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

Optimal. Leaf size=18 \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]

[Out]

-1/(2*b*Sqrt[a + b*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.0104478, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^4)^(3/2),x]

[Out]

-1/(2*b*Sqrt[a + b*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.13264, size = 15, normalized size = 0.83 \[ - \frac{1}{2 b \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*b*sqrt(a + b*x**4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.00953613, size = 18, normalized size = 1. \[ -\frac{1}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*b*Sqrt[a + b*x^4])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 15, normalized size = 0.8 \[ -{\frac{1}{2\,b}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^4+a)^(3/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.44309, size = 19, normalized size = 1.06 \[ -\frac{1}{2 \, \sqrt{b x^{4} + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(sqrt(b*x^4 + a)*b)

_______________________________________________________________________________________

Fricas [A]  time = 0.264633, size = 19, normalized size = 1.06 \[ -\frac{1}{2 \, \sqrt{b x^{4} + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(sqrt(b*x^4 + a)*b)

_______________________________________________________________________________________

Sympy [A]  time = 1.90867, size = 26, normalized size = 1.44 \[ \begin{cases} - \frac{1}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-1/(2*b*sqrt(a + b*x**4)), Ne(b, 0)), (x**4/(4*a**(3/2)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217604, size = 19, normalized size = 1.06 \[ -\frac{1}{2 \, \sqrt{b x^{4} + a} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/(sqrt(b*x^4 + a)*b)