3.2114 \(\int (a+b \sqrt{x}) x \, dx\)

Optimal. Leaf size=19 \[ \frac{a x^2}{2}+\frac{2}{5} b x^{5/2} \]

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

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Rubi [A]  time = 0.0048254, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{a x^2}{2}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])*x,x]

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

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

Mathematica [A]  time = 0.0052623, size = 19, normalized size = 1. \[ \frac{a x^2}{2}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])*x,x]

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

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Maple [A]  time = 0.001, size = 14, normalized size = 0.7 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{2\,b}{5}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2+2/5*b*x^(5/2)

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Maxima [B]  time = 0.957932, size = 86, normalized size = 4.53 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5}}{5 \, b^{4}} - \frac{3 \,{\left (b \sqrt{x} + a\right )}^{4} a}{2 \, b^{4}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3} a^{2}}{b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{2} a^{3}}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44451, size = 36, normalized size = 1.89 \begin{align*} \frac{2}{5} \, b x^{\frac{5}{2}} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 1/2*a*x^2

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Sympy [A]  time = 0.909608, size = 15, normalized size = 0.79 \begin{align*} \frac{a x^{2}}{2} + \frac{2 b x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + 2*b*x**(5/2)/5

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Giac [A]  time = 1.10464, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{5} \, b x^{\frac{5}{2}} + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 1/2*a*x^2