3.1793 \(\int \frac{x^{5/2}}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx\)

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

[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)*S
qrt[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))

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Rubi [A]  time = 0.194291, 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 \[ -\frac{512 b^5}{21 a^6 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{256 b^4}{7 a^5 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}-\frac{64 b^3 \sqrt{x}}{7 a^4 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{32 b^2 x^{3/2}}{21 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{4 b x^{5/2}}{7 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{7/2}}{7 a \left (a+\frac{b}{x}\right )^{3/2}} \]

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)*S
qrt[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))

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Rubi in Sympy [A]  time = 17.8956, size = 134, normalized size = 0.88 \[ \frac{2 x^{\frac{7}{2}}}{7 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{4 b x^{\frac{5}{2}}}{7 a^{2} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} + \frac{32 b^{2} x^{\frac{3}{2}}}{21 a^{3} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{64 b^{3} \sqrt{x}}{7 a^{4} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{256 b^{4}}{7 a^{5} \sqrt{x} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} - \frac{512 b^{5}}{21 a^{6} x^{\frac{3}{2}} \left (a + \frac{b}{x}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(a+b/x)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0740892, size = 82, normalized size = 0.54 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (3 a^5 x^5-6 a^4 b x^4+16 a^3 b^2 x^3-96 a^2 b^3 x^2-384 a b^4 x-256 b^5\right )}{21 a^6 (a x+b)^2} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.011, size = 77, normalized size = 0.5 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,{a}^{5}{x}^{5}-6\,{a}^{4}b{x}^{4}+16\,{a}^{3}{b}^{2}{x}^{3}-96\,{a}^{2}{b}^{3}{x}^{2}-384\,a{b}^{4}x-256\,{b}^{5} \right ) }{21\,{a}^{6}}{x}^{-{\frac{5}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{5}{2}}}} \]

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-25
6*b^5)/a^6/x^(5/2)/((a*x+b)/x)^(5/2)

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Maxima [A]  time = 1.5005, size = 143, normalized size = 0.94 \[ \frac{2 \,{\left (3 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 21 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} b x^{\frac{5}{2}} + 70 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b^{2} x^{\frac{3}{2}} - 210 \, \sqrt{a + \frac{b}{x}} b^{3} \sqrt{x}\right )}}{21 \, a^{6}} - \frac{2 \,{\left (15 \,{\left (a + \frac{b}{x}\right )} b^{4} x - b^{5}\right )}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{6} x^{\frac{3}{2}}} \]

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))

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Fricas [A]  time = 0.236813, size = 109, normalized size = 0.72 \[ \frac{2 \,{\left (3 \, a^{5} x^{5} - 6 \, a^{4} b x^{4} + 16 \, a^{3} b^{2} x^{3} - 96 \, a^{2} b^{3} x^{2} - 384 \, a b^{4} x - 256 \, b^{5}\right )}}{21 \,{\left (a^{7} x + a^{6} b\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}} \]

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)/((a^7*x + a^6*b)*sqrt(x)*sqrt((a*x + b)/x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.241311, size = 112, normalized size = 0.74 \[ \frac{512 \, b^{\frac{7}{2}}}{21 \, a^{6}} + \frac{2 \,{\left (3 \,{\left (a x + b\right )}^{\frac{7}{2}} - 21 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} - 210 \, \sqrt{a x + b} b^{3} - \frac{7 \,{\left (15 \,{\left (a x + b\right )} b^{4} - b^{5}\right )}}{{\left (a x + b\right )}^{\frac{3}{2}}}\right )}}{21 \, a^{6}} \]

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