3.487 \(\int \frac{1}{\left (a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}\right )^{3/2}} \, dx\)

Optimal. Leaf size=300 \[ \frac{3 b^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac{15 b^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}-\frac{30 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{18 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{9 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]

[Out]

(3*b^5*(a + b/x^(1/3)))/(2*a^6*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*
x^(1/3))^2) - (15*b^4*(a + b/x^(1/3)))/(a^6*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(
1/3)]*(b + a*x^(1/3))) + (18*b^2*(a + b/x^(1/3))*x^(1/3))/(a^5*Sqrt[a^2 + b^2/x^
(2/3) + (2*a*b)/x^(1/3)]) - (9*b*(a + b/x^(1/3))*x^(2/3))/(2*a^4*Sqrt[a^2 + b^2/
x^(2/3) + (2*a*b)/x^(1/3)]) + ((a + b/x^(1/3))*x)/(a^3*Sqrt[a^2 + b^2/x^(2/3) +
(2*a*b)/x^(1/3)]) - (30*b^3*(a + b/x^(1/3))*Log[b + a*x^(1/3)])/(a^6*Sqrt[a^2 +
b^2/x^(2/3) + (2*a*b)/x^(1/3)])

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Rubi [A]  time = 0.383812, antiderivative size = 300, 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 \[ \frac{3 b^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac{15 b^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}-\frac{30 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{18 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{9 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^5*(a + b/x^(1/3)))/(2*a^6*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3)]*(b + a*
x^(1/3))^2) - (15*b^4*(a + b/x^(1/3)))/(a^6*Sqrt[a^2 + b^2/x^(2/3) + (2*a*b)/x^(
1/3)]*(b + a*x^(1/3))) + (18*b^2*(a + b/x^(1/3))*x^(1/3))/(a^5*Sqrt[a^2 + b^2/x^
(2/3) + (2*a*b)/x^(1/3)]) - (9*b*(a + b/x^(1/3))*x^(2/3))/(2*a^4*Sqrt[a^2 + b^2/
x^(2/3) + (2*a*b)/x^(1/3)]) + ((a + b/x^(1/3))*x)/(a^3*Sqrt[a^2 + b^2/x^(2/3) +
(2*a*b)/x^(1/3)]) - (30*b^3*(a + b/x^(1/3))*Log[b + a*x^(1/3)])/(a^6*Sqrt[a^2 +
b^2/x^(2/3) + (2*a*b)/x^(1/3)])

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Rubi in Sympy [A]  time = 53.8155, size = 314, normalized size = 1.05 \[ - \frac{3 x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{4 a \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}}} - \frac{15 x}{2 a^{2} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} + \frac{5 x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{a^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} - \frac{15 b x^{\frac{2}{3}} \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{2 a^{4} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} + \frac{30 b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (\frac{1}{\sqrt [3]{x}} \right )}}{a^{6} \left (a + \frac{b}{\sqrt [3]{x}}\right )} - \frac{30 b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (a + \frac{b}{\sqrt [3]{x}} \right )}}{a^{6} \left (a + \frac{b}{\sqrt [3]{x}}\right )} + \frac{30 b^{2} \sqrt [3]{x} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x*(2*a + 2*b/x**(1/3))/(4*a*(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))**(3/2)) -
15*x/(2*a**2*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))) + 5*x*(2*a + 2*b/x**(
1/3))/(a**3*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))) - 15*b*x**(2/3)*(2*a +
2*b/x**(1/3))/(2*a**4*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))) + 30*b**3*sqr
t(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))*log(x**(-1/3))/(a**6*(a + b/x**(1/3)))
- 30*b**3*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))*log(a + b/x**(1/3))/(a**6*
(a + b/x**(1/3))) + 30*b**2*x**(1/3)*sqrt(a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))
/a**6

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Mathematica [A]  time = 0.0864092, size = 126, normalized size = 0.42 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (2 a^5 x^{5/3}-5 a^4 b x^{4/3}+20 a^3 b^2 x+63 a^2 b^3 x^{2/3}+6 a b^4 \sqrt [3]{x}-60 b^3 \left (a \sqrt [3]{x}+b\right )^2 \log \left (a \sqrt [3]{x}+b\right )-27 b^5\right )}{2 a^6 x \left (\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b + a*x^(1/3))*(-27*b^5 + 6*a*b^4*x^(1/3) + 63*a^2*b^3*x^(2/3) + 20*a^3*b^2*x
- 5*a^4*b*x^(4/3) + 2*a^5*x^(5/3) - 60*b^3*(b + a*x^(1/3))^2*Log[b + a*x^(1/3)])
)/(2*a^6*((b + a*x^(1/3))^2/x^(2/3))^(3/2)*x)

