3.483 \(\int \left (a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}\right )^{5/2} \, dx\)

Optimal. Leaf size=291 \[ -\frac{3 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}-\frac{15 a b^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{\sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^2 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}}+\frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}}+\frac{15 a^4 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^3 b^2 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}} \]

[Out]

(-3*b^5*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)])/(2*(a + b/x^(1/3))*x^(2/3)) -
(15*a*b^4*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)])/((a + b/x^(1/3))*x^(1/3))
+ (30*a^3*b^2*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x^(1/3))/(a + b/x^(1/3))
+ (15*a^4*b*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x^(2/3))/(2*(a + b/x^(1/3
))) + (a^5*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x)/(a + b/x^(1/3)) + (30*a^
2*b^3*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*Log[x^(1/3)])/(a + b/x^(1/3))

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Rubi [A]  time = 0.304123, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{3 b^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}-\frac{15 a b^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{\sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^2 b^3 \log \left (\sqrt [3]{x}\right ) \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}}+\frac{a^5 x \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}}+\frac{15 a^4 b x^{2/3} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{2 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{30 a^3 b^2 \sqrt [3]{x} \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}{a+\frac{b}{\sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(5/2),x]

[Out]

(-3*b^5*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)])/(2*(a + b/x^(1/3))*x^(2/3)) -
(15*a*b^4*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)])/((a + b/x^(1/3))*x^(1/3))
+ (30*a^3*b^2*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x^(1/3))/(a + b/x^(1/3))
+ (15*a^4*b*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x^(2/3))/(2*(a + b/x^(1/3
))) + (a^5*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*x)/(a + b/x^(1/3)) + (30*a^
2*b^3*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*Log[x^(1/3)])/(a + b/x^(1/3))

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Rubi in Sympy [A]  time = 32.7387, size = 228, normalized size = 0.78 \[ - \frac{30 a^{2} b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (\frac{1}{\sqrt [3]{x}} \right )}}{a + \frac{b}{\sqrt [3]{x}}} - 30 a b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} - 15 b^{3} \left (a + \frac{b}{\sqrt [3]{x}}\right ) \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} + 10 b^{2} \sqrt [3]{x} \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}} + \frac{5 b x^{\frac{2}{3}} \left (a + \frac{b}{\sqrt [3]{x}}\right ) \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}}}{2} + x \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(5/2),x)

[Out]

-30*a**2*b**3*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))*log(x**(-1/3))/(a + b/
x**(1/3)) - 30*a*b**3*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3)) - 15*b**3*(a +
b/x**(1/3))*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3)) + 10*b**2*x**(1/3)*(a**
2 + 2*a*b/x**(1/3) + b**2/x**(2/3))**(3/2) + 5*b*x**(2/3)*(a + b/x**(1/3))*(a**2
+ 2*a*b/x**(1/3) + b**2/x**(2/3))**(3/2)/2 + x*(a**2 + 2*a*b/x**(1/3) + b**2/x*
*(2/3))**(5/2)

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Mathematica [A]  time = 0.0797503, size = 99, normalized size = 0.34 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (2 a^5 x^{5/3}+15 a^4 b x^{4/3}+60 a^3 b^2 x+20 a^2 b^3 x^{2/3} \log (x)-30 a b^4 \sqrt [3]{x}-3 b^5\right )}{2 x \sqrt{\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(5/2),x]

[Out]

((b + a*x^(1/3))*(-3*b^5 - 30*a*b^4*x^(1/3) + 60*a^3*b^2*x + 15*a^4*b*x^(4/3) +
2*a^5*x^(5/3) + 20*a^2*b^3*x^(2/3)*Log[x]))/(2*Sqrt[(b + a*x^(1/3))^2/x^(2/3)]*x
)

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Maple [A]  time = 0.016, size = 91, normalized size = 0.3 \[{\frac{x}{2} \left ({1 \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}} \right ) ^{{\frac{5}{2}}} \left ( 15\,{a}^{4}b{x}^{4/3}+60\,{a}^{3}{b}^{2}x+20\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{2/3}+2\,{a}^{5}{x}^{5/3}-30\,a{b}^{4}\sqrt [3]{x}-3\,{b}^{5} \right ) \left ( b+a\sqrt [3]{x} \right ) ^{-5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(5/2),x)

[Out]

1/2*((a^2*x^(2/3)+2*a*b*x^(1/3)+b^2)/x^(2/3))^(5/2)*x*(15*a^4*b*x^(4/3)+60*a^3*b
^2*x+20*a^2*b^3*ln(x)*x^(2/3)+2*a^5*x^(5/3)-30*a*b^4*x^(1/3)-3*b^5)/(b+a*x^(1/3)
)^5

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Maxima [A]  time = 0.755331, size = 77, normalized size = 0.26 \[ 10 \, a^{2} b^{3} \log \left (x\right ) + \frac{2 \, a^{5} x^{\frac{5}{3}} + 15 \, a^{4} b x^{\frac{4}{3}} + 60 \, a^{3} b^{2} x - 30 \, a b^{4} x^{\frac{1}{3}} - 3 \, b^{5}}{2 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(5/2),x, algorithm="maxima")

[Out]

10*a^2*b^3*log(x) + 1/2*(2*a^5*x^(5/3) + 15*a^4*b*x^(4/3) + 60*a^3*b^2*x - 30*a*
b^4*x^(1/3) - 3*b^5)/x^(2/3)

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Fricas [A]  time = 0.273202, size = 82, normalized size = 0.28 \[ \frac{2 \, a^{5} x^{\frac{5}{3}} + 60 \, a^{2} b^{3} x^{\frac{2}{3}} \log \left (x^{\frac{1}{3}}\right ) + 60 \, a^{3} b^{2} x - 3 \, b^{5} + 15 \,{\left (a^{4} b x - 2 \, a b^{4}\right )} x^{\frac{1}{3}}}{2 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(5/2),x, algorithm="fricas")

[Out]

1/2*(2*a^5*x^(5/3) + 60*a^2*b^3*x^(2/3)*log(x^(1/3)) + 60*a^3*b^2*x - 3*b^5 + 15
*(a^4*b*x - 2*a*b^4)*x^(1/3))/x^(2/3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.298184, size = 173, normalized size = 0.59 \[ a^{5} x{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right ) + 10 \, a^{2} b^{3}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right ) + \frac{15}{2} \, a^{4} b x^{\frac{2}{3}}{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right ) + 30 \, a^{3} b^{2} x^{\frac{1}{3}}{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right ) - \frac{3 \,{\left (10 \, a b^{4} x^{\frac{1}{3}}{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right ) + b^{5}{\rm sign}\left (a x + b x^{\frac{2}{3}}\right ){\rm sign}\left (x\right )\right )}}{2 \, x^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(5/2),x, algorithm="giac")

[Out]

a^5*x*sign(a*x + b*x^(2/3))*sign(x) + 10*a^2*b^3*ln(abs(x))*sign(a*x + b*x^(2/3)
)*sign(x) + 15/2*a^4*b*x^(2/3)*sign(a*x + b*x^(2/3))*sign(x) + 30*a^3*b^2*x^(1/3
)*sign(a*x + b*x^(2/3))*sign(x) - 3/2*(10*a*b^4*x^(1/3)*sign(a*x + b*x^(2/3))*si
gn(x) + b^5*sign(a*x + b*x^(2/3))*sign(x))/x^(2/3)