3.2126 \(\int \frac{(a+b \sqrt{x})^2}{x^2} \, dx\)

Optimal. Leaf size=24 \[ -\frac{a^2}{x}-\frac{4 a b}{\sqrt{x}}+b^2 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0132717, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{a^2}{x}-\frac{4 a b}{\sqrt{x}}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^2/x^2,x]

[Out]

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

Rule 266

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]]

Rule 43

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*} \int \frac{\left (a+b \sqrt{x}\right )^2}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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]  time = 0.0136935, size = 24, normalized size = 1. \[ -\frac{a^2}{x}-\frac{4 a b}{\sqrt{x}}+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^2/x^2,x]

[Out]

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

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Maple [A]  time = 0.002, size = 23, normalized size = 1. \begin{align*} -{\frac{{a}^{2}}{x}}+{b}^{2}\ln \left ( x \right ) -4\,{\frac{ab}{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.954224, size = 31, normalized size = 1.29 \begin{align*} b^{2} \log \left (x\right ) - \frac{4 \, a b \sqrt{x} + a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.43834, size = 65, normalized size = 2.71 \begin{align*} \frac{2 \, b^{2} x \log \left (\sqrt{x}\right ) - 4 \, a b \sqrt{x} - a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.413131, size = 20, normalized size = 0.83 \begin{align*} - \frac{a^{2}}{x} - \frac{4 a b}{\sqrt{x}} + b^{2} \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/x - 4*a*b/sqrt(x) + b**2*log(x)

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Giac [A]  time = 1.11112, size = 32, normalized size = 1.33 \begin{align*} b^{2} \log \left ({\left | x \right |}\right ) - \frac{4 \, a b \sqrt{x} + a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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