctan \left (\frac{\sqrt{b x^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a} a} + \frac{1}{2 \, \sqrt{b x^{4} + a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x),x, algorithm="giac")

[Out]

1/2*arctan(sqrt(b*x^4 + a)/sqrt(-a))/(sqrt(-a)*a) + 1/2/(sqrt(b*x^4 + a)*a)