3.654 \(\int \left (a+c \sqrt{x}+b x^{2/3}\right )^3 \, dx\)

Optimal. Leaf size=114 \[ a^3 x+\frac{9}{5} a^2 b x^{5/3}+2 a^2 c x^{3/2}+\frac{9}{7} a b^2 x^{7/3}+\frac{36}{13} a b c x^{13/6}+\frac{3}{2} a c^2 x^2+\frac{b^3 x^3}{3}+\frac{18}{17} b^2 c x^{17/6}+\frac{9}{8} b c^2 x^{8/3}+\frac{2}{5} c^3 x^{5/2} \]

[Out]

a^3*x + 2*a^2*c*x^(3/2) + (9*a^2*b*x^(5/3))/5 + (3*a*c^2*x^2)/2 + (36*a*b*c*x^(1
3/6))/13 + (9*a*b^2*x^(7/3))/7 + (2*c^3*x^(5/2))/5 + (9*b*c^2*x^(8/3))/8 + (18*b
^2*c*x^(17/6))/17 + (b^3*x^3)/3

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Rubi [A]  time = 0.339489, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ a^3 x+\frac{9}{5} a^2 b x^{5/3}+2 a^2 c x^{3/2}+\frac{9}{7} a b^2 x^{7/3}+\frac{36}{13} a b c x^{13/6}+\frac{3}{2} a c^2 x^2+\frac{b^3 x^3}{3}+\frac{18}{17} b^2 c x^{17/6}+\frac{9}{8} b c^2 x^{8/3}+\frac{2}{5} c^3 x^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

a^3*x + 2*a^2*c*x^(3/2) + (9*a^2*b*x^(5/3))/5 + (3*a*c^2*x^2)/2 + (36*a*b*c*x^(1
3/6))/13 + (9*a*b^2*x^(7/3))/7 + (2*c^3*x^(5/2))/5 + (9*b*c^2*x^(8/3))/8 + (18*b
^2*c*x^(17/6))/17 + (b^3*x^3)/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{9 a^{2} b x^{\frac{5}{3}}}{5} + 2 a^{2} c x^{\frac{3}{2}} + \frac{9 a b^{2} x^{\frac{7}{3}}}{7} + \frac{36 a b c x^{\frac{13}{6}}}{13} + 3 a c^{2} \int x\, dx + \frac{b^{3} x^{3}}{3} + \frac{18 b^{2} c x^{\frac{17}{6}}}{17} + \frac{9 b c^{2} x^{\frac{8}{3}}}{8} + \frac{2 c^{3} x^{\frac{5}{2}}}{5} + \int a^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

9*a**2*b*x**(5/3)/5 + 2*a**2*c*x**(3/2) + 9*a*b**2*x**(7/3)/7 + 36*a*b*c*x**(13/
6)/13 + 3*a*c**2*Integral(x, x) + b**3*x**3/3 + 18*b**2*c*x**(17/6)/17 + 9*b*c**
2*x**(8/3)/8 + 2*c**3*x**(5/2)/5 + Integral(a**3, x)

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Mathematica [A]  time = 0.056646, size = 114, normalized size = 1. \[ a^3 x+\frac{9}{5} a^2 b x^{5/3}+2 a^2 c x^{3/2}+\frac{9}{7} a b^2 x^{7/3}+\frac{36}{13} a b c x^{13/6}+\frac{3}{2} a c^2 x^2+\frac{b^3 x^3}{3}+\frac{18}{17} b^2 c x^{17/6}+\frac{9}{8} b c^2 x^{8/3}+\frac{2}{5} c^3 x^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

a^3*x + 2*a^2*c*x^(3/2) + (9*a^2*b*x^(5/3))/5 + (3*a*c^2*x^2)/2 + (36*a*b*c*x^(1
3/6))/13 + (9*a*b^2*x^(7/3))/7 + (2*c^3*x^(5/2))/5 + (9*b*c^2*x^(8/3))/8 + (18*b
^2*c*x^(17/6))/17 + (b^3*x^3)/3

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Maple [A]  time = 0.004, size = 86, normalized size = 0.8 \[{\frac{2\,{c}^{3}}{5}{x}^{{\frac{5}{2}}}}+3\,{c}^{2} \left ( 3/8\,{x}^{8/3}b+1/2\,a{x}^{2} \right ) +3\,c \left ({\frac{6\,{b}^{2}}{17}{x}^{{\frac{17}{6}}}}+{\frac{12\,ab}{13}{x}^{{\frac{13}{6}}}}+2/3\,{a}^{2}{x}^{3/2} \right ) +{a}^{3}x+{\frac{{b}^{3}{x}^{3}}{3}}+{\frac{9\,{a}^{2}b}{5}{x}^{{\frac{5}{3}}}}+{\frac{9\,a{b}^{2}}{7}{x}^{{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*c^3*x^(5/2)+3*c^2*(3/8*x^(8/3)*b+1/2*a*x^2)+3*c*(6/17*b^2*x^(17/6)+12/13*a*b
*x^(13/6)+2/3*a^2*x^(3/2))+a^3*x+1/3*b^3*x^3+9/5*a^2*b*x^(5/3)+9/7*a*b^2*x^(7/3)

