3.2955 \(\int \frac{\sqrt{a+b \sqrt{c x^3}}}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac{4}{3} \sqrt{a+b \sqrt{c x^3}}-\frac{4}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.105369, 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 \[ \frac{4}{3} \sqrt{a+b \sqrt{c x^3}}-\frac{4}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0336827, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [A]  time = 0.171, size = 40, normalized size = 0.7 \[ -{\frac{4}{3}{\it Artanh} \left ({1\sqrt{a+b\sqrt{c{x}^{3}}}{\frac{1}{\sqrt{a}}}} \right ) \sqrt{a}}+{\frac{4}{3}\sqrt{a+b\sqrt{c{x}^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/3*arctanh((a+b*(c*x^3)^(1/2))^(1/2)/a^(1/2))*a^(1/2)+4/3*(a+b*(c*x^3)^(1/2))^
(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(sqrt(sqrt(c*x^3)*b + a)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223011, size = 124, normalized size = 2.25 \[ \frac{4 \,{\left (\frac{a c \arctan \left (\frac{\sqrt{\sqrt{c x} b c x + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c}} + \sqrt{\sqrt{c x} b c x + a c} - \frac{a c \arctan \left (\frac{\sqrt{a c}}{\sqrt{-a c}}\right ) + \sqrt{a c} \sqrt{-a c}}{\sqrt{-a c}}\right )}{\left | c \right |}}{3 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/3*(a*c*arctan(sqrt(sqrt(c*x)*b*c*x + a*c)/sqrt(-a*c))/sqrt(-a*c) + sqrt(sqrt(c
*x)*b*c*x + a*c) - (a*c*arctan(sqrt(a*c)/sqrt(-a*c)) + sqrt(a*c)*sqrt(-a*c))/sqr
t(-a*c))*abs(c)/c^(3/2)