3.487 \(\int \frac{x^3}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{a \sqrt{a+b x^2}}{b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0669123, antiderivative size = 36, 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 \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{a \sqrt{a+b x^2}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a + b*x^2],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.92724, size = 29, normalized size = 0.81 \[ - \frac{a \sqrt{a + b x^{2}}}{b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0199465, size = 27, normalized size = 0.75 \[ \frac{\left (b x^2-2 a\right ) \sqrt{a+b x^2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a + b*x^2],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 25, normalized size = 0.7 \[ -{\frac{-b{x}^{2}+2\,a}{3\,{b}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^2+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.226963, size = 31, normalized size = 0.86 \[ \frac{\sqrt{b x^{2} + a}{\left (b x^{2} - 2 \, a\right )}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(b*x^2 + a)*(b*x^2 - 2*a)/b^2

_______________________________________________________________________________________

Sympy [A]  time = 1.74897, size = 44, normalized size = 1.22 \[ \begin{cases} - \frac{2 a \sqrt{a + b x^{2}}}{3 b^{2}} + \frac{x^{2} \sqrt{a + b x^{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.200588, size = 36, normalized size = 1. \[ \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{2} + a} a}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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