3.2064 \(\int \left (a+\frac{b}{x^4}\right )^{3/2} x^3 \, dx\)

Optimal. Leaf size=63 \[ \frac{1}{4} x^4 \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{3}{4} b \sqrt{a+\frac{b}{x^4}}+\frac{3}{4} \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.110076, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{4} x^4 \left (a+\frac{b}{x^4}\right )^{3/2}-\frac{3}{4} b \sqrt{a+\frac{b}{x^4}}+\frac{3}{4} \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^4)^(3/2)*x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.87135, size = 56, normalized size = 0.89 \[ \frac{3 \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{4} - \frac{3 b \sqrt{a + \frac{b}{x^{4}}}}{4} + \frac{x^{4} \left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0977778, size = 80, normalized size = 1.27 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (\left (a x^4-2 b\right ) \sqrt{a x^4+b}+3 \sqrt{a} b x^2 \tanh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{a x^4+b}}\right )\right )}{4 \sqrt{a x^4+b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.028, size = 82, normalized size = 1.3 \[{\frac{{x}^{4}}{4} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( a{x}^{4}\sqrt{a{x}^{4}+b}+3\,\sqrt{a}\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ){x}^{2}b-2\,b\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^4)^(3/2)*x^3,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255415, size = 1, normalized size = 0.02 \[ \left [\frac{3}{8} \, \sqrt{a} b \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right ) + \frac{1}{4} \,{\left (a x^{4} - 2 \, b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}, \frac{3}{4} \, \sqrt{-a} b \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) + \frac{1}{4} \,{\left (a x^{4} - 2 \, b\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 14.6168, size = 95, normalized size = 1.51 \[ \frac{3 \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4} + \frac{a^{2} x^{6}}{4 \sqrt{b} \sqrt{\frac{a x^{4}}{b} + 1}} - \frac{a \sqrt{b} x^{2}}{4 \sqrt{\frac{a x^{4}}{b} + 1}} - \frac{b^{\frac{3}{2}}}{2 x^{2} \sqrt{\frac{a x^{4}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(a)*b*asinh(sqrt(a)*x**2/sqrt(b))/4 + a**2*x**6/(4*sqrt(b)*sqrt(a*x**4/b +
1)) - a*sqrt(b)*x**2/(4*sqrt(a*x**4/b + 1)) - b**(3/2)/(2*x**2*sqrt(a*x**4/b +
1))

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GIAC/XCAS [A]  time = 0.246456, size = 105, normalized size = 1.67 \[ \frac{1}{4} \, \sqrt{a x^{4} + b} a x^{2} - \frac{3}{8} \, \sqrt{a} b{\rm ln}\left ({\left (\sqrt{a} x^{2} - \sqrt{a x^{4} + b}\right )}^{2}\right ) + \frac{\sqrt{a} b^{2}}{{\left (\sqrt{a} x^{2} - \sqrt{a x^{4} + b}\right )}^{2} - b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(a*x^4 + b)*a*x^2 - 3/8*sqrt(a)*b*ln((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2)
+ sqrt(a)*b^2/((sqrt(a)*x^2 - sqrt(a*x^4 + b))^2 - b)