\(\int \frac {a+b x}{\sqrt {x}} \, dx\) [432]

Optimal result

Integrand size = 11, antiderivative size = 19 \[ \int \frac {a+b x}{\sqrt {x}} \, dx=2 a \sqrt {x}+\frac {2}{3} b x^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {a+b x}{\sqrt {x}} \, dx=2 a \sqrt {x}+\frac {2}{3} b x^{3/2} \]

[In]

Int[(a + b*x)/Sqrt[x],x]

[Out]

2*a*Sqrt[x] + (2*b*x^(3/2))/3

Rule 45

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {a+b x}{\sqrt {x}} \, dx=\frac {2}{3} \sqrt {x} (3 a+b x) \]

[In]

Integrate[(a + b*x)/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(3*a + b*x))/3

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68

method	result	size
gosper	\(\frac {2 \sqrt {x}\, \left (b x +3 a \right )}{3}\)	\(13\)
trager	\(\left (\frac {2 b x}{3}+2 a \right ) \sqrt {x}\)	\(13\)
risch	\(\frac {2 \sqrt {x}\, \left (b x +3 a \right )}{3}\)	\(13\)
derivativedivides	\(\frac {2 b \,x^{\frac {3}{2}}}{3}+2 a \sqrt {x}\)	\(14\)
default	\(\frac {2 b \,x^{\frac {3}{2}}}{3}+2 a \sqrt {x}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{\sqrt {x}} \, dx=\frac {2}{3} \, {\left (b x + 3 \, a\right )} \sqrt {x} \]

[In]

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

[Out]

2/3*(b*x + 3*a)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x}{\sqrt {x}} \, dx=2 a \sqrt {x} + \frac {2 b x^{\frac {3}{2}}}{3} \]

[In]

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

[Out]

2*a*sqrt(x) + 2*b*x**(3/2)/3

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {a+b x}{\sqrt {x}} \, dx=\frac {2}{3} \, b x^{\frac {3}{2}} + 2 \, a \sqrt {x} \]

[In]

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

[Out]

2/3*b*x^(3/2) + 2*a*sqrt(x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {a+b x}{\sqrt {x}} \, dx=\frac {2}{3} \, b x^{\frac {3}{2}} + 2 \, a \sqrt {x} \]

[In]

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

[Out]

2/3*b*x^(3/2) + 2*a*sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{\sqrt {x}} \, dx=\frac {2\,\sqrt {x}\,\left (3\,a+b\,x\right )}{3} \]

[In]

int((a + b*x)/x^(1/2),x)

[Out]

(2*x^(1/2)*(3*a + b*x))/3