3.2009 \(\int \left (a+\frac{b}{x^3}\right )^{3/2} x^2 \, dx\)
Optimal. Leaf size=58 \[ \frac{1}{3} x^3 \left (a+\frac{b}{x^3}\right )^{3/2}-b \sqrt{a+\frac{b}{x^3}}+\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right ) \]
[Out]
-(b*Sqrt[a + b/x^3]) + ((a + b/x^3)^(3/2)*x^3)/3 + Sqrt[a]*b*ArcTanh[Sqrt[a + b/
x^3]/Sqrt[a]]
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Rubi [A] time = 0.105789, 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
\[ \frac{1}{3} x^3 \left (a+\frac{b}{x^3}\right )^{3/2}-b \sqrt{a+\frac{b}{x^3}}+\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^3)^(3/2)*x^2,x]
[Out]
-(b*Sqrt[a + b/x^3]) + ((a + b/x^3)^(3/2)*x^3)/3 + Sqrt[a]*b*ArcTanh[Sqrt[a + b/
x^3]/Sqrt[a]]
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Rubi in Sympy [A] time = 8.87008, size = 49, normalized size = 0.84 \[ \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )} - b \sqrt{a + \frac{b}{x^{3}}} + \frac{x^{3} \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**3)**(3/2)*x**2,x)
[Out]
sqrt(a)*b*atanh(sqrt(a + b/x**3)/sqrt(a)) - b*sqrt(a + b/x**3) + x**3*(a + b/x**
3)**(3/2)/3
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Mathematica [A] time = 0.094162, size = 71, normalized size = 1.22 \[ \frac{1}{3} \sqrt{a+\frac{b}{x^3}} \left (\frac{3 \sqrt{a} b x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{\sqrt{a x^3+b}}+a x^3-2 b\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^3)^(3/2)*x^2,x]
[Out]
(Sqrt[a + b/x^3]*(-2*b + a*x^3 + (3*Sqrt[a]*b*x^(3/2)*ArcTanh[(Sqrt[a]*x^(3/2))/
Sqrt[b + a*x^3]])/Sqrt[b + a*x^3]))/3
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Maple [C] time = 0.025, size = 3559, normalized size = 61.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^3)^(3/2)*x^2,x)
[Out]
-1/3*((a*x^3+b)/x^3)^(3/2)*x^3/a*(18*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^2*b-I*(a*x^4+b*x)^(1/2)*(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)*3^(1/2)*x^3*a^2-36*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)*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^3*
a*b+18*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^4*a^2*b-18*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^4*a^2*b-18*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)*Elli
pticPi((-(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^2*b-18*(-(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^4*a^2*b+18*(-(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^4*a^2*b+36*(-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^3*a*b-36*(-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)*El
lipticPi((-(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^3*a*b-18*(-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^2*b+18*(-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^2*b+36*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^3*a*b+3*(a*x^4+b*x)^(1/2)*(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)*x^3*a^2+2*I*(a*x^4+b*x)^(1/2)*(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)*3^(1/2)*a*b-6*(a*x^4+b*x)^(1/2)*(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)*a*b)/(a*x^3+b)/(x*(a*x^3+b))^(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)
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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((a + b/x^3)^(3/2)*x^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.377153, size = 1, normalized size = 0.02 \[ \left [\frac{1}{4} \, \sqrt{a} b \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \,{\left (2 \, a x^{6} + b x^{3}\right )} \sqrt{a} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) + \frac{1}{3} \,{\left (a x^{3} - 2 \, b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}, \frac{1}{2} \, \sqrt{-a} b \arctan \left (\frac{2 \, a x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{{\left (2 \, a x^{3} + b\right )} \sqrt{-a}}\right ) + \frac{1}{3} \,{\left (a x^{3} - 2 \, b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^3)^(3/2)*x^2,x, algorithm="fricas")
[Out]
[1/4*sqrt(a)*b*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3)*sqrt(a)*sq
rt((a*x^3 + b)/x^3)) + 1/3*(a*x^3 - 2*b)*sqrt((a*x^3 + b)/x^3), 1/2*sqrt(-a)*b*a
rctan(2*a*x^3*sqrt((a*x^3 + b)/x^3)/((2*a*x^3 + b)*sqrt(-a))) + 1/3*(a*x^3 - 2*b
)*sqrt((a*x^3 + b)/x^3)]
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Sympy [A] time = 12.0979, size = 100, normalized size = 1.72 \[ \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )} + \frac{a^{2} x^{\frac{9}{2}}}{3 \sqrt{b} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{a \sqrt{b} x^{\frac{3}{2}}}{3 \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{2 b^{\frac{3}{2}}}{3 x^{\frac{3}{2}} \sqrt{\frac{a x^{3}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**3)**(3/2)*x**2,x)
[Out]
sqrt(a)*b*asinh(sqrt(a)*x**(3/2)/sqrt(b)) + a**2*x**(9/2)/(3*sqrt(b)*sqrt(a*x**3
/b + 1)) - a*sqrt(b)*x**(3/2)/(3*sqrt(a*x**3/b + 1)) - 2*b**(3/2)/(3*x**(3/2)*sq
rt(a*x**3/b + 1))
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GIAC/XCAS [A] time = 0.261813, size = 72, normalized size = 1.24 \[ \frac{1}{3} \, \sqrt{a x^{4} + b x} a x - \frac{a b \arctan \left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{2}{3} \, \sqrt{a + \frac{b}{x^{3}}} b \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^3)^(3/2)*x^2,x, algorithm="giac")
[Out]
1/3*sqrt(a*x^4 + b*x)*a*x - a*b*arctan(sqrt(a + b/x^3)/sqrt(-a))/sqrt(-a) - 2/3*
sqrt(a + b/x^3)*b