3.1796 \(\int \frac{x^{3/2}}{(a+\frac{b}{x})^{5/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{256 b^4}{15 a^5 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{128 b^3}{5 a^4 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{32 b^2 \sqrt{x}}{5 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{16 b x^{3/2}}{15 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{5/2}}{5 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

(256*b^4)/(15*a^5*(a + b/x)^(3/2)*x^(3/2)) + (128*b^3)/(5*a^4*(a + b/x)^(3/2)*Sqrt[x]) + (32*b^2*Sqrt[x])/(5*a
^3*(a + b/x)^(3/2)) - (16*b*x^(3/2))/(15*a^2*(a + b/x)^(3/2)) + (2*x^(5/2))/(5*a*(a + b/x)^(3/2))

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Rubi [A]  time = 0.0467119, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{256 b^4}{15 a^5 x^{3/2} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{128 b^3}{5 a^4 \sqrt{x} \left (a+\frac{b}{x}\right )^{3/2}}+\frac{32 b^2 \sqrt{x}}{5 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{16 b x^{3/2}}{15 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{5/2}}{5 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/(a + b/x)^(5/2),x]

[Out]

(256*b^4)/(15*a^5*(a + b/x)^(3/2)*x^(3/2)) + (128*b^3)/(5*a^4*(a + b/x)^(3/2)*Sqrt[x]) + (32*b^2*Sqrt[x])/(5*a
^3*(a + b/x)^(3/2)) - (16*b*x^(3/2))/(15*a^2*(a + b/x)^(3/2)) + (2*x^(5/2))/(5*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^{3/2}}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx &=\frac{2 x^{5/2}}{5 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{(8 b) \int \frac{\sqrt{x}}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx}{5 a}\\ &=-\frac{16 b x^{3/2}}{15 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{5/2}}{5 a \left (a+\frac{b}{x}\right )^{3/2}}+\frac{\left (16 b^2\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} \sqrt{x}} \, dx}{5 a^2}\\ &=\frac{32 b^2 \sqrt{x}}{5 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{16 b x^{3/2}}{15 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{5/2}}{5 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{\left (64 b^3\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} x^{3/2}} \, dx}{5 a^3}\\ &=\frac{128 b^3}{5 a^4 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}}+\frac{32 b^2 \sqrt{x}}{5 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{16 b x^{3/2}}{15 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{5/2}}{5 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^{5/2}} \, dx}{5 a^4}\\ &=\frac{256 b^4}{15 a^5 \left (a+\frac{b}{x}\right )^{3/2} x^{3/2}}+\frac{128 b^3}{5 a^4 \left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}}+\frac{32 b^2 \sqrt{x}}{5 a^3 \left (a+\frac{b}{x}\right )^{3/2}}-\frac{16 b x^{3/2}}{15 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{2 x^{5/2}}{5 a \left (a+\frac{b}{x}\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0281745, size = 71, normalized size = 0.56 \[ \frac{2 \sqrt{x} \sqrt{a+\frac{b}{x}} \left (48 a^2 b^2 x^2-8 a^3 b x^3+3 a^4 x^4+192 a b^3 x+128 b^4\right )}{15 a^5 (a x+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/(a + b/x)^(5/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(128*b^4 + 192*a*b^3*x + 48*a^2*b^2*x^2 - 8*a^3*b*x^3 + 3*a^4*x^4))/(15*a^5*(b + a*x)
^2)

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Maple [A]  time = 0.004, size = 66, normalized size = 0.5 \begin{align*}{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 3\,{x}^{4}{a}^{4}-8\,b{x}^{3}{a}^{3}+48\,{b}^{2}{x}^{2}{a}^{2}+192\,{b}^{3}xa+128\,{b}^{4} \right ) }{15\,{a}^{5}}{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^(3/2)/(a+b/x)^(5/2),x)

[Out]

2/15*(a*x+b)*(3*a^4*x^4-8*a^3*b*x^3+48*a^2*b^2*x^2+192*a*b^3*x+128*b^4)/a^5/x^(5/2)/((a*x+b)/x)^(5/2)

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Maxima [A]  time = 0.982782, size = 120, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (3 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}} x^{\frac{5}{2}} - 20 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} b x^{\frac{3}{2}} + 90 \, \sqrt{a + \frac{b}{x}} b^{2} \sqrt{x}\right )}}{15 \, a^{5}} + \frac{2 \,{\left (12 \,{\left (a + \frac{b}{x}\right )} b^{3} x - b^{4}\right )}}{3 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a^{5} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b/x)^(5/2),x, algorithm="maxima")

[Out]

2/15*(3*(a + b/x)^(5/2)*x^(5/2) - 20*(a + b/x)^(3/2)*b*x^(3/2) + 90*sqrt(a + b/x)*b^2*sqrt(x))/a^5 + 2/3*(12*(
a + b/x)*b^3*x - b^4)/((a + b/x)^(3/2)*a^5*x^(3/2))

