3.2127 \(\int \frac{(a+b \sqrt{x})^2}{x^3} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]

[Out]

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

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Rubi [A]  time = 0.0137169, antiderivative size = 30, 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} \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^2/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0112022, size = 30, normalized size = 1. \[ -\frac{a^2}{2 x^2}-\frac{4 a b}{3 x^{3/2}}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^2/x^3,x]

[Out]

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

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Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{2\,{x}^{2}}}-{\frac{4\,ab}{3}{x}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^2/x^3,x)

[Out]

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

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Maxima [A]  time = 0.960975, size = 32, normalized size = 1.07 \begin{align*} -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^3,x, algorithm="maxima")

[Out]

-1/6*(6*b^2*x + 8*a*b*sqrt(x) + 3*a^2)/x^2

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Fricas [A]  time = 1.46438, size = 59, normalized size = 1.97 \begin{align*} -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^3,x, algorithm="fricas")

[Out]

-1/6*(6*b^2*x + 8*a*b*sqrt(x) + 3*a^2)/x^2

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Sympy [A]  time = 0.61202, size = 26, normalized size = 0.87 \begin{align*} - \frac{a^{2}}{2 x^{2}} - \frac{4 a b}{3 x^{\frac{3}{2}}} - \frac{b^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**2/x**3,x)

[Out]

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

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Giac [A]  time = 1.10237, size = 32, normalized size = 1.07 \begin{align*} -\frac{6 \, b^{2} x + 8 \, a b \sqrt{x} + 3 \, a^{2}}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^3,x, algorithm="giac")

[Out]

-1/6*(6*b^2*x + 8*a*b*sqrt(x) + 3*a^2)/x^2