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

Optimal. Leaf size=130 \[ -\frac{3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{6 a}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]

[Out]

(6*a)/(b^3*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)]) - (3*a^2)/(2*b^3*(a + b*x^(1
/3))*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)]) + (3*(a + b*x^(1/3))*Log[a + b*x^(
1/3)])/(b^3*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])

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Rubi [A]  time = 0.15396, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{6 a}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{3 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]

Antiderivative was successfully verified.

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

[Out]

(6*a)/(b^3*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)]) - (3*a^2)/(2*b^3*(a + b*x^(1
/3))*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)]) + (3*(a + b*x^(1/3))*Log[a + b*x^(
1/3)])/(b^3*Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)])

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Rubi in Sympy [A]  time = 11.9434, size = 131, normalized size = 1.01 \[ - \frac{3 x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right )}{4 b \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{3}{2}}} - \frac{3 \sqrt [3]{x}}{b^{2} \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}} + \frac{3 \left (a + b \sqrt [3]{x}\right ) \log{\left (a + b \sqrt [3]{x} \right )}}{b^{3} \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x**(2/3)*(2*a + 2*b*x**(1/3))/(4*b*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(
3/2)) - 3*x**(1/3)/(b**2*sqrt(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))) + 3*(a + b
*x**(1/3))*log(a + b*x**(1/3))/(b**3*sqrt(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))
)

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Mathematica [A]  time = 0.0469399, size = 72, normalized size = 0.55 \[ \frac{3 a \left (3 a+4 b \sqrt [3]{x}\right )+6 \left (a+b \sqrt [3]{x}\right )^2 \log \left (a+b \sqrt [3]{x}\right )}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*a*(3*a + 4*b*x^(1/3)) + 6*(a + b*x^(1/3))^2*Log[a + b*x^(1/3)])/(2*b^3*(a + b
*x^(1/3))*Sqrt[(a + b*x^(1/3))^2])

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Maple [A]  time = 0.012, size = 92, normalized size = 0.7 \[{\frac{3}{2\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 2\,{x}^{2/3}\ln \left ( a+b\sqrt [3]{x} \right ){b}^{2}+4\,\sqrt [3]{x}\ln \left ( a+b\sqrt [3]{x} \right ) ab+4\,ab\sqrt [3]{x}+2\,{a}^{2}\ln \left ( a+b\sqrt [3]{x} \right ) +3\,{a}^{2} \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)*(2*x^(2/3)*ln(a+b*x^(1/3))*b^2+4*x^(1/
3)*ln(a+b*x^(1/3))*a*b+4*a*b*x^(1/3)+2*a^2*ln(a+b*x^(1/3))+3*a^2)/(a+b*x^(1/3))^
3/b^3

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Maxima [A]  time = 0.74631, size = 88, normalized size = 0.68 \[ \frac{3 \, \log \left (x^{\frac{1}{3}} + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{9 \, a^{2} b^{2}}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{2}} + \frac{6 \, a b x^{\frac{1}{3}}}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*log(x^(1/3) + a/b)/(b^2)^(3/2) + 9/2*a^2*b^2/((b^2)^(7/2)*(x^(1/3) + a/b)^2) +
6*a*b*x^(1/3)/((b^2)^(5/2)*(x^(1/3) + a/b)^2)

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Fricas [A]  time = 0.272356, size = 93, normalized size = 0.72 \[ \frac{3 \,{\left (4 \, a b x^{\frac{1}{3}} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )} \log \left (b x^{\frac{1}{3}} + a\right )\right )}}{2 \,{\left (b^{5} x^{\frac{2}{3}} + 2 \, a b^{4} x^{\frac{1}{3}} + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(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 = 0.284756, size = 86, normalized size = 0.66 \[ \frac{3 \,{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}{\rm sign}\left (b x^{\frac{1}{3}} + a\right )} + \frac{3 \,{\left (4 \, a x^{\frac{1}{3}} + \frac{3 \, a^{2}}{b}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{2}{\rm sign}\left (b x^{\frac{1}{3}} + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*ln(abs(b*x^(1/3) + a))/(b^3*sign(b*x^(1/3) + a)) + 3/2*(4*a*x^(1/3) + 3*a^2/b)
/((b*x^(1/3) + a)^2*b^2*sign(b*x^(1/3) + a))