\(\int \frac {(a+b \sqrt {x})^2}{x^2} \, dx\) [2126]

Optimal result

Integrand size = 15, antiderivative size = 24 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx=-\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, 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 = {272, 45} \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx=-\frac {a^2}{x}-\frac {4 a b}{\sqrt {x}}+b^2 \log (x) \]

[In]

Int[(a + b*Sqrt[x])^2/x^2,x]

[Out]

-(a^2/x) - (4*a*b)/Sqrt[x] + b^2*Log[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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx=-\frac {a \left (a+4 b \sqrt {x}\right )}{x}+b^2 \log (x) \]

[In]

Integrate[(a + b*Sqrt[x])^2/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 5.79 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(-\frac {a^{2}}{x}+b^{2} \ln \left (x \right )-\frac {4 a b}{\sqrt {x}}\)	\(23\)
default	\(-\frac {a^{2}}{x}+b^{2} \ln \left (x \right )-\frac {4 a b}{\sqrt {x}}\)	\(23\)
trager	\(\frac {a^{2} \left (-1+x \right )}{x}-\frac {4 a b}{\sqrt {x}}+b^{2} \ln \left (x \right )\)	\(25\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx=\frac {2 \, b^{2} x \log \left (\sqrt {x}\right ) - 4 \, a b \sqrt {x} - a^{2}}{x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx=- \frac {a^{2}}{x} - \frac {4 a b}{\sqrt {x}} + b^{2} \log {\left (x \right )} \]

[In]

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

[Out]

-a**2/x - 4*a*b/sqrt(x) + b**2*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx=b^{2} \log \left (x\right ) - \frac {4 \, a b \sqrt {x} + a^{2}}{x} \]

[In]

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

[Out]

b^2*log(x) - (4*a*b*sqrt(x) + a^2)/x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx=b^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a b \sqrt {x} + a^{2}}{x} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.76 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \sqrt {x}\right )^2}{x^2} \, dx=2\,b^2\,\ln \left (\sqrt {x}\right )-\frac {a^2+4\,a\,b\,\sqrt {x}}{x} \]

[In]

int((a + b*x^(1/2))^2/x^2,x)

[Out]

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