3.2015 \(\int \frac{x^5}{\sqrt{a+\frac{b}{x^3}}} \, dx\)
Optimal. Leaf size=74 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{b x^3 \sqrt{a+\frac{b}{x^3}}}{4 a^2}+\frac{x^6 \sqrt{a+\frac{b}{x^3}}}{6 a} \]
[Out]
-(b*Sqrt[a + b/x^3]*x^3)/(4*a^2) + (Sqrt[a + b/x^3]*x^6)/(6*a) + (b^2*ArcTanh[Sq
rt[a + b/x^3]/Sqrt[a]])/(4*a^(5/2))
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Rubi [A] time = 0.11931, antiderivative size = 74, 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
\[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 a^{5/2}}-\frac{b x^3 \sqrt{a+\frac{b}{x^3}}}{4 a^2}+\frac{x^6 \sqrt{a+\frac{b}{x^3}}}{6 a} \]
Antiderivative was successfully verified.
[In] Int[x^5/Sqrt[a + b/x^3],x]
[Out]
-(b*Sqrt[a + b/x^3]*x^3)/(4*a^2) + (Sqrt[a + b/x^3]*x^6)/(6*a) + (b^2*ArcTanh[Sq
rt[a + b/x^3]/Sqrt[a]])/(4*a^(5/2))
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Rubi in Sympy [A] time = 9.8833, size = 63, normalized size = 0.85 \[ \frac{x^{6} \sqrt{a + \frac{b}{x^{3}}}}{6 a} - \frac{b x^{3} \sqrt{a + \frac{b}{x^{3}}}}{4 a^{2}} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}}} \]
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*a) - b*x**3*sqrt(a + b/x**3)/(4*a**2) + b**2*atanh(sqrt
(a + b/x**3)/sqrt(a))/(4*a**(5/2))
_______________________________________________________________________________________
Mathematica [A] time = 0.0721158, size = 97, normalized size = 1.31 \[ \frac{\sqrt{a} x^{3/2} \left (2 a^2 x^6-a b x^3-3 b^2\right )+3 b^2 \sqrt{a x^3+b} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{12 a^{5/2} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/Sqrt[a + b/x^3],x]
[Out]
(Sqrt[a]*x^(3/2)*(-3*b^2 - a*b*x^3 + 2*a^2*x^6) + 3*b^2*Sqrt[b + a*x^3]*ArcTanh[
(Sqrt[a]*x^(3/2))/Sqrt[b + a*x^3]])/(12*a^(5/2)*Sqrt[a + b/x^3]*x^(3/2))
_______________________________________________________________________________________
Maple [C] time = 0.043, size = 3567, normalized size = 48.2 \[ \text{output too large to display} \]
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*(a*x^3+b)/a^4*(3*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+18*I*(-a^2*b)^(2/3)*3^(
1/2)*(-(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))*b^2-18*I*(
-a^2*b)^(2/3)*3^(1/2)*(-(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))*b^2-2*I*3^(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*x^4+b*x)^(1/2)*x^4*a^3+36*I*(-a^2*b)^(1/3)*3^(1/2)*
(-(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*b^2-18*I*3^(1/2)*(-(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^2-18*(-(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^2*a^2*b^2+18*I*3^(1/2)*(-(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+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^2*a^2*b^2+36*(-(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^2-36*(-(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)-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))*(-a^2*b)^(2/3)*b^2+
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))*(-a^2*b)^(2/3)*b^2-36*I*(-a^2*b)^(1/3)*3^(1/2)*(-(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*b^2-9*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)
_______________________________________________________________________________________
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(x^5/sqrt(a + b/x^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.369017, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{2} \log \left (-{\left (8 \, a^{2} x^{6} + 8 \, a b x^{3} + b^{2}\right )} \sqrt{a} - 4 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) + 4 \,{\left (2 \, a^{2} x^{6} - 3 \, a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{48 \, a^{3}}, -\frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right ) - 2 \,{\left (2 \, a^{2} x^{6} - 3 \, a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{24 \, a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(a + b/x^3),x, algorithm="fricas")
[Out]
[1/48*(3*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 - 3*a*b*x^3)*sqrt((a*x^3 + b)/x^
3))/a^3, -1/24*(3*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 - 3*a*b*x^3)*sqrt((a*x^3 + b)/x^3))/a^3]
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Sympy [A] time = 15.3022, size = 102, normalized size = 1.38 \[ \frac{x^{\frac{15}{2}}}{6 \sqrt{b} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{\sqrt{b} x^{\frac{9}{2}}}{12 a \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{b^{\frac{3}{2}} x^{\frac{3}{2}}}{4 a^{2} \sqrt{\frac{a x^{3}}{b} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{4 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(a+b/x**3)**(1/2),x)
[Out]
x**(15/2)/(6*sqrt(b)*sqrt(a*x**3/b + 1)) - sqrt(b)*x**(9/2)/(12*a*sqrt(a*x**3/b
+ 1)) - b**(3/2)*x**(3/2)/(4*a**2*sqrt(a*x**3/b + 1)) + b**2*asinh(sqrt(a)*x**(3
/2)/sqrt(b))/(4*a**(5/2))
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GIAC/XCAS [A] time = 0.255942, size = 134, normalized size = 1.81 \[ -\frac{1}{12} \, b^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{\frac{a x^{3} + b}{x^{3}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{5 \, a \sqrt{\frac{a x^{3} + b}{x^{3}}} - \frac{3 \,{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{x^{3}}}{{\left (a - \frac{a x^{3} + b}{x^{3}}\right )}^{2} a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(a + b/x^3),x, algorithm="giac")
[Out]
-1/12*b^2*(3*arctan(sqrt((a*x^3 + b)/x^3)/sqrt(-a))/(sqrt(-a)*a^2) - (5*a*sqrt((
a*x^3 + b)/x^3) - 3*(a*x^3 + b)*sqrt((a*x^3 + b)/x^3)/x^3)/((a - (a*x^3 + b)/x^3
)^2*a^2))