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

Optimal. Leaf size=49 \[ \frac{x \sqrt{a+b x^2}}{2 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0449893, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x \sqrt{a+b x^2}}{2 b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{2 b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[a + b*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.8361, size = 41, normalized size = 0.84 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2 b^{\frac{3}{2}}} + \frac{x \sqrt{a + b x^{2}}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0289751, size = 52, normalized size = 1.06 \[ \frac{x \sqrt{a+b x^2}}{2 b}-\frac{a \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{2 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 39, normalized size = 0.8 \[{\frac{x}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{a}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+a)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235486, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{b x^{2} + a} \sqrt{b} x + a \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{4 \, b^{\frac{3}{2}}}, \frac{\sqrt{b x^{2} + a} \sqrt{-b} x - a \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{2 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*sqrt(b*x^2 + a)*sqrt(b)*x + a*log(2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*
sqrt(b)))/b^(3/2), 1/2*(sqrt(b*x^2 + a)*sqrt(-b)*x - a*arctan(sqrt(-b)*x/sqrt(b*
x^2 + a)))/(sqrt(-b)*b)]

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Sympy [A]  time = 7.30377, size = 42, normalized size = 0.86 \[ \frac{\sqrt{a} x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209541, size = 54, normalized size = 1.1 \[ \frac{\sqrt{b x^{2} + a} x}{2 \, b} + \frac{a{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{2 \, b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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