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

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

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Rubi [A]  time = 0.00, antiderivative size = 19, 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*} 2 a \sqrt {x}+\frac {2}{3} b x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*b*x^(3/2))/3

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} \frac {2}{3} \sqrt {x} (3 a+b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(3*a + b*x))/3

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IntegrateAlgebraic [A]  time = 0.01, size = 20, normalized size = 1.05 \begin {gather*} \frac {2}{3} \left (3 a \sqrt {x}+b x^{3/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3*a*Sqrt[x] + b*x^(3/2)))/3

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fricas [A]  time = 0.92, size = 12, normalized size = 0.63 \begin {gather*} \frac {2}{3} \, {\left (b x + 3 \, a\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]

2/3*(b*x + 3*a)*sqrt(x)

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giac [A]  time = 1.08, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{3} \, b x^{\frac {3}{2}} + 2 \, a \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*x^(3/2) + 2*a*sqrt(x)

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maple [A]  time = 0.00, size = 13, normalized size = 0.68 \begin {gather*} \frac {2 \left (b x +3 a \right ) \sqrt {x}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.32, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{3} \, b x^{\frac {3}{2}} + 2 \, a \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*x^(3/2) + 2*a*sqrt(x)

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mupad [B]  time = 0.03, size = 12, normalized size = 0.63 \begin {gather*} \frac {2\,\sqrt {x}\,\left (3\,a+b\,x\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/x^(1/2),x)

[Out]

(2*x^(1/2)*(3*a + b*x))/3

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sympy [A]  time = 0.16, size = 17, normalized size = 0.89 \begin {gather*} 2 a \sqrt {x} + \frac {2 b x^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) + 2*b*x**(3/2)/3

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