3.406 \(\int \frac{x^5}{\sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 \left (a+b x^3\right )^{3/2}}{9 b^2}-\frac{2 a \sqrt{a+b x^3}}{3 b^2} \]

[Out]

(-2*a*Sqrt[a + b*x^3])/(3*b^2) + (2*(a + b*x^3)^(3/2))/(9*b^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0626581, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 \left (a+b x^3\right )^{3/2}}{9 b^2}-\frac{2 a \sqrt{a+b x^3}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[a + b*x^3],x]

[Out]

(-2*a*Sqrt[a + b*x^3])/(3*b^2) + (2*(a + b*x^3)^(3/2))/(9*b^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.30435, size = 34, normalized size = 0.89 \[ - \frac{2 a \sqrt{a + b x^{3}}}{3 b^{2}} + \frac{2 \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**3+a)**(1/2),x)

[Out]

-2*a*sqrt(a + b*x**3)/(3*b**2) + 2*(a + b*x**3)**(3/2)/(9*b**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0202988, size = 27, normalized size = 0.71 \[ \frac{2 \left (b x^3-2 a\right ) \sqrt{a+b x^3}}{9 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[a + b*x^3],x]

[Out]

(2*(-2*a + b*x^3)*Sqrt[a + b*x^3])/(9*b^2)

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-2\,b{x}^{3}+4\,a}{9\,{b}^{2}}\sqrt{b{x}^{3}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^3+a)^(1/2),x)

[Out]

-2/9*(b*x^3+a)^(1/2)*(-b*x^3+2*a)/b^2

_______________________________________________________________________________________

Maxima [A]  time = 1.43941, size = 41, normalized size = 1.08 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{9 \, b^{2}} - \frac{2 \, \sqrt{b x^{3} + a} a}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(b*x^3 + a),x, algorithm="maxima")

[Out]

2/9*(b*x^3 + a)^(3/2)/b^2 - 2/3*sqrt(b*x^3 + a)*a/b^2

_______________________________________________________________________________________

Fricas [A]  time = 0.231148, size = 31, normalized size = 0.82 \[ \frac{2 \, \sqrt{b x^{3} + a}{\left (b x^{3} - 2 \, a\right )}}{9 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*sqrt(b*x^3 + a)*(b*x^3 - 2*a)/b^2

_______________________________________________________________________________________

Sympy [A]  time = 2.55223, size = 46, normalized size = 1.21 \[ \begin{cases} - \frac{4 a \sqrt{a + b x^{3}}}{9 b^{2}} + \frac{2 x^{3} \sqrt{a + b x^{3}}}{9 b} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**3+a)**(1/2),x)

[Out]

Piecewise((-4*a*sqrt(a + b*x**3)/(9*b**2) + 2*x**3*sqrt(a + b*x**3)/(9*b), Ne(b,
0)), (x**6/(6*sqrt(a)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212993, size = 36, normalized size = 0.95 \[ \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{3} + a} a\right )}}{9 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*((b*x^3 + a)^(3/2) - 3*sqrt(b*x^3 + a)*a)/b^2