3.468 \(\int \frac{1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{5/2}} \, dx\)

Optimal. Leaf size=135 \[ -\frac{3 a^2}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{2 a}{b^3 \left (a+b \sqrt [3]{x}\right )^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]

[Out]

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

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Rubi [A]  time = 0.157291, antiderivative size = 135, 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 \[ -\frac{3 a^2}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{2 a}{b^3 \left (a+b \sqrt [3]{x}\right )^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.86532, size = 75, normalized size = 0.56 \[ \frac{3 x \left (2 a + 2 b \sqrt [3]{x}\right )}{8 a \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}} + \frac{x}{4 a^{2} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{3}{2}}} \]

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))**(5/2),x)

[Out]

3*x*(2*a + 2*b*x**(1/3))/(8*a*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(5/2)) +
x/(4*a**2*(a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(3/2))

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Mathematica [A]  time = 0.0307225, size = 58, normalized size = 0.43 \[ \frac{-a^2-4 a b \sqrt [3]{x}-6 b^2 x^{2/3}}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 54, normalized size = 0.4 \[ -{\frac{1}{4\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 6\,{b}^{2}{x}^{2/3}+4\,ab\sqrt [3]{x}+{a}^{2} \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-5}} \]

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))^(5/2),x)

[Out]

-1/4*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)*(6*b^2*x^(2/3)+4*a*b*x^(1/3)+a^2)/(a+
b*x^(1/3))^5/b^3

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Maxima [A]  time = 0.750146, size = 85, normalized size = 0.63 \[ -\frac{3 \, a^{2} b^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{4}} + \frac{2 \, a b}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{3}} - \frac{3}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{2}} \]

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)^(-5/2),x, algorithm="maxima")

[Out]

-3/4*a^2*b^2/((b^2)^(9/2)*(x^(1/3) + a/b)^4) + 2*a*b/((b^2)^(7/2)*(x^(1/3) + a/b
)^3) - 3/2/((b^2)^(5/2)*(x^(1/3) + a/b)^2)

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Fricas [A]  time = 0.274416, size = 90, normalized size = 0.67 \[ -\frac{6 \, b^{2} x^{\frac{2}{3}} + 4 \, a b x^{\frac{1}{3}} + a^{2}}{4 \,{\left (4 \, a b^{6} x + 6 \, a^{2} b^{5} x^{\frac{2}{3}} + a^{4} b^{3} +{\left (b^{7} x + 4 \, a^{3} b^{4}\right )} x^{\frac{1}{3}}\right )}} \]

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)^(-5/2),x, algorithm="fricas")

[Out]

-1/4*(6*b^2*x^(2/3) + 4*a*b*x^(1/3) + a^2)/(4*a*b^6*x + 6*a^2*b^5*x^(2/3) + a^4*
b^3 + (b^7*x + 4*a^3*b^4)*x^(1/3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}}\, dx \]

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))**(5/2),x)

[Out]

Integral((a**2 + 2*a*b*x**(1/3) + b**2*x**(2/3))**(-5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

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)^(-5/2),x, algorithm="giac")

[Out]

undef