3.1989 \(\int \sqrt{a+\frac{b}{x^3}} x^5 \, dx\)
Optimal. Leaf size=71 \[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{b x^3 \sqrt{a+\frac{b}{x^3}}}{12 a}+\frac{1}{6} x^6 \sqrt{a+\frac{b}{x^3}} \]
[Out]
(b*Sqrt[a + b/x^3]*x^3)/(12*a) + (Sqrt[a + b/x^3]*x^6)/6 - (b^2*ArcTanh[Sqrt[a +
b/x^3]/Sqrt[a]])/(12*a^(3/2))
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Rubi [A] time = 0.116961, antiderivative size = 71, 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
\[ -\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{b x^3 \sqrt{a+\frac{b}{x^3}}}{12 a}+\frac{1}{6} x^6 \sqrt{a+\frac{b}{x^3}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x^3]*x^5,x]
[Out]
(b*Sqrt[a + b/x^3]*x^3)/(12*a) + (Sqrt[a + b/x^3]*x^6)/6 - (b^2*ArcTanh[Sqrt[a +
b/x^3]/Sqrt[a]])/(12*a^(3/2))
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Rubi in Sympy [A] time = 9.57759, size = 60, normalized size = 0.85 \[ \frac{x^{6} \sqrt{a + \frac{b}{x^{3}}}}{6} + \frac{b x^{3} \sqrt{a + \frac{b}{x^{3}}}}{12 a} - \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{12 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5*(a+b/x**3)**(1/2),x)
[Out]
x**6*sqrt(a + b/x**3)/6 + b*x**3*sqrt(a + b/x**3)/(12*a) - b**2*atanh(sqrt(a + b
/x**3)/sqrt(a))/(12*a**(3/2))
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Mathematica [A] time = 0.110671, size = 95, normalized size = 1.34 \[ \frac{x^{3/2} \sqrt{a+\frac{b}{x^3}} \left (\sqrt{a} x^{3/2} \sqrt{a x^3+b} \left (2 a x^3+b\right )-b^2 \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )\right )}{12 a^{3/2} \sqrt{a x^3+b}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x^3]*x^5,x]
[Out]
(Sqrt[a + b/x^3]*x^(3/2)*(Sqrt[a]*x^(3/2)*Sqrt[b + a*x^3]*(b + 2*a*x^3) - b^2*Ar
cTanh[(Sqrt[a]*x^(3/2))/Sqrt[b + a*x^3]]))/(12*a^(3/2)*Sqrt[b + a*x^3])
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Maple [C] time = 0.362, size = 3560, normalized size = 50.1 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5*(a+b/x^3)^(1/2),x)
[Out]
1/12*((a*x^3+b)/x^3)^(1/2)*x^2/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)*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^2*a^2*b^2+12*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))*(-a^2*b)^(1/3)*3^(1/2)*x*a*b^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)*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))*(-a^2*b)^(2/3)*3^(1/2)*b^2-12*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)*Ellipti
cF((-(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))*(-a^2*b)^(1/3)*3^(1/2)*x*a*b^
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)*(a*x^4+b*x)^(1/2)*3^(1
/2)*x*a^2*b+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)*(a*x^4+b*x
)^(1/2)*3^(1/2)*x^4*a^3-6*(-(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^2*a^2*b^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)*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*a^2*b^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^2*a^2*b^2+12*(-(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))*(-a^2*b)^(1/3)*x*a*b^2-12*(-(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))*(-a^2*b)^(1/3)*x*a*b^2-6*x^4*(a*x^4+b*x)^(1/2)*a^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)-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))*(-a^2*b)^(2/3)*b^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))*(-a^2*b)^(2/3)*b^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))*(-a^2*b)^(2/3)*3^(1/2)*b^2-3*b*x*(a*x^4+b*x)^(1/2)*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))/(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(sqrt(a + b/x^3)*x^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.366716, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{a} b^{2} \log \left (-{\left (8 \, a^{2} x^{6} + 8 \, a b x^{3} + b^{2}\right )} \sqrt{a} + 4 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) + 4 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{48 \, a^{2}}, \frac{\sqrt{-a} b^{2} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right ) + 2 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{24 \, a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^3)*x^5,x, algorithm="fricas")
[Out]
[1/48*(sqrt(a)*b^2*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)) + 4*(2*a^2*x^6 + a*b*x^3)*sqrt((a*x^3 + b)/x^3))/
a^2, 1/24*(sqrt(-a)*b^2*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b
)) + 2*(2*a^2*x^6 + a*b*x^3)*sqrt((a*x^3 + b)/x^3))/a^2]
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Sympy [A] time = 14.6212, size = 100, normalized size = 1.41 \[ \frac{a x^{\frac{15}{2}}}{6 \sqrt{b} \sqrt{\frac{a x^{3}}{b} + 1}} + \frac{\sqrt{b} x^{\frac{9}{2}}}{4 \sqrt{\frac{a x^{3}}{b} + 1}} + \frac{b^{\frac{3}{2}} x^{\frac{3}{2}}}{12 a \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{12 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5*(a+b/x**3)**(1/2),x)
[Out]
a*x**(15/2)/(6*sqrt(b)*sqrt(a*x**3/b + 1)) + sqrt(b)*x**(9/2)/(4*sqrt(a*x**3/b +
1)) + b**(3/2)*x**(3/2)/(12*a*sqrt(a*x**3/b + 1)) - b**2*asinh(sqrt(a)*x**(3/2)
/sqrt(b))/(12*a**(3/2))
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GIAC/XCAS [A] time = 0.250734, size = 74, normalized size = 1.04 \[ \frac{1}{12} \, \sqrt{a x^{4} + b x}{\left (2 \, x^{3} + \frac{b}{a}\right )} x + \frac{b^{2} \arctan \left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{-a}}\right )}{12 \, \sqrt{-a} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^3)*x^5,x, algorithm="giac")
[Out]
1/12*sqrt(a*x^4 + b*x)*(2*x^3 + b/a)*x + 1/12*b^2*arctan(sqrt(a + b/x^3)/sqrt(-a
))/(sqrt(-a)*a)