\(\int \frac {a+b x}{x^{5/2}} \, dx\) [434]

Optimal result

Integrand size = 11, antiderivative size = 19 \[ \int \frac {a+b x}{x^{5/2}} \, dx=-\frac {2 a}{3 x^{3/2}}-\frac {2 b}{\sqrt {x}} \]

[Out]

-2/3*a/x^(3/2)-2*b/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} \[ \int \frac {a+b x}{x^{5/2}} \, dx=-\frac {2 a}{3 x^{3/2}}-\frac {2 b}{\sqrt {x}} \]

[In]

Int[(a + b*x)/x^(5/2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x}{x^{5/2}} \, dx=-\frac {2 (a+3 b x)}{3 x^{3/2}} \]

[In]

Integrate[(a + b*x)/x^(5/2),x]

[Out]

(-2*(a + 3*b*x))/(3*x^(3/2))

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63

method	result	size
gosper	\(-\frac {2 \left (3 b x +a \right )}{3 x^{\frac {3}{2}}}\)	\(12\)
trager	\(-\frac {2 \left (3 b x +a \right )}{3 x^{\frac {3}{2}}}\)	\(12\)
risch	\(-\frac {2 \left (3 b x +a \right )}{3 x^{\frac {3}{2}}}\)	\(12\)
derivativedivides	\(-\frac {2 a}{3 x^{\frac {3}{2}}}-\frac {2 b}{\sqrt {x}}\)	\(14\)
default	\(-\frac {2 a}{3 x^{\frac {3}{2}}}-\frac {2 b}{\sqrt {x}}\)	\(14\)

[In]

int((b*x+a)/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \frac {a+b x}{x^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate((b*x+a)/x^(5/2),x, algorithm="fricas")

[Out]

-2/3*(3*b*x + a)/x^(3/2)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{x^{5/2}} \, dx=- \frac {2 a}{3 x^{\frac {3}{2}}} - \frac {2 b}{\sqrt {x}} \]

[In]

integrate((b*x+a)/x**(5/2),x)

[Out]

-2*a/(3*x**(3/2)) - 2*b/sqrt(x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \frac {a+b x}{x^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate((b*x+a)/x^(5/2),x, algorithm="maxima")

[Out]

-2/3*(3*b*x + a)/x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.58 \[ \int \frac {a+b x}{x^{5/2}} \, dx=-\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

integrate((b*x+a)/x^(5/2),x, algorithm="giac")

[Out]

-2/3*(3*b*x + a)/x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {a+b x}{x^{5/2}} \, dx=-\frac {2\,a+6\,b\,x}{3\,x^{3/2}} \]

[In]

int((a + b*x)/x^(5/2),x)

[Out]

-(2*a + 6*b*x)/(3*x^(3/2))