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*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 = 0.165853, 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, 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*}

Mathematica [A]  time = 0.0460942, 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 \begin{align*}{\frac{{c}^{2}{x}^{2}}{2}}+2\,c \left ({\frac{6\,b}{13}{x}^{{\frac{13}{6}}}}+2/3\,a{x}^{3/2} \right ) +x{a}^{2}+{\frac{3\,{b}^{2}}{7}{x}^{{\frac{7}{3}}}}+{\frac{6\,ab}{5}{x}^{{\frac{5}{3}}}} \end{align*}

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

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Maxima [A]  time = 1.11709, size = 61, normalized size = 1. \begin{align*} \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 \end{align*}

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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Fricas [A]  time = 1.70177, size = 130, normalized size = 2.13 \begin{align*} \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 \end{align*}

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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Sympy [A]  time = 1.90677, size = 60, normalized size = 0.98 \begin{align*} 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} \end{align*}

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 [A]  time = 1.14186, size = 58, normalized size = 0.95 \begin{align*} \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 \end{align*}

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