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

Optimal. Leaf size=63 \[ -\frac{\left (a+b x^2\right )^{3/2}}{x}+\frac{3}{2} b x \sqrt{a+b x^2}+\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0534704, antiderivative size = 63, 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 \[ -\frac{\left (a+b x^2\right )^{3/2}}{x}+\frac{3}{2} b x \sqrt{a+b x^2}+\frac{3}{2} a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(3/2)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.09879, size = 56, normalized size = 0.89 \[ \frac{3 a \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{2} + \frac{3 b x \sqrt{a + b x^{2}}}{2} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0449941, size = 58, normalized size = 0.92 \[ \sqrt{a+b x^2} \left (\frac{b x}{2}-\frac{a}{x}\right )+\frac{3}{2} a \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 69, normalized size = 1.1 \[ -{\frac{1}{ax} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{bx}{a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,bx}{2}\sqrt{b{x}^{2}+a}}+{\frac{3\,a}{2}\sqrt{b}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)/x^2,x)

[Out]

-1/a/x*(b*x^2+a)^(5/2)+b/a*x*(b*x^2+a)^(3/2)+3/2*b*x*(b*x^2+a)^(1/2)+3/2*b^(1/2)
*a*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((b*x^2 + a)^(3/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247236, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) + 2 \, \sqrt{b x^{2} + a}{\left (b x^{2} - 2 \, a\right )}}{4 \, x}, \frac{3 \, a \sqrt{-b} x \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) + \sqrt{b x^{2} + a}{\left (b x^{2} - 2 \, a\right )}}{2 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.38128, size = 88, normalized size = 1.4 \[ - \frac{a^{\frac{3}{2}}}{x \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} b x}{2 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2} + \frac{b^{2} x^{3}}{2 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217496, size = 99, normalized size = 1.57 \[ \frac{1}{2} \, \sqrt{b x^{2} + a} b x - \frac{3}{4} \, a \sqrt{b}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \, a^{2} \sqrt{b}}{{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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