3.2938 \(\int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac{2}{3} \sqrt{a+b \left (c x^2\right )^{3/2}}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.109177, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{2}{3} \sqrt{a+b \left (c x^2\right )^{3/2}}-\frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c*x^2)^(3/2)]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.67446, size = 48, normalized size = 0.87 \[ - \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{\sqrt{a}} \right )}}{3} + \frac{2 \sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0467946, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(3/2))^(1/2)/x,x)

[Out]

int((a+b*(c*x^2)^(3/2))^(1/2)/x,x)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223305, size = 1, normalized size = 0.02 \[ \left [\frac{1}{3} \, \sqrt{a} \log \left (\frac{\sqrt{c x^{2}} b c x^{2} - 2 \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + \frac{2}{3} \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a}, -\frac{2}{3} \, \sqrt{-a} \arctan \left (\frac{\sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{\sqrt{-a}}\right ) + \frac{2}{3} \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*(c*x**2)**(3/2))/x, x)

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GIAC/XCAS [A]  time = 0.217552, size = 57, normalized size = 1.04 \[ \frac{2 \, a \arctan \left (\frac{\sqrt{b c^{\frac{3}{2}} x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2}{3} \, \sqrt{b c^{\frac{3}{2}} x^{3} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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