3.2037 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x} \, dx\)
Optimal. Leaf size=46 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{2}{3 a \sqrt{a+\frac{b}{x^3}}} \]
[Out]
-2/(3*a*Sqrt[a + b/x^3]) + (2*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/(3*a^(3/2))
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Rubi [A] time = 0.0831117, antiderivative size = 46, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267
\[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{2}{3 a \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^3)^(3/2)*x),x]
[Out]
-2/(3*a*Sqrt[a + b/x^3]) + (2*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/(3*a^(3/2))
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Rubi in Sympy [A] time = 7.10508, size = 39, normalized size = 0.85 \[ - \frac{2}{3 a \sqrt{a + \frac{b}{x^{3}}}} + \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**3)**(3/2)/x,x)
[Out]
-2/(3*a*sqrt(a + b/x**3)) + 2*atanh(sqrt(a + b/x**3)/sqrt(a))/(3*a**(3/2))
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Mathematica [A] time = 0.0503692, size = 73, normalized size = 1.59 \[ -\frac{2 \left (\sqrt{a} x^{3/2}-\sqrt{a x^3+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )\right )}{3 a^{3/2} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^3)^(3/2)*x),x]
[Out]
(-2*(Sqrt[a]*x^(3/2) - Sqrt[b + a*x^3]*ArcTanh[(Sqrt[a]*x^(3/2))/Sqrt[b + a*x^3]
]))/(3*a^(3/2)*Sqrt[a + b/x^3]*x^(3/2))
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Maple [C] time = 0.023, size = 3438, normalized size = 74.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^3)^(3/2)/x,x)
[Out]
-2/3/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/a^3*(6*I*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)
-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3)
)/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-
a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-
3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(
I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2-6*I*(-(I*
3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(
1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2
)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2
)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),(I*3
^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3)
)^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2-12*I*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)
*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x
+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)
^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*Elliptic
F((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*
(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*
a+12*I*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(
1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*
b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/
(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+
(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/
(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a+6*I*(-a^2*b)
^(2/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2
)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2
)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^
(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))
^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*
(x*(a*x^3+b))^(1/2)-6*I*(-a^2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(
-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2
)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3
))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*
3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+
3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)
-6*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-
a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((
I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3
)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/
2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b)
)^(1/2)*x^2*a^2+6*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)
*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(
1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*
x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^
2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3
^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(1/2)*x^2*a^2+12*(-a^2*b)^(1/3)*(-
(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b
)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(
1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(
1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((
I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(1/
2)*x*a-12*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))
)^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a
^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-
1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a
*x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-
1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(1/2)*x*a-6*(-a^2*b)^(2/3)*
(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2
*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3
^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))
^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),
((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(
1/2)+6*(-a^2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(
1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*
b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/
(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+
(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/
(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(1/2)+I*(1/a^2*x*(-a*x+(-a^2*b
)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/
3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*3^(1/2)*x^2*a^2-3*x^2*a^2*(1/a^2*x*(-a*x+(-a^2*b
)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/
3)-2*a*x-(-a^2*b)^(1/3)))^(1/2))/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I
*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-
a^2*b)^(1/3)))^(1/2)
_______________________________________________________________________________________
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/((a + b/x^3)^(3/2)*x),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.375085, size = 1, normalized size = 0.02 \[ \left [-\frac{4 \, a x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}} -{\left (a x^{3} + b\right )} \sqrt{a} \log \left (-{\left (8 \, a^{2} x^{6} + 8 \, a b x^{3} + b^{2}\right )} \sqrt{a} - 4 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right )}{6 \,{\left (a^{3} x^{3} + a^{2} b\right )}}, -\frac{2 \, a x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}} +{\left (a x^{3} + b\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{3 \,{\left (a^{3} x^{3} + a^{2} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^3)^(3/2)*x),x, algorithm="fricas")
[Out]
[-1/6*(4*a*x^3*sqrt((a*x^3 + b)/x^3) - (a*x^3 + b)*sqrt(a)*log(-(8*a^2*x^6 + 8*a
*b*x^3 + b^2)*sqrt(a) - 4*(2*a^2*x^6 + a*b*x^3)*sqrt((a*x^3 + b)/x^3)))/(a^3*x^3
+ a^2*b), -1/3*(2*a*x^3*sqrt((a*x^3 + b)/x^3) + (a*x^3 + b)*sqrt(-a)*arctan(2*s
qrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b)))/(a^3*x^3 + a^2*b)]
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Sympy [A] time = 7.44187, size = 187, normalized size = 4.07 \[ - \frac{2 a^{3} x^{3} \sqrt{1 + \frac{b}{a x^{3}}}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} - \frac{a^{3} x^{3} \log{\left (\frac{b}{a x^{3}} \right )}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} + \frac{2 a^{3} x^{3} \log{\left (\sqrt{1 + \frac{b}{a x^{3}}} + 1 \right )}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x^{3}} \right )}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x^{3}}} + 1 \right )}}{3 a^{\frac{9}{2}} x^{3} + 3 a^{\frac{7}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**3)**(3/2)/x,x)
[Out]
-2*a**3*x**3*sqrt(1 + b/(a*x**3))/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) - a**3*x**3*l
og(b/(a*x**3))/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) + 2*a**3*x**3*log(sqrt(1 + b/(a*
x**3)) + 1)/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) - a**2*b*log(b/(a*x**3))/(3*a**(9/2
)*x**3 + 3*a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x**3)) + 1)/(3*a**(9/2)*x**3
+ 3*a**(7/2)*b)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^3)^(3/2)*x),x, algorithm="giac")
[Out]
integrate(1/((a + b/x^3)^(3/2)*x), x)