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

Optimal. Leaf size=61 \[ a^2 x+\frac {6}{5} a b x^{5/3}+\frac {4}{3} a c x^{3/2}+\frac {3}{7} b^2 x^{7/3}+\frac {12}{13} b c x^{13/6}+\frac {c^2 x^2}{2} \]

[Out]

a^2*x+4/3*a*c*x^(3/2)+6/5*a*b*x^(5/3)+1/2*c^2*x^2+12/13*b*c*x^(13/6)+3/7*b^2*x^(7/3)

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Rubi [A]  time = 0.17, antiderivative size = 61, 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} \[ a^2 x+\frac {6}{5} a b x^{5/3}+\frac {4}{3} a c x^{3/2}+\frac {3}{7} b^2 x^{7/3}+\frac {12}{13} b c x^{13/6}+\frac {c^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*Sqrt[x] + b*x^(2/3))^2,x]

[Out]

a^2*x + (4*a*c*x^(3/2))/3 + (6*a*b*x^(5/3))/5 + (c^2*x^2)/2 + (12*b*c*x^(13/6))/13 + (3*b^2*x^(7/3))/7

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

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Mathematica [A]  time = 0.05, size = 61, normalized size = 1.00 \[ a^2 x+\frac {6}{5} a b x^{5/3}+\frac {4}{3} a c x^{3/2}+\frac {3}{7} b^2 x^{7/3}+\frac {12}{13} b c x^{13/6}+\frac {c^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*x + (4*a*c*x^(3/2))/3 + (6*a*b*x^(5/3))/5 + (c^2*x^2)/2 + (12*b*c*x^(13/6))/13 + (3*b^2*x^(7/3))/7

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fricas [A]  time = 0.40, size = 43, normalized size = 0.70 \[ \frac {3}{7} \, b^{2} x^{\frac {7}{3}} + \frac {12}{13} \, b c x^{\frac {13}{6}} + \frac {1}{2} \, c^{2} x^{2} + \frac {6}{5} \, a b x^{\frac {5}{3}} + \frac {4}{3} \, a c x^{\frac {3}{2}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*b^2*x^(7/3) + 12/13*b*c*x^(13/6) + 1/2*c^2*x^2 + 6/5*a*b*x^(5/3) + 4/3*a*c*x^(3/2) + a^2*x

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giac [A]  time = 0.43, size = 43, normalized size = 0.70 \[ \frac {3}{7} \, b^{2} x^{\frac {7}{3}} + \frac {12}{13} \, b c x^{\frac {13}{6}} + \frac {1}{2} \, c^{2} x^{2} + \frac {6}{5} \, a b x^{\frac {5}{3}} + \frac {4}{3} \, a c x^{\frac {3}{2}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*b^2*x^(7/3) + 12/13*b*c*x^(13/6) + 1/2*c^2*x^2 + 6/5*a*b*x^(5/3) + 4/3*a*c*x^(3/2) + a^2*x

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maple [A]  time = 0.00, size = 46, normalized size = 0.75 \[ \frac {3 b^{2} x^{\frac {7}{3}}}{7}+\frac {c^{2} x^{2}}{2}+\frac {6 a b \,x^{\frac {5}{3}}}{5}+a^{2} x +2 \left (\frac {6 b \,x^{\frac {13}{6}}}{13}+\frac {2 a \,x^{\frac {3}{2}}}{3}\right ) c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.44, size = 45, normalized size = 0.74 \[ \frac {3}{7} \, b^{2} x^{\frac {7}{3}} + \frac {12}{13} \, b c x^{\frac {13}{6}} + \frac {1}{2} \, c^{2} x^{2} + a^{2} x + \frac {2}{15} \, {\left (9 \, b x^{\frac {5}{3}} + 10 \, c x^{\frac {3}{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/7*b^2*x^(7/3) + 12/13*b*c*x^(13/6) + 1/2*c^2*x^2 + a^2*x + 2/15*(9*b*x^(5/3) + 10*c*x^(3/2))*a

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mupad [B]  time = 3.36, size = 43, normalized size = 0.70 \[ a^2\,x+\frac {3\,b^2\,x^{7/3}}{7}+\frac {c^2\,x^2}{2}+\frac {6\,a\,b\,x^{5/3}}{5}+\frac {4\,a\,c\,x^{3/2}}{3}+\frac {12\,b\,c\,x^{13/6}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(2/3) + c*x^(1/2))^2,x)

[Out]

a^2*x + (3*b^2*x^(7/3))/7 + (c^2*x^2)/2 + (6*a*b*x^(5/3))/5 + (4*a*c*x^(3/2))/3 + (12*b*c*x^(13/6))/13

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sympy [A]  time = 2.42, size = 60, normalized size = 0.98 \[ a^{2} x + \frac {6 a b x^{\frac {5}{3}}}{5} + \frac {4 a c x^{\frac {3}{2}}}{3} + \frac {3 b^{2} x^{\frac {7}{3}}}{7} + \frac {12 b c x^{\frac {13}{6}}}{13} + \frac {c^{2} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x + 6*a*b*x**(5/3)/5 + 4*a*c*x**(3/2)/3 + 3*b**2*x**(7/3)/7 + 12*b*c*x**(13/6)/13 + c**2*x**2/2

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