\(\int \frac {(a+\frac {b}{x})^2}{\sqrt {x}} \, dx\) [1656]

Optimal result

Integrand size = 15, antiderivative size = 32 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=-\frac {2 b^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 a^2 \sqrt {x} \]

[Out]

-2/3*b^2/x^(3/2)-4*a*b/x^(1/2)+2*a^2*x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {269, 45} \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=2 a^2 \sqrt {x}-\frac {4 a b}{\sqrt {x}}-\frac {2 b^2}{3 x^{3/2}} \]

[In]

Int[(a + b/x)^2/Sqrt[x],x]

[Out]

(-2*b^2)/(3*x^(3/2)) - (4*a*b)/Sqrt[x] + 2*a^2*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])

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a x)^2}{x^{5/2}} \, dx \\ & = \int \left (\frac {b^2}{x^{5/2}}+\frac {2 a b}{x^{3/2}}+\frac {a^2}{\sqrt {x}}\right ) \, dx \\ & = -\frac {2 b^2}{3 x^{3/2}}-\frac {4 a b}{\sqrt {x}}+2 a^2 \sqrt {x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=-\frac {2 \left (b^2+6 a b x-3 a^2 x^2\right )}{3 x^{3/2}} \]

[In]

Integrate[(a + b/x)^2/Sqrt[x],x]

[Out]

(-2*(b^2 + 6*a*b*x - 3*a^2*x^2))/(3*x^(3/2))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
gosper	\(\frac {-4 a b x -\frac {2}{3} b^{2}+2 a^{2} x^{2}}{x^{\frac {3}{2}}}\)	\(25\)
derivativedivides	\(-\frac {2 b^{2}}{3 x^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {x}}+2 a^{2} \sqrt {x}\)	\(25\)
default	\(-\frac {2 b^{2}}{3 x^{\frac {3}{2}}}-\frac {4 a b}{\sqrt {x}}+2 a^{2} \sqrt {x}\)	\(25\)
trager	\(\frac {-4 a b x -\frac {2}{3} b^{2}+2 a^{2} x^{2}}{x^{\frac {3}{2}}}\)	\(25\)
risch	\(\frac {-4 a b x -\frac {2}{3} b^{2}+2 a^{2} x^{2}}{x^{\frac {3}{2}}}\)	\(25\)

[In]

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

[Out]

2/3*(3*a^2*x^2-6*a*b*x-b^2)/x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=\frac {2 \, {\left (3 \, a^{2} x^{2} - 6 \, a b x - b^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

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

[Out]

2/3*(3*a^2*x^2 - 6*a*b*x - b^2)/x^(3/2)

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=2 a^{2} \sqrt {x} - \frac {4 a b}{\sqrt {x}} - \frac {2 b^{2}}{3 x^{\frac {3}{2}}} \]

[In]

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

[Out]

2*a**2*sqrt(x) - 4*a*b/sqrt(x) - 2*b**2/(3*x**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=2 \, a^{2} \sqrt {x} - \frac {4 \, a b}{\sqrt {x}} - \frac {2 \, b^{2}}{3 \, x^{\frac {3}{2}}} \]

[In]

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

[Out]

2*a^2*sqrt(x) - 4*a*b/sqrt(x) - 2/3*b^2/x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=2 \, a^{2} \sqrt {x} - \frac {2 \, {\left (6 \, a b x + b^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

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

[Out]

2*a^2*sqrt(x) - 2/3*(6*a*b*x + b^2)/x^(3/2)

Mupad [B] (verification not implemented)

Time = 5.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^2}{\sqrt {x}} \, dx=-\frac {-6\,a^2\,x^2+12\,a\,b\,x+2\,b^2}{3\,x^{3/2}} \]

[In]

int((a + b/x)^2/x^(1/2),x)

[Out]

-(2*b^2 - 6*a^2*x^2 + 12*a*b*x)/(3*x^(3/2))