3.2128 \(\int \frac{(a+b \sqrt{x})^2}{x^4} \, dx\)

Optimal. Leaf size=32 \[ -\frac{a^2}{3 x^3}-\frac{4 a b}{5 x^{5/2}}-\frac{b^2}{2 x^2} \]

[Out]

-a^2/(3*x^3) - (4*a*b)/(5*x^(5/2)) - b^2/(2*x^2)

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Rubi [A]  time = 0.0136014, antiderivative size = 32, 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} \[ -\frac{a^2}{3 x^3}-\frac{4 a b}{5 x^{5/2}}-\frac{b^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^2/x^4,x]

[Out]

-a^2/(3*x^3) - (4*a*b)/(5*x^(5/2)) - b^2/(2*x^2)

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^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^7} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^2}{x^7}+\frac{2 a b}{x^6}+\frac{b^2}{x^5}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^2}{3 x^3}-\frac{4 a b}{5 x^{5/2}}-\frac{b^2}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0108061, size = 28, normalized size = 0.88 \[ -\frac{10 a^2+24 a b \sqrt{x}+15 b^2 x}{30 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^2/x^4,x]

[Out]

-(10*a^2 + 24*a*b*Sqrt[x] + 15*b^2*x)/(30*x^3)

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{3\,{x}^{3}}}-{\frac{4\,ab}{5}{x}^{-{\frac{5}{2}}}}-{\frac{{b}^{2}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^2/x^4,x)

[Out]

-1/3*a^2/x^3-4/5*a*b/x^(5/2)-1/2*b^2/x^2

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Maxima [A]  time = 0.981268, size = 32, normalized size = 1. \begin{align*} -\frac{15 \, b^{2} x + 24 \, a b \sqrt{x} + 10 \, a^{2}}{30 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^4,x, algorithm="maxima")

[Out]

-1/30*(15*b^2*x + 24*a*b*sqrt(x) + 10*a^2)/x^3

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Fricas [A]  time = 1.40874, size = 65, normalized size = 2.03 \begin{align*} -\frac{15 \, b^{2} x + 24 \, a b \sqrt{x} + 10 \, a^{2}}{30 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^4,x, algorithm="fricas")

[Out]

-1/30*(15*b^2*x + 24*a*b*sqrt(x) + 10*a^2)/x^3

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Sympy [A]  time = 0.922679, size = 29, normalized size = 0.91 \begin{align*} - \frac{a^{2}}{3 x^{3}} - \frac{4 a b}{5 x^{\frac{5}{2}}} - \frac{b^{2}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**2/x**4,x)

[Out]

-a**2/(3*x**3) - 4*a*b/(5*x**(5/2)) - b**2/(2*x**2)

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Giac [A]  time = 1.13645, size = 32, normalized size = 1. \begin{align*} -\frac{15 \, b^{2} x + 24 \, a b \sqrt{x} + 10 \, a^{2}}{30 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^4,x, algorithm="giac")

[Out]

-1/30*(15*b^2*x + 24*a*b*sqrt(x) + 10*a^2)/x^3