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

Optimal. Leaf size=137 \[ \frac{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^4}{5 b^3}+\frac{3 a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^3}{4 b^3} \]

[Out]

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

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Rubi [A]  time = 0.122619, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^4}{5 b^3}+\frac{3 a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^3}{4 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.0208, size = 122, normalized size = 0.89 \[ \frac{a^{2} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{8 b^{3}} - \frac{a \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{5 b^{3}} + \frac{x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0332571, size = 65, normalized size = 0.47 \[ \frac{x \sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (20 a^3+45 a^2 b \sqrt [3]{x}+36 a b^2 x^{2/3}+10 b^3 x\right )}{20 \left (a+b \sqrt [3]{x}\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^(1/3))^2]*x*(20*a^3 + 45*a^2*b*x^(1/3) + 36*a*b^2*x^(2/3) + 10*b^
3*x))/(20*(a + b*x^(1/3)))

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Maple [A]  time = 0.004, size = 65, normalized size = 0.5 \[{\frac{1}{20}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 36\,a{x}^{5/3}{b}^{2}+45\,{x}^{4/3}{a}^{2}b+10\,{b}^{3}{x}^{2}+20\,{a}^{3}x \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(1/2)*(36*a*x^(5/3)*b^2+45*x^(4/3)*a^2*b+10
*b^3*x^2+20*a^3*x)/(a+b*x^(1/3))

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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((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272695, size = 43, normalized size = 0.31 \[ \frac{1}{2} \, b^{3} x^{2} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}} + a^{3} x \]

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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 = 0.282533, size = 86, normalized size = 0.63 \[ \frac{1}{2} \, b^{3} x^{2}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + a^{3} x{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) \]

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]

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