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

Optimal. Leaf size=54 \[ a^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a \sqrt{a+b x^2}+\frac{1}{3} \left (a+b x^2\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0991253, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ a^{3/2} \left (-\tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )\right )+a \sqrt{a+b x^2}+\frac{1}{3} \left (a+b x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(3/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.44854, size = 44, normalized size = 0.81 \[ - a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + a \sqrt{a + b x^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0603539, size = 62, normalized size = 1.15 \[ -a^{3/2} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+a^{3/2} \log (x)+\left (\frac{4 a}{3}+\frac{b x^2}{3}\right ) \sqrt{a+b x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 52, normalized size = 1. \[{\frac{1}{3} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{a}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) +a\sqrt{b{x}^{2}+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247842, size = 1, normalized size = 0.02 \[ \left [\frac{1}{2} \, a^{\frac{3}{2}} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + \frac{1}{3} \,{\left (b x^{2} + 4 \, a\right )} \sqrt{b x^{2} + a}, -\sqrt{-a} a \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) + \frac{1}{3} \,{\left (b x^{2} + 4 \, a\right )} \sqrt{b x^{2} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.93548, size = 78, normalized size = 1.44 \[ \frac{4 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}}{3} + \frac{a^{\frac{3}{2}} \log{\left (\frac{b x^{2}}{a} \right )}}{2} - a^{\frac{3}{2}} \log{\left (\sqrt{1 + \frac{b x^{2}}{a}} + 1 \right )} + \frac{\sqrt{a} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208983, size = 65, normalized size = 1.2 \[ \frac{a^{2} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{1}{3} \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} + \sqrt{b x^{2} + a} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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