3.1786 \(\int \frac{x^{5/2}}{(a+\frac{b}{x})^{3/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{32 b^2 x^{3/2}}{35 a^3 \sqrt{a+\frac{b}{x}}}-\frac{256 b^4}{35 a^5 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{128 b^3 \sqrt{x}}{35 a^4 \sqrt{a+\frac{b}{x}}}-\frac{16 b x^{5/2}}{35 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}} \]

[Out]

(-256*b^4)/(35*a^5*Sqrt[a + b/x]*Sqrt[x]) - (128*b^3*Sqrt[x])/(35*a^4*Sqrt[a + b/x]) + (32*b^2*x^(3/2))/(35*a^
3*Sqrt[a + b/x]) - (16*b*x^(5/2))/(35*a^2*Sqrt[a + b/x]) + (2*x^(7/2))/(7*a*Sqrt[a + b/x])

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Rubi [A]  time = 0.0450926, 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} \[ \frac{32 b^2 x^{3/2}}{35 a^3 \sqrt{a+\frac{b}{x}}}-\frac{256 b^4}{35 a^5 \sqrt{x} \sqrt{a+\frac{b}{x}}}-\frac{128 b^3 \sqrt{x}}{35 a^4 \sqrt{a+\frac{b}{x}}}-\frac{16 b x^{5/2}}{35 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)/(a + b/x)^(3/2),x]

[Out]

(-256*b^4)/(35*a^5*Sqrt[a + b/x]*Sqrt[x]) - (128*b^3*Sqrt[x])/(35*a^4*Sqrt[a + b/x]) + (32*b^2*x^(3/2))/(35*a^
3*Sqrt[a + b/x]) - (16*b*x^(5/2))/(35*a^2*Sqrt[a + b/x]) + (2*x^(7/2))/(7*a*Sqrt[a + b/x])

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 )^{3/2}} \, dx &=\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}}-\frac{(8 b) \int \frac{x^{3/2}}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx}{7 a}\\ &=-\frac{16 b x^{5/2}}{35 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}}+\frac{\left (48 b^2\right ) \int \frac{\sqrt{x}}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx}{35 a^2}\\ &=\frac{32 b^2 x^{3/2}}{35 a^3 \sqrt{a+\frac{b}{x}}}-\frac{16 b x^{5/2}}{35 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}}-\frac{\left (64 b^3\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} \sqrt{x}} \, dx}{35 a^3}\\ &=-\frac{128 b^3 \sqrt{x}}{35 a^4 \sqrt{a+\frac{b}{x}}}+\frac{32 b^2 x^{3/2}}{35 a^3 \sqrt{a+\frac{b}{x}}}-\frac{16 b x^{5/2}}{35 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}}+\frac{\left (128 b^4\right ) \int \frac{1}{\left (a+\frac{b}{x}\right )^{3/2} x^{3/2}} \, dx}{35 a^4}\\ &=-\frac{256 b^4}{35 a^5 \sqrt{a+\frac{b}{x}} \sqrt{x}}-\frac{128 b^3 \sqrt{x}}{35 a^4 \sqrt{a+\frac{b}{x}}}+\frac{32 b^2 x^{3/2}}{35 a^3 \sqrt{a+\frac{b}{x}}}-\frac{16 b x^{5/2}}{35 a^2 \sqrt{a+\frac{b}{x}}}+\frac{2 x^{7/2}}{7 a \sqrt{a+\frac{b}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0166502, size = 64, normalized size = 0.51 \[ \frac{2 \left (16 a^2 b^2 x^2-8 a^3 b x^3+5 a^4 x^4-64 a b^3 x-128 b^4\right )}{35 a^5 \sqrt{x} \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)/(a + b/x)^(3/2),x]

[Out]

(2*(-128*b^4 - 64*a*b^3*x + 16*a^2*b^2*x^2 - 8*a^3*b*x^3 + 5*a^4*x^4))/(35*a^5*Sqrt[a + b/x]*Sqrt[x])

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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 ( 5\,{x}^{4}{a}^{4}-8\,b{x}^{3}{a}^{3}+16\,{b}^{2}{x}^{2}{a}^{2}-64\,{b}^{3}xa-128\,{b}^{4} \right ) }{35\,{a}^{5}}{x}^{-{\frac{3}{2}}} \left ({\frac{ax+b}{x}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(a+b/x)^(3/2),x)

[Out]

2/35*(a*x+b)*(5*a^4*x^4-8*a^3*b*x^3+16*a^2*b^2*x^2-64*a*b^3*x-128*b^4)/a^5/x^(3/2)/((a*x+b)/x)^(3/2)

