3.392 \(\int \frac{\left (a+b x^3\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=58 \[ -\frac{\left (a+b x^3\right )^{3/2}}{3 x^3}+b \sqrt{a+b x^3}-\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0917193, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{\left (a+b x^3\right )^{3/2}}{3 x^3}+b \sqrt{a+b x^3}-\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(3/2)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.43797, size = 49, normalized size = 0.84 \[ - \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )} + b \sqrt{a + b x^{3}} - \frac{\left (a + b x^{3}\right )^{\frac{3}{2}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(3/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.18158, size = 58, normalized size = 1. \[ \sqrt{a+b x^3} \left (-\frac{b \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a}{3 x^3}+\frac{2 b}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(3/2)/x^4,x]

[Out]

Sqrt[a + b*x^3]*((2*b)/3 - a/(3*x^3) - (b*ArcTanh[Sqrt[1 + (b*x^3)/a]])/Sqrt[1 +
(b*x^3)/a])

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Maple [A]  time = 0.027, size = 49, normalized size = 0.8 \[ -{\frac{a}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{2\,b}{3}\sqrt{b{x}^{3}+a}}-b{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) \sqrt{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(3/2)/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222428, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{a} b x^{3} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \,{\left (2 \, b x^{3} - a\right )} \sqrt{b x^{3} + a}}{6 \, x^{3}}, -\frac{3 \, \sqrt{-a} b x^{3} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right ) -{\left (2 \, b x^{3} - a\right )} \sqrt{b x^{3} + a}}{3 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(3/2)/x^4,x, algorithm="fricas")

[Out]

[1/6*(3*sqrt(a)*b*x^3*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3) + 2*(2*
b*x^3 - a)*sqrt(b*x^3 + a))/x^3, -1/3*(3*sqrt(-a)*b*x^3*arctan(sqrt(b*x^3 + a)/s
qrt(-a)) - (2*b*x^3 - a)*sqrt(b*x^3 + a))/x^3]

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Sympy [A]  time = 8.82524, size = 100, normalized size = 1.72 \[ - \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )} - \frac{a^{2}}{3 \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{a \sqrt{b}}{3 x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 b^{\frac{3}{2}} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(3/2)/x**4,x)

[Out]

-sqrt(a)*b*asinh(sqrt(a)/(sqrt(b)*x**(3/2))) - a**2/(3*sqrt(b)*x**(9/2)*sqrt(a/(
b*x**3) + 1)) + a*sqrt(b)/(3*x**(3/2)*sqrt(a/(b*x**3) + 1)) + 2*b**(3/2)*x**(3/2
)/(3*sqrt(a/(b*x**3) + 1))

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GIAC/XCAS [A]  time = 0.252048, size = 77, normalized size = 1.33 \[ \frac{1}{3} \,{\left (\frac{3 \, a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b x^{3} + a} - \frac{\sqrt{b x^{3} + a} a}{b x^{3}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(3/2)/x^4,x, algorithm="giac")

[Out]

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