3.2129 \(\int \frac{(a+b \sqrt{x})^2}{x^5} \, dx\)

Optimal. Leaf size=32 \[ -\frac{a^2}{4 x^4}-\frac{4 a b}{7 x^{7/2}}-\frac{b^2}{3 x^3} \]

[Out]

-a^2/(4*x^4) - (4*a*b)/(7*x^(7/2)) - b^2/(3*x^3)

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Rubi [A]  time = 0.0142292, 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} \[ -\frac{a^2}{4 x^4}-\frac{4 a b}{7 x^{7/2}}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^2/x^5,x]

[Out]

-a^2/(4*x^4) - (4*a*b)/(7*x^(7/2)) - b^2/(3*x^3)

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^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^9} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^2}{x^9}+\frac{2 a b}{x^8}+\frac{b^2}{x^7}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^2}{4 x^4}-\frac{4 a b}{7 x^{7/2}}-\frac{b^2}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0109708, size = 28, normalized size = 0.88 \[ -\frac{21 a^2+48 a b \sqrt{x}+28 b^2 x}{84 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^2/x^5,x]

[Out]

-(21*a^2 + 48*a*b*Sqrt[x] + 28*b^2*x)/(84*x^4)

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{a}^{2}}{4\,{x}^{4}}}-{\frac{4\,ab}{7}{x}^{-{\frac{7}{2}}}}-{\frac{{b}^{2}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^2/x^5,x)

[Out]

-1/4*a^2/x^4-4/7*a*b/x^(7/2)-1/3*b^2/x^3

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Maxima [A]  time = 0.977787, size = 32, normalized size = 1. \begin{align*} -\frac{28 \, b^{2} x + 48 \, a b \sqrt{x} + 21 \, a^{2}}{84 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^5,x, algorithm="maxima")

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4

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Fricas [A]  time = 1.47106, size = 65, normalized size = 2.03 \begin{align*} -\frac{28 \, b^{2} x + 48 \, a b \sqrt{x} + 21 \, a^{2}}{84 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^5,x, algorithm="fricas")

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4

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Sympy [A]  time = 1.34353, size = 29, normalized size = 0.91 \begin{align*} - \frac{a^{2}}{4 x^{4}} - \frac{4 a b}{7 x^{\frac{7}{2}}} - \frac{b^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**2/x**5,x)

[Out]

-a**2/(4*x**4) - 4*a*b/(7*x**(7/2)) - b**2/(3*x**3)

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Giac [A]  time = 1.09991, size = 32, normalized size = 1. \begin{align*} -\frac{28 \, b^{2} x + 48 \, a b \sqrt{x} + 21 \, a^{2}}{84 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^2/x^5,x, algorithm="giac")

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4