3.2113 \(\int (a+b \sqrt{x}) x^2 \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^3}{3}+\frac{2}{7} b x^{7/2} \]

[Out]

(a*x^3)/3 + (2*b*x^(7/2))/7

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Rubi [A]  time = 0.0048315, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{a x^3}{3}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])*x^2,x]

[Out]

(a*x^3)/3 + (2*b*x^(7/2))/7

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0063185, size = 19, normalized size = 1. \[ \frac{a x^3}{3}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])*x^2,x]

[Out]

(a*x^3)/3 + (2*b*x^(7/2))/7

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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \begin{align*}{\frac{a{x}^{3}}{3}}+{\frac{2\,b}{7}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.97545, size = 132, normalized size = 6.95 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7}}{7 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{6} a}{3 \, b^{6}} + \frac{4 \,{\left (b \sqrt{x} + a\right )}^{5} a^{2}}{b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{4} a^{3}}{b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{3} a^{4}}{3 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{2} a^{5}}{b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.46357, size = 36, normalized size = 1.89 \begin{align*} \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*b*x^(7/2) + 1/3*a*x^3

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Sympy [A]  time = 0.965971, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{3}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + 2*b*x**(7/2)/7

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Giac [A]  time = 1.08505, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*b*x^(7/2) + 1/3*a*x^3