\(\int \frac {a+b x}{x^{3/2}} \, dx\) [433]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {a+b x}{x^{3/2}} \, dx=-\frac {2 a}{\sqrt {x}}+2 b \sqrt {x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, 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} \[ \int \frac {a+b x}{x^{3/2}} \, dx=2 b \sqrt {x}-\frac {2 a}{\sqrt {x}} \]

[In]

Int[(a + b*x)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + 2*b*Sqrt[x]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^{3/2}} \, dx=-\frac {2 (a-b x)}{\sqrt {x}} \]

[In]

Integrate[(a + b*x)/x^(3/2),x]

[Out]

(-2*(a - b*x))/Sqrt[x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71

method	result	size
gosper	\(-\frac {2 \left (-b x +a \right )}{\sqrt {x}}\)	\(12\)
trager	\(-\frac {2 \left (-b x +a \right )}{\sqrt {x}}\)	\(12\)
risch	\(-\frac {2 \left (-b x +a \right )}{\sqrt {x}}\)	\(12\)
derivativedivides	\(-\frac {2 a}{\sqrt {x}}+2 b \sqrt {x}\)	\(14\)
default	\(-\frac {2 a}{\sqrt {x}}+2 b \sqrt {x}\)	\(14\)

[In]

int((b*x+a)/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x}{x^{3/2}} \, dx=\frac {2 \, {\left (b x - a\right )}}{\sqrt {x}} \]

[In]

integrate((b*x+a)/x^(3/2),x, algorithm="fricas")

[Out]

2*(b*x - a)/sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {a+b x}{x^{3/2}} \, dx=- \frac {2 a}{\sqrt {x}} + 2 b \sqrt {x} \]

[In]

integrate((b*x+a)/x**(3/2),x)

[Out]

-2*a/sqrt(x) + 2*b*sqrt(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^{3/2}} \, dx=2 \, b \sqrt {x} - \frac {2 \, a}{\sqrt {x}} \]

[In]

integrate((b*x+a)/x^(3/2),x, algorithm="maxima")

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{x^{3/2}} \, dx=2 \, b \sqrt {x} - \frac {2 \, a}{\sqrt {x}} \]

[In]

integrate((b*x+a)/x^(3/2),x, algorithm="giac")

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x}{x^{3/2}} \, dx=-\frac {2\,\left (a-b\,x\right )}{\sqrt {x}} \]

[In]

int((a + b*x)/x^(3/2),x)

[Out]

-(2*(a - b*x))/x^(1/2)