\(\int (a+\frac {b}{x})^2 \sqrt {x} \, dx\) [1655]

Optimal result

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

[Out]

2/3*a^2*x^(3/2)-2*b^2/x^(1/2)+4*a*b*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} \[ \int \left (a+\frac {b}{x}\right )^2 \sqrt {x} \, dx=\frac {2}{3} a^2 x^{3/2}+4 a b \sqrt {x}-\frac {2 b^2}{\sqrt {x}} \]

[In]

Int[(a + b/x)^2*Sqrt[x],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (a+\frac {b}{x}\right )^2 \sqrt {x} \, dx=-\frac {2 \left (3 b^2-6 a b x-a^2 x^2\right )}{3 \sqrt {x}} \]

[In]

Integrate[(a + b/x)^2*Sqrt[x],x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75

method	result	size
gosper	\(\frac {-2 b^{2}+\frac {2}{3} a^{2} x^{2}+4 a b x}{\sqrt {x}}\)	\(24\)
trager	\(\frac {-2 b^{2}+\frac {2}{3} a^{2} x^{2}+4 a b x}{\sqrt {x}}\)	\(24\)
risch	\(\frac {-2 b^{2}+\frac {2}{3} a^{2} x^{2}+4 a b x}{\sqrt {x}}\)	\(24\)
derivativedivides	\(\frac {2 a^{2} x^{\frac {3}{2}}}{3}-\frac {2 b^{2}}{\sqrt {x}}+4 a b \sqrt {x}\)	\(25\)
default	\(\frac {2 a^{2} x^{\frac {3}{2}}}{3}-\frac {2 b^{2}}{\sqrt {x}}+4 a b \sqrt {x}\)	\(25\)

[In]

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

[Out]

2/3*(a^2*x^2+6*a*b*x-3*b^2)/x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \left (a+\frac {b}{x}\right )^2 \sqrt {x} \, dx=\frac {2 \, {\left (a^{2} x^{2} + 6 \, a b x - 3 \, b^{2}\right )}}{3 \, \sqrt {x}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \left (a+\frac {b}{x}\right )^2 \sqrt {x} \, dx=\frac {2 a^{2} x^{\frac {3}{2}}}{3} + 4 a b \sqrt {x} - \frac {2 b^{2}}{\sqrt {x}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \left (a+\frac {b}{x}\right )^2 \sqrt {x} \, dx=\frac {2}{3} \, {\left (a^{2} + \frac {6 \, a b}{x}\right )} x^{\frac {3}{2}} - \frac {2 \, b^{2}}{\sqrt {x}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \left (a+\frac {b}{x}\right )^2 \sqrt {x} \, dx=\frac {2}{3} \, a^{2} x^{\frac {3}{2}} + 4 \, a b \sqrt {x} - \frac {2 \, b^{2}}{\sqrt {x}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \left (a+\frac {b}{x}\right )^2 \sqrt {x} \, dx=\frac {2\,a^2\,x^2+12\,a\,b\,x-6\,b^2}{3\,\sqrt {x}} \]

[In]

int(x^(1/2)*(a + b/x)^2,x)

[Out]

(2*a^2*x^2 - 6*b^2 + 12*a*b*x)/(3*x^(1/2))