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

Optimal. Leaf size=21 \[ \frac{2}{3} a x^{3/2}+\frac{2}{5} b x^{5/2} \]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(5/2))/5

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Rubi [A]  time = 0.0041047, antiderivative size = 21, 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 = {43} \[ \frac{2}{3} a x^{3/2}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + b*x),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(5/2))/5

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

Mathematica [A]  time = 0.004014, size = 17, normalized size = 0.81 \[ \frac{2}{15} x^{3/2} (5 a+3 b x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + b*x),x]

[Out]

(2*x^(3/2)*(5*a + 3*b*x))/15

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \begin{align*}{\frac{6\,bx+10\,a}{15}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*x^(3/2)*(3*b*x+5*a)

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Maxima [A]  time = 1.00839, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{5} \, b x^{\frac{5}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 2/3*a*x^(3/2)

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Fricas [A]  time = 1.57422, size = 43, normalized size = 2.05 \begin{align*} \frac{2}{15} \,{\left (3 \, b x^{2} + 5 \, a x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*b*x^2 + 5*a*x)*sqrt(x)

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Sympy [A]  time = 2.04054, size = 19, normalized size = 0.9 \begin{align*} \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(3/2)/3 + 2*b*x**(5/2)/5

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Giac [A]  time = 1.25034, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{5} \, b x^{\frac{5}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 2/3*a*x^(3/2)