3.2132 \(\int (a+b \sqrt{x})^3 x^2 \, dx\)

Optimal. Leaf size=47 \[ \frac{6}{7} a^2 b x^{7/2}+\frac{a^3 x^3}{3}+\frac{3}{4} a b^2 x^4+\frac{2}{9} b^3 x^{9/2} \]

[Out]

(a^3*x^3)/3 + (6*a^2*b*x^(7/2))/7 + (3*a*b^2*x^4)/4 + (2*b^3*x^(9/2))/9

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Rubi [A]  time = 0.0277525, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{6}{7} a^2 b x^{7/2}+\frac{a^3 x^3}{3}+\frac{3}{4} a b^2 x^4+\frac{2}{9} b^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3*x^2,x]

[Out]

(a^3*x^3)/3 + (6*a^2*b*x^(7/2))/7 + (3*a*b^2*x^4)/4 + (2*b^3*x^(9/2))/9

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0173192, size = 47, normalized size = 1. \[ \frac{6}{7} a^2 b x^{7/2}+\frac{a^3 x^3}{3}+\frac{3}{4} a b^2 x^4+\frac{2}{9} b^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3*x^2,x]

[Out]

(a^3*x^3)/3 + (6*a^2*b*x^(7/2))/7 + (3*a*b^2*x^4)/4 + (2*b^3*x^(9/2))/9

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{3}}{3}}+{\frac{6\,b{a}^{2}}{7}{x}^{{\frac{7}{2}}}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{2\,{b}^{3}}{9}{x}^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3+6/7*a^2*b*x^(7/2)+3/4*a*b^2*x^4+2/9*b^3*x^(9/2)

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Maxima [B]  time = 0.957928, size = 132, normalized size = 2.81 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{9}}{9 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{8} a}{4 \, b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )}^{7} a^{2}}{7 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{6} a^{3}}{3 \, b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5} a^{4}}{b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{5}}{2 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b*sqrt(x) + a)^9/b^6 - 5/4*(b*sqrt(x) + a)^8*a/b^6 + 20/7*(b*sqrt(x) + a)^7*a^2/b^6 - 10/3*(b*sqrt(x) + a
)^6*a^3/b^6 + 2*(b*sqrt(x) + a)^5*a^4/b^6 - 1/2*(b*sqrt(x) + a)^4*a^5/b^6

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Fricas [A]  time = 1.44282, size = 96, normalized size = 2.04 \begin{align*} \frac{3}{4} \, a b^{2} x^{4} + \frac{1}{3} \, a^{3} x^{3} + \frac{2}{63} \,{\left (7 \, b^{3} x^{4} + 27 \, a^{2} b x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*a*b^2*x^4 + 1/3*a^3*x^3 + 2/63*(7*b^3*x^4 + 27*a^2*b*x^3)*sqrt(x)

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Sympy [A]  time = 1.43242, size = 44, normalized size = 0.94 \begin{align*} \frac{a^{3} x^{3}}{3} + \frac{6 a^{2} b x^{\frac{7}{2}}}{7} + \frac{3 a b^{2} x^{4}}{4} + \frac{2 b^{3} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + 6*a**2*b*x**(7/2)/7 + 3*a*b**2*x**4/4 + 2*b**3*x**(9/2)/9

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Giac [A]  time = 1.14254, size = 47, normalized size = 1. \begin{align*} \frac{2}{9} \, b^{3} x^{\frac{9}{2}} + \frac{3}{4} \, a b^{2} x^{4} + \frac{6}{7} \, a^{2} b x^{\frac{7}{2}} + \frac{1}{3} \, a^{3} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*b^3*x^(9/2) + 3/4*a*b^2*x^4 + 6/7*a^2*b*x^(7/2) + 1/3*a^3*x^3