\(\int \frac {x^{5/2}}{(a+\frac {b}{x})^{5/2}} \, dx\) [1795]

Optimal result

Integrand size = 17, antiderivative size = 152 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {512 b^5}{21 a^6 \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}-\frac {256 b^4}{7 a^5 \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}}-\frac {64 b^3 \sqrt {x}}{7 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {32 b^2 x^{3/2}}{21 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b x^{5/2}}{7 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2 x^{7/2}}{7 a \left (a+\frac {b}{x}\right )^{3/2}} \]

[Out]

-512/21*b^5/a^6/(a+b/x)^(3/2)/x^(3/2)+32/21*b^2*x^(3/2)/a^3/(a+b/x)^(3/2)-4/7*b*x^(5/2)/a^2/(a+b/x)^(3/2)+2/7*
x^(7/2)/a/(a+b/x)^(3/2)-256/7*b^4/a^5/(a+b/x)^(3/2)/x^(1/2)-64/7*b^3*x^(1/2)/a^4/(a+b/x)^(3/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270} \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {512 b^5}{21 a^6 x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}-\frac {256 b^4}{7 a^5 \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}}-\frac {64 b^3 \sqrt {x}}{7 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {32 b^2 x^{3/2}}{21 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b x^{5/2}}{7 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2 x^{7/2}}{7 a \left (a+\frac {b}{x}\right )^{3/2}} \]

[In]

Int[x^(5/2)/(a + b/x)^(5/2),x]

[Out]

(-512*b^5)/(21*a^6*(a + b/x)^(3/2)*x^(3/2)) - (256*b^4)/(7*a^5*(a + b/x)^(3/2)*Sqrt[x]) - (64*b^3*Sqrt[x])/(7*
a^4*(a + b/x)^(3/2)) + (32*b^2*x^(3/2))/(21*a^3*(a + b/x)^(3/2)) - (4*b*x^(5/2))/(7*a^2*(a + b/x)^(3/2)) + (2*
x^(7/2))/(7*a*(a + b/x)^(3/2))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 x^{7/2}}{7 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {(10 b) \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx}{7 a} \\ & = -\frac {4 b x^{5/2}}{7 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2 x^{7/2}}{7 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {\left (16 b^2\right ) \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx}{7 a^2} \\ & = \frac {32 b^2 x^{3/2}}{21 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b x^{5/2}}{7 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2 x^{7/2}}{7 a \left (a+\frac {b}{x}\right )^{3/2}}-\frac {\left (32 b^3\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} \sqrt {x}} \, dx}{7 a^3} \\ & = -\frac {64 b^3 \sqrt {x}}{7 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {32 b^2 x^{3/2}}{21 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b x^{5/2}}{7 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2 x^{7/2}}{7 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {\left (128 b^4\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{3/2}} \, dx}{7 a^4} \\ & = -\frac {256 b^4}{7 a^5 \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}}-\frac {64 b^3 \sqrt {x}}{7 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {32 b^2 x^{3/2}}{21 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b x^{5/2}}{7 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2 x^{7/2}}{7 a \left (a+\frac {b}{x}\right )^{3/2}}+\frac {\left (256 b^5\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{5/2}} \, dx}{7 a^5} \\ & = -\frac {512 b^5}{21 a^6 \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}-\frac {256 b^4}{7 a^5 \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}}-\frac {64 b^3 \sqrt {x}}{7 a^4 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {32 b^2 x^{3/2}}{21 a^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 b x^{5/2}}{7 a^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {2 x^{7/2}}{7 a \left (a+\frac {b}{x}\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.34 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.54 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} \left (-256 b^5-384 a b^4 x-96 a^2 b^3 x^2+16 a^3 b^2 x^3-6 a^4 b x^4+3 a^5 x^5\right )}{21 a^6 (b+a x)^2} \]

[In]

Integrate[x^(5/2)/(a + b/x)^(5/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(-256*b^5 - 384*a*b^4*x - 96*a^2*b^3*x^2 + 16*a^3*b^2*x^3 - 6*a^4*b*x^4 + 3*a^5*x^5))
/(21*a^6*(b + a*x)^2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.51

method	result	size
gosper	\(\frac {2 \left (a x +b \right ) \left (3 a^{5} x^{5}-6 a^{4} b \,x^{4}+16 a^{3} b^{2} x^{3}-96 a^{2} b^{3} x^{2}-384 b^{4} x a -256 b^{5}\right )}{21 a^{6} x^{\frac {5}{2}} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}\)	\(77\)
default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (3 a^{5} x^{5}-6 a^{4} b \,x^{4}+16 a^{3} b^{2} x^{3}-96 a^{2} b^{3} x^{2}-384 b^{4} x a -256 b^{5}\right )}{21 \left (a x +b \right )^{2} a^{6}}\)	\(79\)
risch	\(\frac {2 \left (3 a^{3} x^{3}-12 a^{2} b \,x^{2}+37 a \,b^{2} x -158 b^{3}\right ) \left (a x +b \right )}{21 a^{6} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}-\frac {2 b^{4} \left (15 a x +14 b \right )}{3 a^{6} \left (a x +b \right ) \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\)	\(93\)

[In]

int(x^(5/2)/(a+b/x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/21*(a*x+b)*(3*a^5*x^5-6*a^4*b*x^4+16*a^3*b^2*x^3-96*a^2*b^3*x^2-384*a*b^4*x-256*b^5)/a^6/x^(5/2)/((a*x+b)/x)
^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.61 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 \, {\left (3 \, a^{5} x^{5} - 6 \, a^{4} b x^{4} + 16 \, a^{3} b^{2} x^{3} - 96 \, a^{2} b^{3} x^{2} - 384 \, a b^{4} x - 256 \, b^{5}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{21 \, {\left (a^{8} x^{2} + 2 \, a^{7} b x + a^{6} b^{2}\right )}} \]

[In]

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

[Out]

2/21*(3*a^5*x^5 - 6*a^4*b*x^4 + 16*a^3*b^2*x^3 - 96*a^2*b^3*x^2 - 384*a*b^4*x - 256*b^5)*sqrt(x)*sqrt((a*x + b
)/x)/(a^8*x^2 + 2*a^7*b*x + a^6*b^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (134) = 268\).

