3.2965 \(\int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac{4}{9} \sqrt{a+b \left (c x^3\right )^{3/2}}-\frac{4}{9} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right ) \]

[Out]

(4*Sqrt[a + b*(c*x^3)^(3/2)])/9 - (4*Sqrt[a]*ArcTanh[Sqrt[a + b*(c*x^3)^(3/2)]/S
qrt[a]])/9

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Rubi [A]  time = 0.0977762, 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 \[ \frac{4}{9} \sqrt{a+b \left (c x^3\right )^{3/2}}-\frac{4}{9} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*(c*x^3)^(3/2)])/9 - (4*Sqrt[a]*ArcTanh[Sqrt[a + b*(c*x^3)^(3/2)]/S
qrt[a]])/9

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0472468, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^3\right )^{3/2}}}{x} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt{a+b \left ( c{x}^{3} \right ) ^{{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22673, size = 159, normalized size = 2.89 \[ \frac{4 \,{\left (\frac{a c^{2} \arctan \left (\frac{\sqrt{\sqrt{c x} b c^{4} x^{4} + a c^{3}}}{\sqrt{-a c} c}\right )}{\sqrt{-a c}} + \sqrt{\sqrt{c x} b c^{4} x^{4} + a c^{3}} - \frac{a c^{2} \arctan \left (\frac{\sqrt{a c^{3}}}{\sqrt{-a c} c}\right ) + \sqrt{a c^{3}} \sqrt{-a c}}{\sqrt{-a c}}\right )}{\left | c \right |}}{9 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/9*(a*c^2*arctan(sqrt(sqrt(c*x)*b*c^4*x^4 + a*c^3)/(sqrt(-a*c)*c))/sqrt(-a*c) +
sqrt(sqrt(c*x)*b*c^4*x^4 + a*c^3) - (a*c^2*arctan(sqrt(a*c^3)/(sqrt(-a*c)*c)) +
sqrt(a*c^3)*sqrt(-a*c))/sqrt(-a*c))*abs(c)/c^(5/2)