3.855 \(\int \frac{x^5}{\left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}-\frac{x^2}{2 b \sqrt{a+b x^4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0744296, antiderivative size = 52, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 b^{3/2}}-\frac{x^2}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^4)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.89916, size = 42, normalized size = 0.81 \[ - \frac{x^{2}}{2 b \sqrt{a + b x^{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{2 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**4+a)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0421878, size = 55, normalized size = 1.06 \[ \frac{\log \left (\sqrt{b} \sqrt{a+b x^4}+b x^2\right )}{2 b^{3/2}}-\frac{x^2}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^4)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 42, normalized size = 0.8 \[ -{\frac{{x}^{2}}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}}+{\frac{1}{2}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^4+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.327374, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, \sqrt{b x^{4} + a} \sqrt{b} x^{2} -{\left (b x^{4} + a\right )} \log \left (-2 \, \sqrt{b x^{4} + a} b x^{2} -{\left (2 \, b x^{4} + a\right )} \sqrt{b}\right )}{4 \,{\left (b^{2} x^{4} + a b\right )} \sqrt{b}}, -\frac{\sqrt{b x^{4} + a} \sqrt{-b} x^{2} -{\left (b x^{4} + a\right )} \arctan \left (\frac{\sqrt{-b} x^{2}}{\sqrt{b x^{4} + a}}\right )}{2 \,{\left (b^{2} x^{4} + a b\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^4 + a)^(3/2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 5.97639, size = 44, normalized size = 0.85 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} - \frac{x^{2}}{2 \sqrt{a} b \sqrt{1 + \frac{b x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**4+a)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.239285, size = 58, normalized size = 1.12 \[ -\frac{x^{2}}{2 \, \sqrt{b x^{4} + a} b} - \frac{{\rm ln}\left ({\left | -\sqrt{b} x^{2} + \sqrt{b x^{4} + a} \right |}\right )}{2 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^4 + a)^(3/2),x, algorithm="giac")

[Out]

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