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

Optimal. Leaf size=55 \[ -\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0906742, antiderivative size = 55, 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 \[ -\frac{a^2}{b^3 \sqrt{a+b x^2}}-\frac{2 a \sqrt{a+b x^2}}{b^3}+\frac{\left (a+b x^2\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^2)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.6797, size = 48, normalized size = 0.87 \[ - \frac{a^{2}}{b^{3} \sqrt{a + b x^{2}}} - \frac{2 a \sqrt{a + b x^{2}}}{b^{3}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0281339, size = 38, normalized size = 0.69 \[ \frac{-8 a^2-4 a b x^2+b^2 x^4}{3 b^3 \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-8*a^2 - 4*a*b*x^2 + b^2*x^4)/(3*b^3*Sqrt[a + b*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 36, normalized size = 0.7 \[ -{\frac{-{b}^{2}{x}^{4}+4\,ab{x}^{2}+8\,{a}^{2}}{3\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)^(3/2),x)

[Out]

-1/3*(-b^2*x^4+4*a*b*x^2+8*a^2)/(b*x^2+a)^(1/2)/b^3

_______________________________________________________________________________________

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^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.226775, size = 62, normalized size = 1.13 \[ \frac{{\left (b^{2} x^{4} - 4 \, a b x^{2} - 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b^2*x^4 - 4*a*b*x^2 - 8*a^2)*sqrt(b*x^2 + a)/(b^4*x^2 + a*b^3)

_______________________________________________________________________________________

Sympy [A]  time = 3.01689, size = 68, normalized size = 1.24 \[ \begin{cases} - \frac{8 a^{2}}{3 b^{3} \sqrt{a + b x^{2}}} - \frac{4 a x^{2}}{3 b^{2} \sqrt{a + b x^{2}}} + \frac{x^{4}}{3 b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-8*a**2/(3*b**3*sqrt(a + b*x**2)) - 4*a*x**2/(3*b**2*sqrt(a + b*x**2)
) + x**4/(3*b*sqrt(a + b*x**2)), Ne(b, 0)), (x**6/(6*a**(3/2)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220025, size = 55, normalized size = 1. \[ \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{b x^{2} + a} a - \frac{3 \, a^{2}}{\sqrt{b x^{2} + a}}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b*x^2 + a)^(3/2) - 6*sqrt(b*x^2 + a)*a - 3*a^2/sqrt(b*x^2 + a))/b^3