3.362 \(\int x^2 \sqrt{a+b x^2} \, dx\)

Optimal. Leaf size=70 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{a x \sqrt{a+b x^2}}{8 b}+\frac{1}{4} x^3 \sqrt{a+b x^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0727705, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{8 b^{3/2}}+\frac{a x \sqrt{a+b x^2}}{8 b}+\frac{1}{4} x^3 \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[a + b*x^2],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.28896, size = 60, normalized size = 0.86 \[ - \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{8 b^{\frac{3}{2}}} + \frac{a x \sqrt{a + b x^{2}}}{8 b} + \frac{x^{3} \sqrt{a + b x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0377583, size = 64, normalized size = 0.91 \[ \sqrt{a+b x^2} \left (\frac{a x}{8 b}+\frac{x^3}{4}\right )-\frac{a^2 \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{8 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 57, normalized size = 0.8 \[{\frac{x}{4\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{ax}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{{a}^{2}}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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(sqrt(b*x^2 + a)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.246921, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (2 \, b x^{3} + a x\right )} \sqrt{b x^{2} + a} \sqrt{b}}{16 \, b^{\frac{3}{2}}}, -\frac{a^{2} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (2 \, b x^{3} + a x\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{8 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(a^2*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(2*b*x^3 + a*x
)*sqrt(b*x^2 + a)*sqrt(b))/b^(3/2), -1/8*(a^2*arctan(sqrt(-b)*x/sqrt(b*x^2 + a))
- (2*b*x^3 + a*x)*sqrt(b*x^2 + a)*sqrt(-b))/(sqrt(-b)*b)]

_______________________________________________________________________________________

Sympy [A]  time = 11.5336, size = 92, normalized size = 1.31 \[ \frac{a^{\frac{3}{2}} x}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 \sqrt{a} x^{3}}{8 \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{b x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*x/(8*b*sqrt(1 + b*x**2/a)) + 3*sqrt(a)*x**3/(8*sqrt(1 + b*x**2/a)) - a*
*2*asinh(sqrt(b)*x/sqrt(a))/(8*b**(3/2)) + b*x**5/(4*sqrt(a)*sqrt(1 + b*x**2/a))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212077, size = 68, normalized size = 0.97 \[ \frac{1}{8} \, \sqrt{b x^{2} + a}{\left (2 \, x^{2} + \frac{a}{b}\right )} x + \frac{a^{2}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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