3.2139 \(\int \frac{(a+b \sqrt{x})^3}{x^5} \, dx\)

Optimal. Leaf size=45 \[ -\frac{6 a^2 b}{7 x^{7/2}}-\frac{a^3}{4 x^4}-\frac{a b^2}{x^3}-\frac{2 b^3}{5 x^{5/2}} \]

[Out]

-a^3/(4*x^4) - (6*a^2*b)/(7*x^(7/2)) - (a*b^2)/x^3 - (2*b^3)/(5*x^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0203438, antiderivative size = 45, 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} \[ -\frac{6 a^2 b}{7 x^{7/2}}-\frac{a^3}{4 x^4}-\frac{a b^2}{x^3}-\frac{2 b^3}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3/x^5,x]

[Out]

-a^3/(4*x^4) - (6*a^2*b)/(7*x^(7/2)) - (a*b^2)/x^3 - (2*b^3)/(5*x^(5/2))

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

Mathematica [A]  time = 0.0161908, size = 41, normalized size = 0.91 \[ -\frac{120 a^2 b \sqrt{x}+35 a^3+140 a b^2 x+56 b^3 x^{3/2}}{140 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x^5,x]

[Out]

-(35*a^3 + 120*a^2*b*Sqrt[x] + 140*a*b^2*x + 56*b^3*x^(3/2))/(140*x^4)

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{6\,b{a}^{2}}{7}{x}^{-{\frac{7}{2}}}}-{\frac{{b}^{2}a}{{x}^{3}}}-{\frac{2\,{b}^{3}}{5}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^3/x^5,x)

[Out]

-1/4*a^3/x^4-6/7*a^2*b/x^(7/2)-a*b^2/x^3-2/5*b^3/x^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 0.973464, size = 47, normalized size = 1.04 \begin{align*} -\frac{56 \, b^{3} x^{\frac{3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt{x} + 35 \, a^{3}}{140 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^5,x, algorithm="maxima")

[Out]

-1/140*(56*b^3*x^(3/2) + 140*a*b^2*x + 120*a^2*b*sqrt(x) + 35*a^3)/x^4

________________________________________________________________________________________

Fricas [A]  time = 1.46931, size = 92, normalized size = 2.04 \begin{align*} -\frac{140 \, a b^{2} x + 35 \, a^{3} + 8 \,{\left (7 \, b^{3} x + 15 \, a^{2} b\right )} \sqrt{x}}{140 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^5,x, algorithm="fricas")

[Out]

-1/140*(140*a*b^2*x + 35*a^3 + 8*(7*b^3*x + 15*a^2*b)*sqrt(x))/x^4

________________________________________________________________________________________

Sympy [A]  time = 1.92839, size = 42, normalized size = 0.93 \begin{align*} - \frac{a^{3}}{4 x^{4}} - \frac{6 a^{2} b}{7 x^{\frac{7}{2}}} - \frac{a b^{2}}{x^{3}} - \frac{2 b^{3}}{5 x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**3/x**5,x)

[Out]

-a**3/(4*x**4) - 6*a**2*b/(7*x**(7/2)) - a*b**2/x**3 - 2*b**3/(5*x**(5/2))

________________________________________________________________________________________

Giac [A]  time = 1.1176, size = 47, normalized size = 1.04 \begin{align*} -\frac{56 \, b^{3} x^{\frac{3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt{x} + 35 \, a^{3}}{140 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x^5,x, algorithm="giac")

[Out]

-1/140*(56*b^3*x^(3/2) + 140*a*b^2*x + 120*a^2*b*sqrt(x) + 35*a^3)/x^4