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

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

[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^(13/6))/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 = 0.192508, 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, Rules used = {6741, 6742} \[ \frac{9}{5} a^2 b x^{5/3}+2 a^2 c x^{3/2}+a^3 x+\frac{9}{7} a b^2 x^{7/3}+\frac{36}{13} a b c x^{13/6}+\frac{3}{2} a c^2 x^2+\frac{18}{17} b^2 c x^{17/6}+\frac{b^3 x^3}{3}+\frac{9}{8} b c^2 x^{8/3}+\frac{2}{5} c^3 x^{5/2} \]

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^(13/6))/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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin{align*} \int \left (a+c \sqrt{x}+b x^{2/3}\right )^3 \, dx &=6 \operatorname{Subst}\left (\int x^5 \left (a+x^3 (c+b x)\right )^3 \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname{Subst}\left (\int x^5 \left (a+c x^3+b x^4\right )^3 \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname{Subst}\left (\int \left (a^3 x^5+3 a^2 c x^8+3 a^2 b x^9+3 a c^2 x^{11}+6 a b c x^{12}+3 a b^2 x^{13}+c^3 x^{14}+3 b c^2 x^{15}+3 b^2 c x^{16}+b^3 x^{17}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=a^3 x+2 a^2 c x^{3/2}+\frac{9}{5} a^2 b x^{5/3}+\frac{3}{2} a c^2 x^2+\frac{36}{13} a b c x^{13/6}+\frac{9}{7} a b^2 x^{7/3}+\frac{2}{5} c^3 x^{5/2}+\frac{9}{8} b c^2 x^{8/3}+\frac{18}{17} b^2 c x^{17/6}+\frac{b^3 x^3}{3}\\ \end{align*}

Mathematica [A]  time = 0.076006, size = 114, normalized size = 1. \[ \frac{9}{5} a^2 b x^{5/3}+2 a^2 c x^{3/2}+a^3 x+\frac{9}{7} a b^2 x^{7/3}+\frac{36}{13} a b c x^{13/6}+\frac{3}{2} a c^2 x^2+\frac{18}{17} b^2 c x^{17/6}+\frac{b^3 x^3}{3}+\frac{9}{8} b c^2 x^{8/3}+\frac{2}{5} c^3 x^{5/2} \]

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^(13/6))/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 = 0.003, size = 86, normalized size = 0.8 \begin{align*}{\frac{2\,{c}^{3}}{5}{x}^{{\frac{5}{2}}}}+3\,{c}^{2} \left ( 3/8\,b{x}^{8/3}+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}}}} \end{align*}

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*b*x^(8/3)+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 = 1.10472, size = 115, normalized size = 1.01 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(2/3)+c*x^(1/2))^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 = 1.70376, size = 243, normalized size = 2.13 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(2/3)+c*x^(1/2))^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 = 2.75444, size = 116, normalized size = 1.02 \begin{align*} 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} \end{align*}

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 [A]  time = 1.1156, size = 113, normalized size = 0.99 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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