3.1795 \(\int \frac{x^{5/2}}{(a+\frac{b}{x})^{5/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{512 b^5}{21 a^6 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{32 b^2 x^{3/2}}{21 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{256 b^4}{7 a^5 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{64 b^3 \sqrt{x}}{7 a^4 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 b x^{5/2}}{7 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{7/2}}{7 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(-512*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))

________________________________________________________________________________________

Rubi [A]  time = 0.062057, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ -\frac{512 b^5}{21 a^6 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{32 b^2 x^{3/2}}{21 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{256 b^4}{7 a^5 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{64 b^3 \sqrt{x}}{7 a^4 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 b x^{5/2}}{7 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{7/2}}{7 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[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 271

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]

Rule 264

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]

Rubi steps

\begin{align*} \int \frac{x^{5/2}}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx &=\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]  time = 0.0219127, size = 82, normalized size = 0.54 \[ \frac{2 \left (-96 a^2 b^3 x^2+16 a^3 b^2 x^3-6 a^4 b x^4+3 a^5 x^5-384 a b^4 x-256 b^5\right )}{21 a^6 \sqrt{x} \sqrt{a+\frac{b}{x}} (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/(a + b/x)^(5/2),x]

[Out]

(2*(-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*Sqrt[a + b/x]
*Sqrt[x]*(b + a*x))

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 77, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2\,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\,a{b}^{4}x-256\,{b}^{5} \right ) }{21\,{a}^{6}}{x}^{-{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

________________________________________________________________________________________

Maxima [A]  time = 0.985402, size = 143, normalized size = 0.94 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Fricas [A]  time = 1.4817, size = 200, normalized size = 1.32 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 88.9246, size = 799, normalized size = 5.26 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]  time = 1.25494, size = 112, normalized size = 0.74 \begin{align*} \frac{512 \, b^{\frac{7}{2}}}{21 \, a^{6}} + \frac{2 \,{\left (3 \,{\left (a x + b\right )}^{\frac{7}{2}} - 21 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} - 210 \, \sqrt{a x + b} b^{3} - \frac{7 \,{\left (15 \,{\left (a x + b\right )} b^{4} - b^{5}\right )}}{{\left (a x + b\right )}^{\frac{3}{2}}}\right )}}{21 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

512/21*b^(7/2)/a^6 + 2/21*(3*(a*x + b)^(7/2) - 21*(a*x + b)^(5/2)*b + 70*(a*x + b)^(3/2)*b^2 - 210*sqrt(a*x +
b)*b^3 - 7*(15*(a*x + b)*b^4 - b^5)/(a*x + b)^(3/2))/a^6