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

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

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{3} a x^{3/2}+\frac {2}{5} b x^{5/2} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.81 \begin {gather*} \frac {2}{15} x^{3/2} (5 a+3 b x) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} \frac {2}{15} \left (5 a x^{3/2}+3 b x^{5/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[x]*(a + b*x),x]

[Out]

(2*(5*a*x^(3/2) + 3*b*x^(5/2)))/15

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fricas [A]  time = 0.99, size = 16, normalized size = 0.76 \begin {gather*} \frac {2}{15} \, {\left (3 \, b x^{2} + 5 \, a x\right )} \sqrt {x} \end {gather*}

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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giac [A]  time = 0.86, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{5} \, b x^{\frac {5}{2}} + \frac {2}{3} \, a x^{\frac {3}{2}} \end {gather*}

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)

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maple [A]  time = 0.00, size = 14, normalized size = 0.67 \begin {gather*} \frac {2 \left (3 b x +5 a \right ) x^{\frac {3}{2}}}{15} \end {gather*}

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 = 1.34, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{5} \, b x^{\frac {5}{2}} + \frac {2}{3} \, a x^{\frac {3}{2}} \end {gather*}

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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mupad [B]  time = 0.02, size = 13, normalized size = 0.62 \begin {gather*} \frac {2\,x^{3/2}\,\left (5\,a+3\,b\,x\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b*x),x)

[Out]

(2*x^(3/2)*(5*a + 3*b*x))/15

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sympy [A]  time = 1.62, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 a x^{\frac {3}{2}}}{3} + \frac {2 b x^{\frac {5}{2}}}{5} \end {gather*}

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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