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Fricas [A]  time = 1.50965, size = 177, normalized size = 1.4 \begin{align*} \frac{2 \,{\left (3 \, a^{4} x^{4} - 8 \, a^{3} b x^{3} + 48 \, a^{2} b^{2} x^{2} + 192 \, a b^{3} x + 128 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{15 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b/x)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*a^4*x^4 - 8*a^3*b*x^3 + 48*a^2*b^2*x^2 + 192*a*b^3*x + 128*b^4)*sqrt(x)*sqrt((a*x + b)/x)/(a^7*x^2 + 2
*a^6*b*x + a^5*b^2)

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Sympy [B]  time = 34.2835, size = 536, normalized size = 4.25 \begin{align*} \frac{6 a^{6} b^{\frac{33}{2}} x^{6} \sqrt{\frac{a x}{b} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{3} + 90 a^{7} b^{18} x^{2} + 60 a^{6} b^{19} x + 15 a^{5} b^{20}} - \frac{4 a^{5} b^{\frac{35}{2}} x^{5} \sqrt{\frac{a x}{b} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{3} + 90 a^{7} b^{18} x^{2} + 60 a^{6} b^{19} x + 15 a^{5} b^{20}} + \frac{70 a^{4} b^{\frac{37}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{3} + 90 a^{7} b^{18} x^{2} + 60 a^{6} b^{19} x + 15 a^{5} b^{20}} + \frac{560 a^{3} b^{\frac{39}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{3} + 90 a^{7} b^{18} x^{2} + 60 a^{6} b^{19} x + 15 a^{5} b^{20}} + \frac{1120 a^{2} b^{\frac{41}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{3} + 90 a^{7} b^{18} x^{2} + 60 a^{6} b^{19} x + 15 a^{5} b^{20}} + \frac{896 a b^{\frac{43}{2}} x \sqrt{\frac{a x}{b} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{3} + 90 a^{7} b^{18} x^{2} + 60 a^{6} b^{19} x + 15 a^{5} b^{20}} + \frac{256 b^{\frac{45}{2}} \sqrt{\frac{a x}{b} + 1}}{15 a^{9} b^{16} x^{4} + 60 a^{8} b^{17} x^{3} + 90 a^{7} b^{18} x^{2} + 60 a^{6} b^{19} x + 15 a^{5} b^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)/(a+b/x)**(5/2),x)

[Out]

6*a**6*b**(33/2)*x**6*sqrt(a*x/b + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**3 + 90*a**7*b**18*x**2 + 60*a**6*
b**19*x + 15*a**5*b**20) - 4*a**5*b**(35/2)*x**5*sqrt(a*x/b + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**3 + 90
*a**7*b**18*x**2 + 60*a**6*b**19*x + 15*a**5*b**20) + 70*a**4*b**(37/2)*x**4*sqrt(a*x/b + 1)/(15*a**9*b**16*x*
*4 + 60*a**8*b**17*x**3 + 90*a**7*b**18*x**2 + 60*a**6*b**19*x + 15*a**5*b**20) + 560*a**3*b**(39/2)*x**3*sqrt
(a*x/b + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**3 + 90*a**7*b**18*x**2 + 60*a**6*b**19*x + 15*a**5*b**20) +
1120*a**2*b**(41/2)*x**2*sqrt(a*x/b + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**3 + 90*a**7*b**18*x**2 + 60*a
**6*b**19*x + 15*a**5*b**20) + 896*a*b**(43/2)*x*sqrt(a*x/b + 1)/(15*a**9*b**16*x**4 + 60*a**8*b**17*x**3 + 90
*a**7*b**18*x**2 + 60*a**6*b**19*x + 15*a**5*b**20) + 256*b**(45/2)*sqrt(a*x/b + 1)/(15*a**9*b**16*x**4 + 60*a
**8*b**17*x**3 + 90*a**7*b**18*x**2 + 60*a**6*b**19*x + 15*a**5*b**20)

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Giac [A]  time = 1.20168, size = 96, normalized size = 0.76 \begin{align*} -\frac{256 \, b^{\frac{5}{2}}}{15 \, a^{5}} + \frac{2 \,{\left (3 \,{\left (a x + b\right )}^{\frac{5}{2}} - 20 \,{\left (a x + b\right )}^{\frac{3}{2}} b + 90 \, \sqrt{a x + b} b^{2} + \frac{5 \,{\left (12 \,{\left (a x + b\right )} b^{3} - b^{4}\right )}}{{\left (a x + b\right )}^{\frac{3}{2}}}\right )}}{15 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b/x)^(5/2),x, algorithm="giac")

[Out]

-256/15*b^(5/2)/a^5 + 2/15*(3*(a*x + b)^(5/2) - 20*(a*x + b)^(3/2)*b + 90*sqrt(a*x + b)*b^2 + 5*(12*(a*x + b)*
b^3 - b^4)/(a*x + b)^(3/2))/a^5