3.2080 \(\int \frac{x^3}{\sqrt{a+\frac{b}{x^4}}} \, dx\)

Optimal. Leaf size=50 \[ \frac{x^4 \sqrt{a+\frac{b}{x^4}}}{4 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0938088, antiderivative size = 50, 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 \[ \frac{x^4 \sqrt{a+\frac{b}{x^4}}}{4 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a + b/x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.99084, size = 41, normalized size = 0.82 \[ \frac{x^{4} \sqrt{a + \frac{b}{x^{4}}}}{4 a} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0616594, size = 78, normalized size = 1.56 \[ \frac{\sqrt{a} x^2 \left (a x^4+b\right )-b \sqrt{a x^4+b} \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )}{4 a^{3/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a + b/x^4],x]

[Out]

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

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Maple [A]  time = 0.02, size = 70, normalized size = 1.4 \[ -{\frac{1}{4\,{x}^{2}}\sqrt{a{x}^{4}+b} \left ( -{x}^{2}\sqrt{a{x}^{4}+b}{a}^{{\frac{3}{2}}}+b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) a \right ){\frac{1}{\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^4)^(1/2),x)

[Out]

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

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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/sqrt(a + b/x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253831, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{a} b \log \left (2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right )}{8 \, a^{2}}, \frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{-a} b \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{4 \, a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*a*x^4*sqrt((a*x^4 + b)/x^4) + sqrt(a)*b*log(2*a*x^4*sqrt((a*x^4 + b)/x^4
) - (2*a*x^4 + b)*sqrt(a)))/a^2, 1/4*(a*x^4*sqrt((a*x^4 + b)/x^4) + sqrt(-a)*b*a
rctan(sqrt(-a)/sqrt((a*x^4 + b)/x^4)))/a^2]

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Sympy [A]  time = 9.0067, size = 46, normalized size = 0.92 \[ \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{4}}{b} + 1}}{4 a} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.256846, size = 90, normalized size = 1.8 \[ \frac{1}{4} \, b{\left (\frac{\arctan \left (\frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{\frac{a x^{4} + b}{x^{4}}}}{{\left (a - \frac{a x^{4} + b}{x^{4}}\right )} a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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