3.2008 \(\int \left (a+\frac{b}{x^3}\right )^{3/2} x^5 \, dx\)
Optimal. Leaf size=68 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{1}{4} b x^3 \sqrt{a+\frac{b}{x^3}}+\frac{1}{6} x^6 \left (a+\frac{b}{x^3}\right )^{3/2} \]
[Out]
(b*Sqrt[a + b/x^3]*x^3)/4 + ((a + b/x^3)^(3/2)*x^6)/6 + (b^2*ArcTanh[Sqrt[a + b/
x^3]/Sqrt[a]])/(4*Sqrt[a])
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Rubi [A] time = 0.112587, antiderivative size = 68, 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
\[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{1}{4} b x^3 \sqrt{a+\frac{b}{x^3}}+\frac{1}{6} x^6 \left (a+\frac{b}{x^3}\right )^{3/2} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^3)^(3/2)*x^5,x]
[Out]
(b*Sqrt[a + b/x^3]*x^3)/4 + ((a + b/x^3)^(3/2)*x^6)/6 + (b^2*ArcTanh[Sqrt[a + b/
x^3]/Sqrt[a]])/(4*Sqrt[a])
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Rubi in Sympy [A] time = 9.61194, size = 58, normalized size = 0.85 \[ \frac{b x^{3} \sqrt{a + \frac{b}{x^{3}}}}{4} + \frac{x^{6} \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{6} + \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**3)**(3/2)*x**5,x)
[Out]
b*x**3*sqrt(a + b/x**3)/4 + x**6*(a + b/x**3)**(3/2)/6 + b**2*atanh(sqrt(a + b/x
**3)/sqrt(a))/(4*sqrt(a))
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Mathematica [A] time = 0.146723, size = 81, normalized size = 1.19 \[ \frac{1}{12} x^{3/2} \sqrt{a+\frac{b}{x^3}} \left (\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{\sqrt{a} \sqrt{a x^3+b}}+x^{3/2} \left (2 a x^3+5 b\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^3)^(3/2)*x^5,x]
[Out]
(Sqrt[a + b/x^3]*x^(3/2)*(x^(3/2)*(5*b + 2*a*x^3) + (3*b^2*ArcTanh[(Sqrt[a]*x^(3
/2))/Sqrt[b + a*x^3]])/(Sqrt[a]*Sqrt[b + a*x^3])))/12
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Maple [C] time = 0.019, size = 3569, normalized size = 52.5 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^3)^(3/2)*x^5,x)
[Out]
-1/12*((a*x^3+b)/x^3)^(3/2)*x^5/a^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)*Elliptic
Pi((-(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^2-36*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^2-18*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^2-5*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*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*(-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^2-2*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^3-18*(-(
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)-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+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+15*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))/(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^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.372458, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{a} b^{2} \log \left (-{\left (8 \, a^{2} x^{6} + 8 \, a b x^{3} + b^{2}\right )} \sqrt{a} - 4 \,{\left (2 \, a^{2} x^{6} + a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) + 4 \,{\left (2 \, a^{2} x^{6} + 5 \, a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{48 \, a}, -\frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right ) - 2 \,{\left (2 \, a^{2} x^{6} + 5 \, a b x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{24 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^3)^(3/2)*x^5,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 + 5*a*b*x^3)*sqrt((a*x^3 + b)/x^
3))/a, -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 + 5*a*b*x^3)*sqrt((a*x^3 + b)/x^3))/a]
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Sympy [A] time = 18.6384, size = 76, normalized size = 1.12 \[ \frac{a \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{6} + \frac{5 b^{\frac{3}{2}} x^{\frac{3}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{12} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{4 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**3)**(3/2)*x**5,x)
[Out]
a*sqrt(b)*x**(9/2)*sqrt(a*x**3/b + 1)/6 + 5*b**(3/2)*x**(3/2)*sqrt(a*x**3/b + 1)
/12 + b**2*asinh(sqrt(a)*x**(3/2)/sqrt(b))/(4*sqrt(a))
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GIAC/XCAS [A] time = 0.250621, size = 69, normalized size = 1.01 \[ \frac{1}{12} \, \sqrt{a x^{4} + b x}{\left (2 \, a x^{3} + 5 \, b\right )} x - \frac{b^{2} \arctan \left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^3)^(3/2)*x^5,x, algorithm="giac")
[Out]
1/12*sqrt(a*x^4 + b*x)*(2*a*x^3 + 5*b)*x - 1/4*b^2*arctan(sqrt(a + b/x^3)/sqrt(-
a))/sqrt(-a)