3.1719 \(\int \frac{(a+\frac{b}{x})^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3} \]

[Out]

(-2*a^2*(a + b/x)^(7/2))/(7*b^3) + (4*a*(a + b/x)^(9/2))/(9*b^3) - (2*(a + b/x)^(11/2))/(11*b^3)

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Rubi [A]  time = 0.0242511, antiderivative size = 59, 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} \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)/x^4,x]

[Out]

(-2*a^2*(a + b/x)^(7/2))/(7*b^3) + (4*a*(a + b/x)^(9/2))/(9*b^3) - (2*(a + b/x)^(11/2))/(11*b^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+\frac{b}{x}\right )^{5/2}}{x^4} \, dx &=-\operatorname{Subst}\left (\int x^2 (a+b x)^{5/2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{5/2}}{b^2}-\frac{2 a (a+b x)^{7/2}}{b^2}+\frac{(a+b x)^{9/2}}{b^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 a^2 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^3}+\frac{4 a \left (a+\frac{b}{x}\right )^{9/2}}{9 b^3}-\frac{2 \left (a+\frac{b}{x}\right )^{11/2}}{11 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0220551, size = 47, normalized size = 0.8 \[ -\frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^3 \left (8 a^2 x^2-28 a b x+63 b^2\right )}{693 b^3 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)/x^4,x]

[Out]

(-2*Sqrt[a + b/x]*(b + a*x)^3*(63*b^2 - 28*a*b*x + 8*a^2*x^2))/(693*b^3*x^5)

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Maple [A]  time = 0.004, size = 44, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}-28\,xab+63\,{b}^{2} \right ) }{693\,{b}^{3}{x}^{3}} \left ({\frac{ax+b}{x}} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(5/2)/x^4,x)

[Out]

-2/693*(a*x+b)*(8*a^2*x^2-28*a*b*x+63*b^2)*((a*x+b)/x)^(5/2)/b^3/x^3

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Maxima [A]  time = 1.03701, size = 63, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}}}{11 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a}{9 \, b^{3}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{2}}{7 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/x^4,x, algorithm="maxima")

[Out]

-2/11*(a + b/x)^(11/2)/b^3 + 4/9*(a + b/x)^(9/2)*a/b^3 - 2/7*(a + b/x)^(7/2)*a^2/b^3

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Fricas [A]  time = 1.68553, size = 161, normalized size = 2.73 \begin{align*} -\frac{2 \,{\left (8 \, a^{5} x^{5} - 4 \, a^{4} b x^{4} + 3 \, a^{3} b^{2} x^{3} + 113 \, a^{2} b^{3} x^{2} + 161 \, a b^{4} x + 63 \, b^{5}\right )} \sqrt{\frac{a x + b}{x}}}{693 \, b^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/x^4,x, algorithm="fricas")

[Out]

-2/693*(8*a^5*x^5 - 4*a^4*b*x^4 + 3*a^3*b^2*x^3 + 113*a^2*b^3*x^2 + 161*a*b^4*x + 63*b^5)*sqrt((a*x + b)/x)/(b
^3*x^5)

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Sympy [B]  time = 2.5966, size = 1073, normalized size = 18.19 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(5/2)/x**4,x)

[Out]

-16*a**(27/2)*b**(9/2)*x**8*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 20
79*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 40*a**(25/2)*b**(11/2)*x**7*sqrt(a*x/b + 1)/(69
3*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**
10*x**(11/2)) - 30*a**(23/2)*b**(13/2)*x**6*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**
8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 236*a**(21/2)*b**(15/2)*x**5*sq
rt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) +
693*a**(11/2)*b**10*x**(11/2)) - 1010*a**(19/2)*b**(17/2)*x**4*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) +
2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 1776*a**(17/
2)*b**(19/2)*x**3*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/
2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 1570*a**(15/2)*b**(21/2)*x**2*sqrt(a*x/b + 1)/(693*a**(17
/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(1
1/2)) - 700*a**(13/2)*b**(23/2)*x*sqrt(a*x/b + 1)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2
) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) - 126*a**(11/2)*b**(25/2)*sqrt(a*x/b + 1)/(
693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b
**10*x**(11/2)) + 16*a**14*b**4*x**(17/2)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079
*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) + 48*a**13*b**5*x**(15/2)/(693*a**(17/2)*b**7*x**(1
7/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2)) + 48*a**
12*b**6*x**(13/2)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b**8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2
) + 693*a**(11/2)*b**10*x**(11/2)) + 16*a**11*b**7*x**(11/2)/(693*a**(17/2)*b**7*x**(17/2) + 2079*a**(15/2)*b*
*8*x**(15/2) + 2079*a**(13/2)*b**9*x**(13/2) + 693*a**(11/2)*b**10*x**(11/2))

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Giac [B]  time = 2.23017, size = 365, normalized size = 6.19 \begin{align*} \frac{2 \,{\left (924 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{8} a^{4} \mathrm{sgn}\left (x\right ) + 4851 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} b \mathrm{sgn}\left (x\right ) + 11781 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b^{2} \mathrm{sgn}\left (x\right ) + 16863 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{3} \mathrm{sgn}\left (x\right ) + 15345 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{4} \mathrm{sgn}\left (x\right ) + 9009 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{5} \mathrm{sgn}\left (x\right ) + 3311 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{6} \mathrm{sgn}\left (x\right ) + 693 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{7} \mathrm{sgn}\left (x\right ) + 63 \, b^{8} \mathrm{sgn}\left (x\right )\right )}}{693 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/x^4,x, algorithm="giac")

[Out]

2/693*(924*(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*sgn(x) + 4851*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*b*sgn
(x) + 11781*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6*a^3*b^2*sgn(x) + 16863*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)
*b^3*sgn(x) + 15345*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*b^4*sgn(x) + 9009*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*
a^(3/2)*b^5*sgn(x) + 3311*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^6*sgn(x) + 693*(sqrt(a)*x - sqrt(a*x^2 + b*x))
*sqrt(a)*b^7*sgn(x) + 63*b^8*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^11