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

Optimal. Leaf size=88 \[ \frac{3 b x^{4/3} \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{4 \left (a+b \sqrt [3]{x}\right )}+\frac{a x \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{a+b \sqrt [3]{x}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0908995, antiderivative size = 88, 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 \[ \frac{3 b x^{4/3} \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{4 \left (a+b \sqrt [3]{x}\right )}+\frac{a x \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{a+b \sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.32815, size = 70, normalized size = 0.8 \[ \frac{a x \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}}{4 \left (a + b \sqrt [3]{x}\right )} + \frac{3 x \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}}{4} \]

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))**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0162209, size = 43, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (4 a x+3 b x^{4/3}\right )}{4 \left (a+b \sqrt [3]{x}\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 43, normalized size = 0.5 \[{\frac{1}{4}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 3\,b{x}^{4/3}+4\,ax \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \]

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))^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.276833, size = 14, normalized size = 0.16 \[ \frac{3}{4} \, b x^{\frac{4}{3}} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/4*b*x^(4/3) + a*x

_______________________________________________________________________________________

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

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))**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270966, size = 35, normalized size = 0.4 \[ \frac{3}{4} \, b x^{\frac{4}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + a x{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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