Time = 6.12 (sec) , antiderivative size = 799, normalized size of antiderivative = 5.26 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {6 a^{8} b^{\frac {51}{2}} x^{8} \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} + \frac {6 a^{7} b^{\frac {53}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} + \frac {14 a^{6} b^{\frac {55}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} - \frac {126 a^{5} b^{\frac {57}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} - \frac {1260 a^{4} b^{\frac {59}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} - \frac {3360 a^{3} b^{\frac {61}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} - \frac {4032 a^{2} b^{\frac {63}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} - \frac {2304 a b^{\frac {65}{2}} x \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} - \frac {512 b^{\frac {67}{2}} \sqrt {\frac {a x}{b} + 1}}{21 a^{11} b^{25} x^{5} + 105 a^{10} b^{26} x^{4} + 210 a^{9} b^{27} x^{3} + 210 a^{8} b^{28} x^{2} + 105 a^{7} b^{29} x + 21 a^{6} b^{30}} \]

[In]

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

[Out]

6*a**8*b**(51/2)*x**8*sqrt(a*x/b + 1)/(21*a**11*b**25*x**5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3 + 210*
a**8*b**28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30) + 6*a**7*b**(53/2)*x**7*sqrt(a*x/b + 1)/(21*a**11*b**25*x*
*5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3 + 210*a**8*b**28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30) + 14
*a**6*b**(55/2)*x**6*sqrt(a*x/b + 1)/(21*a**11*b**25*x**5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3 + 210*a
**8*b**28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30) - 126*a**5*b**(57/2)*x**5*sqrt(a*x/b + 1)/(21*a**11*b**25*x
**5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3 + 210*a**8*b**28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30) - 1
260*a**4*b**(59/2)*x**4*sqrt(a*x/b + 1)/(21*a**11*b**25*x**5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3 + 21
0*a**8*b**28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30) - 3360*a**3*b**(61/2)*x**3*sqrt(a*x/b + 1)/(21*a**11*b**
25*x**5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3 + 210*a**8*b**28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30)
- 4032*a**2*b**(63/2)*x**2*sqrt(a*x/b + 1)/(21*a**11*b**25*x**5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3
+ 210*a**8*b**28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30) - 2304*a*b**(65/2)*x*sqrt(a*x/b + 1)/(21*a**11*b**25
*x**5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3 + 210*a**8*b**28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30) -
512*b**(67/2)*sqrt(a*x/b + 1)/(21*a**11*b**25*x**5 + 105*a**10*b**26*x**4 + 210*a**9*b**27*x**3 + 210*a**8*b*
*28*x**2 + 105*a**7*b**29*x + 21*a**6*b**30)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.70 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=\frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} x^{\frac {7}{2}} - 21 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b x^{\frac {5}{2}} + 70 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}} - 210 \, \sqrt {a + \frac {b}{x}} b^{3} \sqrt {x}\right )}}{21 \, a^{6}} - \frac {2 \, {\left (15 \, {\left (a + \frac {b}{x}\right )} b^{4} x - b^{5}\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{6} x^{\frac {3}{2}}} \]

[In]

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

[Out]

2/21*(3*(a + b/x)^(7/2)*x^(7/2) - 21*(a + b/x)^(5/2)*b*x^(5/2) + 70*(a + b/x)^(3/2)*b^2*x^(3/2) - 210*sqrt(a +
b/x)*b^3*sqrt(x))/a^6 - 2/3*(15*(a + b/x)*b^4*x - b^5)/((a + b/x)^(3/2)*a^6*x^(3/2))

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.59 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (a x + b\right )} b^{4} - b^{5}\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{6}} + \frac {2 \, {\left (3 \, {\left (a x + b\right )}^{\frac {7}{2}} a^{36} - 21 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{36} b + 70 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{36} b^{2} - 210 \, \sqrt {a x + b} a^{36} b^{3}\right )}}{21 \, a^{42}} \]

[In]

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

[Out]

-2/3*(15*(a*x + b)*b^4 - b^5)/((a*x + b)^(3/2)*a^6) + 2/21*(3*(a*x + b)^(7/2)*a^36 - 21*(a*x + b)^(5/2)*a^36*b
+ 70*(a*x + b)^(3/2)*a^36*b^2 - 210*sqrt(a*x + b)*a^36*b^3)/a^42

Mupad [B] (verification not implemented)

Time = 6.67 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.51 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{5/2}} \, dx=-\frac {2\,\sqrt {x}\,\sqrt {\frac {b+a\,x}{x}}\,\left (-3\,a^5\,x^5+6\,a^4\,b\,x^4-16\,a^3\,b^2\,x^3+96\,a^2\,b^3\,x^2+384\,a\,b^4\,x+256\,b^5\right )}{21\,a^6\,{\left (b+a\,x\right )}^2} \]

[In]

int(x^(5/2)/(a + b/x)^(5/2),x)

[Out]

-(2*x^(1/2)*((b + a*x)/x)^(1/2)*(256*b^5 - 3*a^5*x^5 + 6*a^4*b*x^4 + 96*a^2*b^3*x^2 - 16*a^3*b^2*x^3 + 384*a*b
^4*x))/(21*a^6*(b + a*x)^2)