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

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

[Out]

32/35*b^2*x^(3/2)/a^3/(a+b/x)^(1/2)-16/35*b*x^(5/2)/a^2/(a+b/x)^(1/2)+2/7*x^(7/2)/a/(a+b/x)^(1/2)-256/35*b^4/a
^5/(a+b/x)^(1/2)/x^(1/2)-128/35*b^3*x^(1/2)/a^4/(a+b/x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 64, normalized size = 0.51 \[ \frac {2 \left (5 a^4 x^4-8 a^3 b x^3+16 a^2 b^2 x^2-64 a b^3 x-128 b^4\right )}{35 a^5 \sqrt {x} \sqrt {a+\frac {b}{x}}} \]

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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fricas [A]  time = 0.59, size = 70, normalized size = 0.56 \[ \frac {2 \, {\left (5 \, a^{4} x^{4} - 8 \, a^{3} b x^{3} + 16 \, a^{2} b^{2} x^{2} - 64 \, a b^{3} x - 128 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{35 \, {\left (a^{6} x + a^{5} b\right )}} \]

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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giac [A]  time = 0.18, size = 85, normalized size = 0.67 \[ \frac {256 \, b^{\frac {7}{2}}}{35 \, a^{5}} - \frac {2 \, b^{4}}{\sqrt {a x + b} a^{5}} + \frac {2 \, {\left (5 \, {\left (a x + b\right )}^{\frac {7}{2}} a^{30} - 28 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{30} b + 70 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{30} b^{2} - 140 \, \sqrt {a x + b} a^{30} b^{3}\right )}}{35 \, a^{35}} \]

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*b^4/(sqrt(a*x + b)*a^5) + 2/35*(5*(a*x + b)^(7/2)*a^30 - 28*(a*x + b)^(5/2)*a^30*b + 70
*(a*x + b)^(3/2)*a^30*b^2 - 140*sqrt(a*x + b)*a^30*b^3)/a^35

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maple [A]  time = 0.01, size = 66, normalized size = 0.52 \[ \frac {2 \left (a x +b \right ) \left (5 a^{4} x^{4}-8 a^{3} x^{3} b +16 a^{2} x^{2} b^{2}-64 a x \,b^{3}-128 b^{4}\right )}{35 \left (\frac {a x +b}{x}\right )^{\frac {3}{2}} a^{5} x^{\frac {3}{2}}} \]

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 = 0.99, size = 90, normalized size = 0.71 \[ -\frac {2 \, b^{4}}{\sqrt {a + \frac {b}{x}} a^{5} \sqrt {x}} + \frac {2 \, {\left (5 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} x^{\frac {7}{2}} - 28 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b x^{\frac {5}{2}} + 70 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}} - 140 \, \sqrt {a + \frac {b}{x}} b^{3} \sqrt {x}\right )}}{35 \, a^{5}} \]

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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mupad [B]  time = 1.63, size = 67, normalized size = 0.53 \[ -\frac {2\,\sqrt {x}\,\sqrt {\frac {b+a\,x}{x}}\,\left (-5\,a^4\,x^4+8\,a^3\,b\,x^3-16\,a^2\,b^2\,x^2+64\,a\,b^3\,x+128\,b^4\right )}{35\,a^5\,\left (b+a\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)/(a + b/x)^(3/2),x)

[Out]

-(2*x^(1/2)*((b + a*x)/x)^(1/2)*(128*b^4 - 5*a^4*x^4 + 8*a^3*b*x^3 - 16*a^2*b^2*x^2 + 64*a*b^3*x))/(35*a^5*(b
+ a*x))

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sympy [B]  time = 14.00, size = 614, normalized size = 4.87 \[ \frac {10 a^{7} b^{\frac {33}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} + \frac {14 a^{6} b^{\frac {35}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} + \frac {14 a^{5} b^{\frac {37}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {70 a^{4} b^{\frac {39}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {560 a^{3} b^{\frac {41}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {1120 a^{2} b^{\frac {43}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {896 a b^{\frac {45}{2}} x \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {256 b^{\frac {47}{2}} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} \]

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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