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

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*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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Rubi [A]  time = 0.279751, antiderivative size = 61, 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 \[ a^2 x+\frac{6}{5} a b x^{5/3}+\frac{4}{3} a c x^{3/2}+\frac{3}{7} b^2 x^{7/3}+\frac{12}{13} b c x^{13/6}+\frac{c^2 x^2}{2} \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{6 a b x^{\frac{5}{3}}}{5} + \frac{4 a c x^{\frac{3}{2}}}{3} + \frac{3 b^{2} x^{\frac{7}{3}}}{7} + \frac{12 b c x^{\frac{13}{6}}}{13} + c^{2} \int x\, dx + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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*Integral(x, x) + Integral(a**2, x)

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Mathematica [A]  time = 0.0289994, size = 61, normalized size = 1. \[ a^2 x+\frac{6}{5} a b x^{5/3}+\frac{4}{3} a c x^{3/2}+\frac{3}{7} b^2 x^{7/3}+\frac{12}{13} b c x^{13/6}+\frac{c^2 x^2}{2} \]

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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Maple [A]  time = 0.003, size = 46, normalized size = 0.8 \[{\frac{{c}^{2}{x}^{2}}{2}}+2\,c \left ({\frac{6\,b}{13}{x}^{{\frac{13}{6}}}}+2/3\,a{x}^{3/2} \right ) +{a}^{2}x+{\frac{3\,{b}^{2}}{7}{x}^{{\frac{7}{3}}}}+{\frac{6\,ab}{5}{x}^{{\frac{5}{3}}}} \]

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 = 0.723466, size = 61, normalized size = 1. \[ \frac{3}{7} \, b^{2} x^{\frac{7}{3}} + \frac{12}{13} \, b c x^{\frac{13}{6}} + \frac{1}{2} \, c^{2} x^{2} + a^{2} x + \frac{2}{15} \,{\left (9 \, b x^{\frac{5}{3}} + 10 \, c x^{\frac{3}{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(2/3) + c*sqrt(x) + a)^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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Fricas [A]  time = 0.261377, size = 58, normalized size = 0.95 \[ \frac{3}{7} \, b^{2} x^{\frac{7}{3}} + \frac{12}{13} \, b c x^{\frac{13}{6}} + \frac{1}{2} \, c^{2} x^{2} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{4}{3} \, a c x^{\frac{3}{2}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(2/3) + c*sqrt(x) + a)^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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Sympy [A]  time = 1.22844, 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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GIAC/XCAS [A]  time = 0.268326, size = 58, normalized size = 0.95 \[ \frac{3}{7} \, b^{2} x^{\frac{7}{3}} + \frac{12}{13} \, b c x^{\frac{13}{6}} + \frac{1}{2} \, c^{2} x^{2} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{4}{3} \, a c x^{\frac{3}{2}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(2/3) + c*sqrt(x) + a)^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