3.2140 \(\int \frac{(a+b \sqrt{x})^3}{x^6} \, dx\)

Optimal. Leaf size=47 \[ -\frac{2 a^2 b}{3 x^{9/2}}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{4 x^4}-\frac{2 b^3}{7 x^{7/2}} \]

[Out]

-a^3/(5*x^5) - (2*a^2*b)/(3*x^(9/2)) - (3*a*b^2)/(4*x^4) - (2*b^3)/(7*x^(7/2))

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Rubi [A]  time = 0.0197708, antiderivative size = 47, 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} \[ -\frac{2 a^2 b}{3 x^{9/2}}-\frac{a^3}{5 x^5}-\frac{3 a b^2}{4 x^4}-\frac{2 b^3}{7 x^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3/x^6,x]

[Out]

-a^3/(5*x^5) - (2*a^2*b)/(3*x^(9/2)) - (3*a*b^2)/(4*x^4) - (2*b^3)/(7*x^(7/2))

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 )^3}{x^6} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^{11}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{a^3}{x^{11}}+\frac{3 a^2 b}{x^{10}}+\frac{3 a b^2}{x^9}+\frac{b^3}{x^8}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^3}{5 x^5}-\frac{2 a^2 b}{3 x^{9/2}}-\frac{3 a b^2}{4 x^4}-\frac{2 b^3}{7 x^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0165871, size = 41, normalized size = 0.87 \[ -\frac{280 a^2 b \sqrt{x}+84 a^3+315 a b^2 x+120 b^3 x^{3/2}}{420 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x^6,x]

[Out]

-(84*a^3 + 280*a^2*b*Sqrt[x] + 315*a*b^2*x + 120*b^3*x^(3/2))/(420*x^5)

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Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{5\,{x}^{5}}}-{\frac{2\,b{a}^{2}}{3}{x}^{-{\frac{9}{2}}}}-{\frac{3\,{b}^{2}a}{4\,{x}^{4}}}-{\frac{2\,{b}^{3}}{7}{x}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^3/x^6,x)

[Out]

-1/5*a^3/x^5-2/3*a^2*b/x^(9/2)-3/4*a*b^2/x^4-2/7*b^3/x^(7/2)

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Maxima [A]  time = 0.977384, size = 47, normalized size = 1. \begin{align*} -\frac{120 \, b^{3} x^{\frac{3}{2}} + 315 \, a b^{2} x + 280 \, a^{2} b \sqrt{x} + 84 \, a^{3}}{420 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^6,x, algorithm="maxima")

[Out]

-1/420*(120*b^3*x^(3/2) + 315*a*b^2*x + 280*a^2*b*sqrt(x) + 84*a^3)/x^5

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Fricas [A]  time = 1.48485, size = 92, normalized size = 1.96 \begin{align*} -\frac{315 \, a b^{2} x + 84 \, a^{3} + 40 \,{\left (3 \, b^{3} x + 7 \, a^{2} b\right )} \sqrt{x}}{420 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^6,x, algorithm="fricas")

[Out]

-1/420*(315*a*b^2*x + 84*a^3 + 40*(3*b^3*x + 7*a^2*b)*sqrt(x))/x^5

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Sympy [A]  time = 2.953, size = 46, normalized size = 0.98 \begin{align*} - \frac{a^{3}}{5 x^{5}} - \frac{2 a^{2} b}{3 x^{\frac{9}{2}}} - \frac{3 a b^{2}}{4 x^{4}} - \frac{2 b^{3}}{7 x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**3/x**6,x)

[Out]

-a**3/(5*x**5) - 2*a**2*b/(3*x**(9/2)) - 3*a*b**2/(4*x**4) - 2*b**3/(7*x**(7/2))

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Giac [A]  time = 1.11883, size = 47, normalized size = 1. \begin{align*} -\frac{120 \, b^{3} x^{\frac{3}{2}} + 315 \, a b^{2} x + 280 \, a^{2} b \sqrt{x} + 84 \, a^{3}}{420 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^6,x, algorithm="giac")

[Out]

-1/420*(120*b^3*x^(3/2) + 315*a*b^2*x + 280*a^2*b*sqrt(x) + 84*a^3)/x^5