3.1720 \(\int \frac{(a+\frac{b}{x})^{5/2}}{x^5} \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4}+\frac{2 a^3 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{13/2}}{13 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4} \]

[Out]

(2*a^3*(a + b/x)^(7/2))/(7*b^4) - (2*a^2*(a + b/x)^(9/2))/(3*b^4) + (6*a*(a + b/x)^(11/2))/(11*b^4) - (2*(a +
b/x)^(13/2))/(13*b^4)

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Rubi [A]  time = 0.0320359, antiderivative size = 80, 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} \[ -\frac{2 a^2 \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4}+\frac{2 a^3 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{13/2}}{13 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)/x^5,x]

[Out]

(2*a^3*(a + b/x)^(7/2))/(7*b^4) - (2*a^2*(a + b/x)^(9/2))/(3*b^4) + (6*a*(a + b/x)^(11/2))/(11*b^4) - (2*(a +
b/x)^(13/2))/(13*b^4)

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^5} \, dx &=-\operatorname{Subst}\left (\int x^3 (a+b x)^{5/2} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{5/2}}{b^3}+\frac{3 a^2 (a+b x)^{7/2}}{b^3}-\frac{3 a (a+b x)^{9/2}}{b^3}+\frac{(a+b x)^{11/2}}{b^3}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 a^3 \left (a+\frac{b}{x}\right )^{7/2}}{7 b^4}-\frac{2 a^2 \left (a+\frac{b}{x}\right )^{9/2}}{3 b^4}+\frac{6 a \left (a+\frac{b}{x}\right )^{11/2}}{11 b^4}-\frac{2 \left (a+\frac{b}{x}\right )^{13/2}}{13 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0283743, size = 58, normalized size = 0.72 \[ \frac{2 \sqrt{a+\frac{b}{x}} (a x+b)^3 \left (-56 a^2 b x^2+16 a^3 x^3+126 a b^2 x-231 b^3\right )}{3003 b^4 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)/x^5,x]

[Out]

(2*Sqrt[a + b/x]*(b + a*x)^3*(-231*b^3 + 126*a*b^2*x - 56*a^2*b*x^2 + 16*a^3*x^3))/(3003*b^4*x^6)

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Maple [A]  time = 0.005, size = 55, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 16\,{a}^{3}{x}^{3}-56\,{a}^{2}b{x}^{2}+126\,xa{b}^{2}-231\,{b}^{3} \right ) }{3003\,{x}^{4}{b}^{4}} \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^5,x)

[Out]

2/3003*(a*x+b)*(16*a^3*x^3-56*a^2*b*x^2+126*a*b^2*x-231*b^3)*((a*x+b)/x)^(5/2)/x^4/b^4

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Maxima [A]  time = 0.992197, size = 86, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{13}{2}}}{13 \, b^{4}} + \frac{6 \,{\left (a + \frac{b}{x}\right )}^{\frac{11}{2}} a}{11 \, b^{4}} - \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{9}{2}} a^{2}}{3 \, b^{4}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} a^{3}}{7 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/13*(a + b/x)^(13/2)/b^4 + 6/11*(a + b/x)^(11/2)*a/b^4 - 2/3*(a + b/x)^(9/2)*a^2/b^4 + 2/7*(a + b/x)^(7/2)*a
^3/b^4

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Fricas [A]  time = 1.70187, size = 185, normalized size = 2.31 \begin{align*} \frac{2 \,{\left (16 \, a^{6} x^{6} - 8 \, a^{5} b x^{5} + 6 \, a^{4} b^{2} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{2} b^{4} x^{2} - 567 \, a b^{5} x - 231 \, b^{6}\right )} \sqrt{\frac{a x + b}{x}}}{3003 \, b^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(16*a^6*x^6 - 8*a^5*b*x^5 + 6*a^4*b^2*x^4 - 5*a^3*b^3*x^3 - 371*a^2*b^4*x^2 - 567*a*b^5*x - 231*b^6)*sq
rt((a*x + b)/x)/(b^4*x^6)

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Sympy [B]  time = 3.96058, size = 2562, normalized size = 32.02 \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**5,x)

[Out]

