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

Optimal. Leaf size=53 \[ \frac{x^2 \sqrt{a+b x^4}}{4 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0764817, antiderivative size = 53, 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 \[ \frac{x^2 \sqrt{a+b x^4}}{4 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{4 b^{3/2}} \]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.04486, size = 44, normalized size = 0.83 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{4 b^{\frac{3}{2}}} + \frac{x^{2} \sqrt{a + b x^{4}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0428265, size = 56, normalized size = 1.06 \[ \frac{x^2 \sqrt{a+b x^4}}{4 b}-\frac{a \log \left (\sqrt{b} \sqrt{a+b x^4}+b x^2\right )}{4 b^{3/2}} \]

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

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 43, normalized size = 0.8 \[{\frac{{x}^{2}}{4\,b}\sqrt{b{x}^{4}+a}}-{\frac{a}{4}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^4+a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Fricas [A]  time = 0.262007, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{b x^{4} + a} \sqrt{b} x^{2} + a \log \left (2 \, \sqrt{b x^{4} + a} b x^{2} -{\left (2 \, b x^{4} + a\right )} \sqrt{b}\right )}{8 \, b^{\frac{3}{2}}}, \frac{\sqrt{b x^{4} + a} \sqrt{-b} x^{2} - a \arctan \left (\frac{\sqrt{-b} x^{2}}{\sqrt{b x^{4} + a}}\right )}{4 \, \sqrt{-b} b}\right ] \]

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

_______________________________________________________________________________________

Sympy [A]  time = 7.68053, size = 46, normalized size = 0.87 \[ \frac{\sqrt{a} x^{2} \sqrt{1 + \frac{b x^{4}}{a}}}{4 b} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{4 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.231578, size = 59, normalized size = 1.11 \[ \frac{\sqrt{b x^{4} + a} x^{2}}{4 \, b} + \frac{a{\rm ln}\left ({\left | -\sqrt{b} x^{2} + \sqrt{b x^{4} + a} \right |}\right )}{4 \, b^{\frac{3}{2}}} \]

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)))/b^(3/2
)