3.426 \(\int \frac{1}{x^4 \left (a+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{\sqrt{a+b x^3}}{a^2 x^3}+\frac{2}{3 a x^3 \sqrt{a+b x^3}} \]

[Out]

2/(3*a*x^3*Sqrt[a + b*x^3]) - Sqrt[a + b*x^3]/(a^2*x^3) + (b*ArcTanh[Sqrt[a + b*
x^3]/Sqrt[a]])/a^(5/2)

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Rubi [A]  time = 0.102586, antiderivative size = 66, 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 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{\sqrt{a+b x^3}}{a^2 x^3}+\frac{2}{3 a x^3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^3)^(3/2)),x]

[Out]

2/(3*a*x^3*Sqrt[a + b*x^3]) - Sqrt[a + b*x^3]/(a^2*x^3) + (b*ArcTanh[Sqrt[a + b*
x^3]/Sqrt[a]])/a^(5/2)

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Rubi in Sympy [A]  time = 10.1661, size = 58, normalized size = 0.88 \[ \frac{2}{3 a x^{3} \sqrt{a + b x^{3}}} - \frac{\sqrt{a + b x^{3}}}{a^{2} x^{3}} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**3+a)**(3/2),x)

[Out]

2/(3*a*x**3*sqrt(a + b*x**3)) - sqrt(a + b*x**3)/(a**2*x**3) + b*atanh(sqrt(a +
b*x**3)/sqrt(a))/a**(5/2)

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Mathematica [A]  time = 0.338247, size = 64, normalized size = 0.97 \[ -\frac{-3 b x^3 \sqrt{\frac{b x^3}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+a+3 b x^3}{3 a^2 x^3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^3)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.033, size = 57, normalized size = 0.9 \[ -{\frac{1}{3\,{x}^{3}{a}^{2}}\sqrt{b{x}^{3}+a}}-{\frac{2\,b}{3\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}+{b{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^3+a)^(3/2),x)

[Out]

-1/3*(b*x^3+a)^(1/2)/x^3/a^2-2/3*b/a^2/((x^3+a/b)*b)^(1/2)+b*arctanh((b*x^3+a)^(
1/2)/a^(1/2))/a^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244232, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{b x^{3} + a} b x^{3} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) - 2 \,{\left (3 \, b x^{3} + a\right )} \sqrt{a}}{6 \, \sqrt{b x^{3} + a} a^{\frac{5}{2}} x^{3}}, -\frac{3 \, \sqrt{b x^{3} + a} b x^{3} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (3 \, b x^{3} + a\right )} \sqrt{-a}}{3 \, \sqrt{b x^{3} + a} \sqrt{-a} a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(3/2)*x^4),x, algorithm="fricas")

[Out]

[1/6*(3*sqrt(b*x^3 + a)*b*x^3*log(((b*x^3 + 2*a)*sqrt(a) + 2*sqrt(b*x^3 + a)*a)/
x^3) - 2*(3*b*x^3 + a)*sqrt(a))/(sqrt(b*x^3 + a)*a^(5/2)*x^3), -1/3*(3*sqrt(b*x^
3 + a)*b*x^3*arctan(a/(sqrt(b*x^3 + a)*sqrt(-a))) + (3*b*x^3 + a)*sqrt(-a))/(sqr
t(b*x^3 + a)*sqrt(-a)*a^2*x^3)]

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Sympy [A]  time = 11.9643, size = 75, normalized size = 1.14 \[ - \frac{1}{3 a \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b}}{a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**3+a)**(3/2),x)

[Out]

-1/(3*a*sqrt(b)*x**(9/2)*sqrt(a/(b*x**3) + 1)) - sqrt(b)/(a**2*x**(3/2)*sqrt(a/(
b*x**3) + 1)) + b*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/a**(5/2)

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GIAC/XCAS [A]  time = 0.212928, size = 89, normalized size = 1.35 \[ -\frac{1}{3} \, b{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \, b x^{3} + a}{{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{3} + a} a\right )} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(3/2)*x^4),x, algorithm="giac")

[Out]

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