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Maple [A]  time = 0.017, size = 141, normalized size = 0.5 \[{\frac{1}{2\,x{a}^{6}} \left ( 2\,{a}^{5}{x}^{5/3}-5\,{a}^{4}b{x}^{4/3}-60\,{x}^{2/3}\ln \left ( b+a\sqrt [3]{x} \right ){a}^{2}{b}^{3}+63\,{x}^{2/3}{a}^{2}{b}^{3}-120\,\sqrt [3]{x}\ln \left ( b+a\sqrt [3]{x} \right ) a{b}^{4}+6\,a{b}^{4}\sqrt [3]{x}-60\,\ln \left ( b+a\sqrt [3]{x} \right ){b}^{5}+20\,{a}^{3}{b}^{2}x-27\,{b}^{5} \right ) \left ( b+a\sqrt [3]{x} \right ) \left ({1 \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/((a^2*x^(2/3)+2*a*b*x^(1/3)+b^2)/x^(2/3))^(3/2)/x*(2*a^5*x^(5/3)-5*a^4*b*x^(
4/3)-60*x^(2/3)*ln(b+a*x^(1/3))*a^2*b^3+63*x^(2/3)*a^2*b^3-120*x^(1/3)*ln(b+a*x^
(1/3))*a*b^4+6*a*b^4*x^(1/3)-60*ln(b+a*x^(1/3))*b^5+20*a^3*b^2*x-27*b^5)*(b+a*x^
(1/3))/a^6

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Maxima [A]  time = 0.752276, size = 131, normalized size = 0.44 \[ \frac{2 \, a^{5} x^{\frac{5}{3}} - 5 \, a^{4} b x^{\frac{4}{3}} + 20 \, a^{3} b^{2} x + 63 \, a^{2} b^{3} x^{\frac{2}{3}} + 6 \, a b^{4} x^{\frac{1}{3}} - 27 \, b^{5}}{2 \,{\left (a^{8} x^{\frac{2}{3}} + 2 \, a^{7} b x^{\frac{1}{3}} + a^{6} b^{2}\right )}} - \frac{30 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right )}{a^{6}} \]

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))^(-3/2),x, algorithm="maxima")

[Out]

1/2*(2*a^5*x^(5/3) - 5*a^4*b*x^(4/3) + 20*a^3*b^2*x + 63*a^2*b^3*x^(2/3) + 6*a*b
^4*x^(1/3) - 27*b^5)/(a^8*x^(2/3) + 2*a^7*b*x^(1/3) + a^6*b^2) - 30*b^3*log(a*x^
(1/3) + b)/a^6

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Fricas [A]  time = 0.27443, size = 154, normalized size = 0.51 \[ \frac{20 \, a^{3} b^{2} x - 27 \, b^{5} - 60 \,{\left (a^{2} b^{3} x^{\frac{2}{3}} + 2 \, a b^{4} x^{\frac{1}{3}} + b^{5}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) +{\left (2 \, a^{5} x + 63 \, a^{2} b^{3}\right )} x^{\frac{2}{3}} -{\left (5 \, a^{4} b x - 6 \, a b^{4}\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{8} x^{\frac{2}{3}} + 2 \, a^{7} b x^{\frac{1}{3}} + a^{6} b^{2}\right )}} \]

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))^(-3/2),x, algorithm="fricas")

[Out]

1/2*(20*a^3*b^2*x - 27*b^5 - 60*(a^2*b^3*x^(2/3) + 2*a*b^4*x^(1/3) + b^5)*log(a*
x^(1/3) + b) + (2*a^5*x + 63*a^2*b^3)*x^(2/3) - (5*a^4*b*x - 6*a*b^4)*x^(1/3))/(
a^8*x^(2/3) + 2*a^7*b*x^(1/3) + a^6*b^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a**2 + 2*a*b/x**(1/3) + b**2/x**(2/3))**(-3/2), x)

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GIAC/XCAS [A]  time = 0.317177, size = 163, normalized size = 0.54 \[ -\frac{30 \, b^{3}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{6}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} - \frac{3 \,{\left (10 \, a b^{4} x^{\frac{1}{3}} + 9 \, b^{5}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{6}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} + \frac{2 \, a^{6} x - 9 \, a^{5} b x^{\frac{2}{3}} + 36 \, a^{4} b^{2} x^{\frac{1}{3}}}{2 \, a^{9}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} \]

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))^(-3/2),x, algorithm="giac")

[Out]

-30*b^3*ln(abs(a*x^(1/3) + b))/(a^6*sign(a*x^(2/3) + b*x^(1/3))) - 3/2*(10*a*b^4
*x^(1/3) + 9*b^5)/((a*x^(1/3) + b)^2*a^6*sign(a*x^(2/3) + b*x^(1/3))) + 1/2*(2*a
^6*x - 9*a^5*b*x^(2/3) + 36*a^4*b^2*x^(1/3))/(a^9*sign(a*x^(2/3) + b*x^(1/3)))