3.834 \(\int \frac{x^5}{\sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=55 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{4 b^{3/2}}-\frac{x^2 \sqrt{a-b x^4}}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0814901, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{4 b^{3/2}}-\frac{x^2 \sqrt{a-b x^4}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[a - b*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 8.75837, size = 44, normalized size = 0.8 \[ \frac{a \operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} \right )}}{4 b^{\frac{3}{2}}} - \frac{x^{2} \sqrt{a - b x^{4}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-b*x**4+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0556924, size = 55, normalized size = 1. \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{4 b^{3/2}}-\frac{x^2 \sqrt{a-b x^4}}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[a - b*x^4],x]

[Out]

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

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Maple [A]  time = 0.019, size = 44, normalized size = 0.8 \[{\frac{a}{4}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){b}^{-{\frac{3}{2}}}}-{\frac{{x}^{2}}{4\,b}\sqrt{-b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-b*x^4+a)^(1/2),x)

[Out]

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

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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^5/sqrt(-b*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260498, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{-b x^{4} + a} \sqrt{-b} x^{2} - a \log \left (2 \, \sqrt{-b x^{4} + a} b x^{2} +{\left (2 \, b x^{4} - a\right )} \sqrt{-b}\right )}{8 \, \sqrt{-b} b}, -\frac{\sqrt{-b x^{4} + a} \sqrt{b} x^{2} - a \arctan \left (\frac{\sqrt{b} x^{2}}{\sqrt{-b x^{4} + a}}\right )}{4 \, b^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-b*x^4 + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.83894, size = 128, normalized size = 2.33 \[ \begin{cases} - \frac{i \sqrt{a} x^{2} \sqrt{-1 + \frac{b x^{4}}{a}}}{4 b} - \frac{i a \operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} & \text{for}\: \left |{\frac{b x^{4}}{a}}\right | > 1 \\- \frac{\sqrt{a} x^{2}}{4 b \sqrt{1 - \frac{b x^{4}}{a}}} + \frac{a \operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} + \frac{x^{6}}{4 \sqrt{a} \sqrt{1 - \frac{b x^{4}}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-I*sqrt(a)*x**2*sqrt(-1 + b*x**4/a)/(4*b) - I*a*acosh(sqrt(b)*x**2/sq
rt(a))/(4*b**(3/2)), Abs(b*x**4/a) > 1), (-sqrt(a)*x**2/(4*b*sqrt(1 - b*x**4/a))
+ a*asin(sqrt(b)*x**2/sqrt(a))/(4*b**(3/2)) + x**6/(4*sqrt(a)*sqrt(1 - b*x**4/a
)), True))

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GIAC/XCAS [A]  time = 0.235467, size = 72, normalized size = 1.31 \[ -\frac{\sqrt{-b x^{4} + a} x^{2}}{4 \, b} - \frac{a{\rm ln}\left ({\left | -\sqrt{-b} x^{2} + \sqrt{-b x^{4} + a} \right |}\right )}{4 \, \sqrt{-b} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-b*x^4 + a),x, algorithm="giac")

[Out]

-1/4*sqrt(-b*x^4 + a)*x^2/b - 1/4*a*ln(abs(-sqrt(-b)*x^2 + sqrt(-b*x^4 + a)))/(s
qrt(-b)*b)