32*a**(37/2)*b**(23/2)*x**12*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2)
+ 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018
*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) + 176*a**(35/2)*b**(25/2)*x**11*sqrt(a*x/b + 1)/(
3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**
(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b*
*21*x**(13/2)) + 396*a**(33/2)*b**(27/2)*x**10*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/
2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19
*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) + 462*a**(31/2)*b**(29/2)*x**9*
sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(
21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) +
3003*a**(13/2)*b**21*x**(13/2)) - 462*a**(29/2)*b**(31/2)*x**8*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2)
+ 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045
*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 5544*a**(27/2
)*b**(33/2)*x**7*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**
(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b
**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 18480*a**(25/2)*b**(35/2)*x**6*sqrt(a*x/b + 1)/(3003*a**(25
/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**1
8*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/
2)) - 34716*a**(23/2)*b**(37/2)*x**5*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x
**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2)
+ 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 40788*a**(21/2)*b**(39/2)*x**4*sqrt(a*x
/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) +
60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**
(13/2)*b**21*x**(13/2)) - 30712*a**(19/2)*b**(41/2)*x**3*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 180
18*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(1
7/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 14476*a**(17/2)*b**
(43/2)*x**2*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2
)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*
x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 3906*a**(15/2)*b**(45/2)*x*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15
*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/
2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 462
*a**(13/2)*b**(47/2)*sqrt(a*x/b + 1)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045
*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/
2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 32*a**19*b**11*x**(25/2)/(3003*a**(25/2)*b**15*x**(25/2
) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 4504
5*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 192*a**18*b*
*12*x**(23/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21
/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 30
03*a**(13/2)*b**21*x**(13/2)) - 480*a**17*b**13*x**(21/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b*
*16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(
17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 640*a**16*b**14*x**(19/2)/(3003*a*
*(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*
b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**
(13/2)) - 480*a**15*b**15*x**(17/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*
a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2
)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 192*a**14*b**16*x**(15/2)/(3003*a**(25/2)*b**15*x**(25/2
) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/2) + 60060*a**(19/2)*b**18*x**(19/2) + 4504
5*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 3003*a**(13/2)*b**21*x**(13/2)) - 32*a**13*b**
17*x**(13/2)/(3003*a**(25/2)*b**15*x**(25/2) + 18018*a**(23/2)*b**16*x**(23/2) + 45045*a**(21/2)*b**17*x**(21/
2) + 60060*a**(19/2)*b**18*x**(19/2) + 45045*a**(17/2)*b**19*x**(17/2) + 18018*a**(15/2)*b**20*x**(15/2) + 300
3*a**(13/2)*b**21*x**(13/2))

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Giac [B]  time = 1.5008, size = 406, normalized size = 5.08 \begin{align*} \frac{2 \,{\left (6006 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{9} a^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 36036 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{8} a^{4} b \mathrm{sgn}\left (x\right ) + 99099 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7} a^{\frac{7}{2}} b^{2} \mathrm{sgn}\left (x\right ) + 161733 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{6} a^{3} b^{3} \mathrm{sgn}\left (x\right ) + 171171 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5} a^{\frac{5}{2}} b^{4} \mathrm{sgn}\left (x\right ) + 121121 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} b^{5} \mathrm{sgn}\left (x\right ) + 57057 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b^{6} \mathrm{sgn}\left (x\right ) + 17199 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{7} \mathrm{sgn}\left (x\right ) + 3003 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{8} \mathrm{sgn}\left (x\right ) + 231 \, b^{9} \mathrm{sgn}\left (x\right )\right )}}{3003 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3003*(6006*(sqrt(a)*x - sqrt(a*x^2 + b*x))^9*a^(9/2)*sgn(x) + 36036*(sqrt(a)*x - sqrt(a*x^2 + b*x))^8*a^4*b*
sgn(x) + 99099*(sqrt(a)*x - sqrt(a*x^2 + b*x))^7*a^(7/2)*b^2*sgn(x) + 161733*(sqrt(a)*x - sqrt(a*x^2 + b*x))^6
*a^3*b^3*sgn(x) + 171171*(sqrt(a)*x - sqrt(a*x^2 + b*x))^5*a^(5/2)*b^4*sgn(x) + 121121*(sqrt(a)*x - sqrt(a*x^2
+ b*x))^4*a^2*b^5*sgn(x) + 57057*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^6*sgn(x) + 17199*(sqrt(a)*x - sq
rt(a*x^2 + b*x))^2*a*b^7*sgn(x) + 3003*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^8*sgn(x) + 231*b^9*sgn(x))/(s
qrt(a)*x - sqrt(a*x^2 + b*x))^13