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Maxima [A]  time = 0.722573, size = 115, normalized size = 1.01 \[ \frac{1}{3} \, b^{3} x^{3} + \frac{18}{17} \, b^{2} c x^{\frac{17}{6}} + \frac{9}{8} \, b c^{2} x^{\frac{8}{3}} + \frac{2}{5} \, c^{3} x^{\frac{5}{2}} + a^{3} x + \frac{1}{5} \,{\left (9 \, b x^{\frac{5}{3}} + 10 \, c x^{\frac{3}{2}}\right )} a^{2} + \frac{3}{182} \,{\left (78 \, b^{2} x^{\frac{7}{3}} + 168 \, b c x^{\frac{13}{6}} + 91 \, c^{2} x^{2}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^3*x^3 + 18/17*b^2*c*x^(17/6) + 9/8*b*c^2*x^(8/3) + 2/5*c^3*x^(5/2) + a^3*x
+ 1/5*(9*b*x^(5/3) + 10*c*x^(3/2))*a^2 + 3/182*(78*b^2*x^(7/3) + 168*b*c*x^(13/
6) + 91*c^2*x^2)*a

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Fricas [A]  time = 0.265884, size = 123, normalized size = 1.08 \[ \frac{1}{3} \, b^{3} x^{3} + \frac{18}{17} \, b^{2} c x^{\frac{17}{6}} + \frac{9}{7} \, a b^{2} x^{\frac{7}{3}} + \frac{36}{13} \, a b c x^{\frac{13}{6}} + \frac{3}{2} \, a c^{2} x^{2} + a^{3} x + \frac{9}{40} \,{\left (5 \, b c^{2} x^{2} + 8 \, a^{2} b x\right )} x^{\frac{2}{3}} + \frac{2}{5} \,{\left (c^{3} x^{2} + 5 \, a^{2} c x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^3*x^3 + 18/17*b^2*c*x^(17/6) + 9/7*a*b^2*x^(7/3) + 36/13*a*b*c*x^(13/6) +
3/2*a*c^2*x^2 + a^3*x + 9/40*(5*b*c^2*x^2 + 8*a^2*b*x)*x^(2/3) + 2/5*(c^3*x^2 +
5*a^2*c*x)*sqrt(x)

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Sympy [A]  time = 1.28097, size = 116, normalized size = 1.02 \[ a^{3} x + \frac{9 a^{2} b x^{\frac{5}{3}}}{5} + 2 a^{2} c x^{\frac{3}{2}} + \frac{9 a b^{2} x^{\frac{7}{3}}}{7} + \frac{36 a b c x^{\frac{13}{6}}}{13} + \frac{3 a c^{2} x^{2}}{2} + \frac{b^{3} x^{3}}{3} + \frac{18 b^{2} c x^{\frac{17}{6}}}{17} + \frac{9 b c^{2} x^{\frac{8}{3}}}{8} + \frac{2 c^{3} x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x + 9*a**2*b*x**(5/3)/5 + 2*a**2*c*x**(3/2) + 9*a*b**2*x**(7/3)/7 + 36*a*b*
c*x**(13/6)/13 + 3*a*c**2*x**2/2 + b**3*x**3/3 + 18*b**2*c*x**(17/6)/17 + 9*b*c*
*2*x**(8/3)/8 + 2*c**3*x**(5/2)/5

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GIAC/XCAS [A]  time = 0.26422, size = 113, normalized size = 0.99 \[ \frac{1}{3} \, b^{3} x^{3} + \frac{18}{17} \, b^{2} c x^{\frac{17}{6}} + \frac{9}{8} \, b c^{2} x^{\frac{8}{3}} + \frac{2}{5} \, c^{3} x^{\frac{5}{2}} + \frac{9}{7} \, a b^{2} x^{\frac{7}{3}} + \frac{36}{13} \, a b c x^{\frac{13}{6}} + \frac{3}{2} \, a c^{2} x^{2} + \frac{9}{5} \, a^{2} b x^{\frac{5}{3}} + 2 \, a^{2} c x^{\frac{3}{2}} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^3*x^3 + 18/17*b^2*c*x^(17/6) + 9/8*b*c^2*x^(8/3) + 2/5*c^3*x^(5/2) + 9/7*a
*b^2*x^(7/3) + 36/13*a*b*c*x^(13/6) + 3/2*a*c^2*x^2 + 9/5*a^2*b*x^(5/3) + 2*a^2*
c*x^(3/2) + a^3*x