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

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

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Rubi [A]  time = 0.19, antiderivative size = 114, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.09, size = 114, normalized size = 1.00 \[ a^3 x+\frac {9}{5} a^2 b x^{5/3}+2 a^2 c x^{3/2}+\frac {9}{7} a b^2 x^{7/3}+\frac {36}{13} a b c x^{13/6}+\frac {3}{2} a c^2 x^2+\frac {b^3 x^3}{3}+\frac {18}{17} b^2 c x^{17/6}+\frac {9}{8} b c^2 x^{8/3}+\frac {2}{5} c^3 x^{5/2} \]

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^(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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fricas [A]  time = 0.42, size = 91, normalized size = 0.80 \[ \frac {1}{3} \, b^{3} x^{3} + \frac {18}{17} \, b^{2} c x^{\frac {17}{6}} + \frac {9}{7} \, a b^{2} x^{\frac {7}{3}} + \frac {36}{13} \, a b c x^{\frac {13}{6}} + \frac {3}{2} \, a c^{2} x^{2} + a^{3} x + \frac {9}{40} \, {\left (5 \, b c^{2} x^{2} + 8 \, a^{2} b x\right )} x^{\frac {2}{3}} + \frac {2}{5} \, {\left (c^{3} x^{2} + 5 \, a^{2} c x\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [A]  time = 0.34, size = 84, normalized size = 0.74 \[ \frac {1}{3} \, b^{3} x^{3} + \frac {18}{17} \, b^{2} c x^{\frac {17}{6}} + \frac {9}{8} \, b c^{2} x^{\frac {8}{3}} + \frac {2}{5} \, c^{3} x^{\frac {5}{2}} + \frac {9}{7} \, a b^{2} x^{\frac {7}{3}} + \frac {36}{13} \, a b c x^{\frac {13}{6}} + \frac {3}{2} \, a c^{2} x^{2} + \frac {9}{5} \, a^{2} b x^{\frac {5}{3}} + 2 \, a^{2} c x^{\frac {3}{2}} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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

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maple [A]  time = 0.00, size = 86, normalized size = 0.75 \[ \frac {b^{3} x^{3}}{3}+\frac {2 c^{3} x^{\frac {5}{2}}}{5}+\frac {9 a \,b^{2} x^{\frac {7}{3}}}{7}+\frac {9 a^{2} b \,x^{\frac {5}{3}}}{5}+a^{3} x +3 \left (\frac {3 b \,x^{\frac {8}{3}}}{8}+\frac {a \,x^{2}}{2}\right ) c^{2}+3 \left (\frac {6 b^{2} x^{\frac {17}{6}}}{17}+\frac {12 a b \,x^{\frac {13}{6}}}{13}+\frac {2 a^{2} x^{\frac {3}{2}}}{3}\right ) c \]

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 = 0.44, size = 85, normalized size = 0.75 \[ \frac {1}{3} \, b^{3} x^{3} + \frac {18}{17} \, b^{2} c x^{\frac {17}{6}} + \frac {9}{8} \, b c^{2} x^{\frac {8}{3}} + \frac {2}{5} \, c^{3} x^{\frac {5}{2}} + a^{3} x + \frac {1}{5} \, {\left (9 \, b x^{\frac {5}{3}} + 10 \, c x^{\frac {3}{2}}\right )} a^{2} + \frac {3}{182} \, {\left (78 \, b^{2} x^{\frac {7}{3}} + 168 \, b c x^{\frac {13}{6}} + 91 \, c^{2} x^{2}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [B]  time = 0.06, size = 84, normalized size = 0.74 \[ a^3\,x+\frac {b^3\,x^3}{3}+\frac {2\,c^3\,x^{5/2}}{5}+\frac {9\,a^2\,b\,x^{5/3}}{5}+\frac {9\,a\,b^2\,x^{7/3}}{7}+\frac {3\,a\,c^2\,x^2}{2}+2\,a^2\,c\,x^{3/2}+\frac {9\,b\,c^2\,x^{8/3}}{8}+\frac {18\,b^2\,c\,x^{17/6}}{17}+\frac {36\,a\,b\,c\,x^{13/6}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(2/3) + c*x^(1/2))^3,x)

[Out]

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

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sympy [A]  time = 3.45, 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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