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

Optimal. Leaf size=38 \[ \frac{2 \left (a+b x^3\right )^{5/2}}{15 b^2}-\frac{2 a \left (a+b x^3\right )^{3/2}}{9 b^2} \]

[Out]

(-2*a*(a + b*x^3)^(3/2))/(9*b^2) + (2*(a + b*x^3)^(5/2))/(15*b^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0585054, 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 \[ \frac{2 \left (a+b x^3\right )^{5/2}}{15 b^2}-\frac{2 a \left (a+b x^3\right )^{3/2}}{9 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[a + b*x^3],x]

[Out]

(-2*a*(a + b*x^3)^(3/2))/(9*b^2) + (2*(a + b*x^3)^(5/2))/(15*b^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.35061, size = 34, normalized size = 0.89 \[ - \frac{2 a \left (a + b x^{3}\right )^{\frac{3}{2}}}{9 b^{2}} + \frac{2 \left (a + b x^{3}\right )^{\frac{5}{2}}}{15 b^{2}} \]

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*(a + b*x**3)**(3/2)/(9*b**2) + 2*(a + b*x**3)**(5/2)/(15*b**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0189852, size = 38, normalized size = 1. \[ \frac{2 \sqrt{a+b x^3} \left (-2 a^2+a b x^3+3 b^2 x^6\right )}{45 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x^3]*(-2*a^2 + a*b*x^3 + 3*b^2*x^6))/(45*b^2)

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{-6\,b{x}^{3}+4\,a}{45\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^3+a)^(1/2),x)

[Out]

-2/45*(b*x^3+a)^(3/2)*(-3*b*x^3+2*a)/b^2

_______________________________________________________________________________________

Maxima [A]  time = 1.44205, size = 41, normalized size = 1.08 \[ \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}}}{15 \, b^{2}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a}{9 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(b*x^3 + a)^(5/2)/b^2 - 2/9*(b*x^3 + a)^(3/2)*a/b^2

_______________________________________________________________________________________

Fricas [A]  time = 0.217539, size = 46, normalized size = 1.21 \[ \frac{2 \,{\left (3 \, b^{2} x^{6} + a b x^{3} - 2 \, a^{2}\right )} \sqrt{b x^{3} + a}}{45 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*(3*b^2*x^6 + a*b*x^3 - 2*a^2)*sqrt(b*x^3 + a)/b^2

_______________________________________________________________________________________

Sympy [A]  time = 1.62163, size = 66, normalized size = 1.74 \[ \begin{cases} - \frac{4 a^{2} \sqrt{a + b x^{3}}}{45 b^{2}} + \frac{2 a x^{3} \sqrt{a + b x^{3}}}{45 b} + \frac{2 x^{6} \sqrt{a + b x^{3}}}{15} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases} \]

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**2*sqrt(a + b*x**3)/(45*b**2) + 2*a*x**3*sqrt(a + b*x**3)/(45*b)
+ 2*x**6*sqrt(a + b*x**3)/15, Ne(b, 0)), (sqrt(a)*x**6/6, True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.264269, size = 39, normalized size = 1.03 \[ \frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a\right )}}{45 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*(3*(b*x^3 + a)^(5/2) - 5*(b*x^3 + a)^(3/2)*a)/b^2