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Maxima [A]  time = 1.00376, size = 122, normalized size = 0.97 \begin{align*} -\frac{2 \, b^{4}}{\sqrt{a + \frac{b}{x}} a^{5} \sqrt{x}} + \frac{2 \,{\left (5 \,{\left (a + \frac{b}{x}\right )}^{\frac{7}{2}} x^{\frac{7}{2}} - 28 \,{\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}} - 140 \, \sqrt{a + \frac{b}{x}} b^{3} \sqrt{x}\right )}}{35 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^4/(sqrt(a + b/x)*a^5*sqrt(x)) + 2/35*(5*(a + b/x)^(7/2)*x^(7/2) - 28*(a + b/x)^(5/2)*b*x^(5/2) + 70*(a +
b/x)^(3/2)*b^2*x^(3/2) - 140*sqrt(a + b/x)*b^3*sqrt(x))/a^5

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Fricas [A]  time = 1.47347, size = 154, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (5 \, a^{4} x^{4} - 8 \, a^{3} b x^{3} + 16 \, a^{2} b^{2} x^{2} - 64 \, a b^{3} x - 128 \, b^{4}\right )} \sqrt{x} \sqrt{\frac{a x + b}{x}}}{35 \,{\left (a^{6} x + a^{5} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*a^4*x^4 - 8*a^3*b*x^3 + 16*a^2*b^2*x^2 - 64*a*b^3*x - 128*b^4)*sqrt(x)*sqrt((a*x + b)/x)/(a^6*x + a^5*
b)

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Sympy [B]  time = 88.6523, size = 614, normalized size = 4.87 \begin{align*} \frac{10 a^{7} b^{\frac{33}{2}} x^{7} \sqrt{\frac{a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} + \frac{14 a^{6} b^{\frac{35}{2}} x^{6} \sqrt{\frac{a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} + \frac{14 a^{5} b^{\frac{37}{2}} x^{5} \sqrt{\frac{a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac{70 a^{4} b^{\frac{39}{2}} x^{4} \sqrt{\frac{a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac{560 a^{3} b^{\frac{41}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac{1120 a^{2} b^{\frac{43}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac{896 a b^{\frac{45}{2}} x \sqrt{\frac{a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac{256 b^{\frac{47}{2}} \sqrt{\frac{a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*a**7*b**(33/2)*x**7*sqrt(a*x/b + 1)/(35*a**9*b**16*x**4 + 140*a**8*b**17*x**3 + 210*a**7*b**18*x**2 + 140*a
**6*b**19*x + 35*a**5*b**20) + 14*a**6*b**(35/2)*x**6*sqrt(a*x/b + 1)/(35*a**9*b**16*x**4 + 140*a**8*b**17*x**
3 + 210*a**7*b**18*x**2 + 140*a**6*b**19*x + 35*a**5*b**20) + 14*a**5*b**(37/2)*x**5*sqrt(a*x/b + 1)/(35*a**9*
b**16*x**4 + 140*a**8*b**17*x**3 + 210*a**7*b**18*x**2 + 140*a**6*b**19*x + 35*a**5*b**20) - 70*a**4*b**(39/2)
*x**4*sqrt(a*x/b + 1)/(35*a**9*b**16*x**4 + 140*a**8*b**17*x**3 + 210*a**7*b**18*x**2 + 140*a**6*b**19*x + 35*
a**5*b**20) - 560*a**3*b**(41/2)*x**3*sqrt(a*x/b + 1)/(35*a**9*b**16*x**4 + 140*a**8*b**17*x**3 + 210*a**7*b**
18*x**2 + 140*a**6*b**19*x + 35*a**5*b**20) - 1120*a**2*b**(43/2)*x**2*sqrt(a*x/b + 1)/(35*a**9*b**16*x**4 + 1
40*a**8*b**17*x**3 + 210*a**7*b**18*x**2 + 140*a**6*b**19*x + 35*a**5*b**20) - 896*a*b**(45/2)*x*sqrt(a*x/b +
1)/(35*a**9*b**16*x**4 + 140*a**8*b**17*x**3 + 210*a**7*b**18*x**2 + 140*a**6*b**19*x + 35*a**5*b**20) - 256*b
**(47/2)*sqrt(a*x/b + 1)/(35*a**9*b**16*x**4 + 140*a**8*b**17*x**3 + 210*a**7*b**18*x**2 + 140*a**6*b**19*x +
35*a**5*b**20)

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Giac [A]  time = 1.18841, size = 95, normalized size = 0.75 \begin{align*} \frac{256 \, b^{\frac{7}{2}}}{35 \, a^{5}} + \frac{2 \,{\left (5 \,{\left (a x + b\right )}^{\frac{7}{2}} - 28 \,{\left (a x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (a x + b\right )}^{\frac{3}{2}} b^{2} - 140 \, \sqrt{a x + b} b^{3} - \frac{35 \, b^{4}}{\sqrt{a x + b}}\right )}}{35 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

256/35*b^(7/2)/a^5 + 2/35*(5*(a*x + b)^(7/2) - 28*(a*x + b)^(5/2)*b + 70*(a*x + b)^(3/2)*b^2 - 140*sqrt(a*x +
b)*b^3 - 35*b^4/sqrt(a*x + b))/a^5