3.1779 \(\int \frac {x^{3/2}}{\sqrt {a+\frac {b}{x}}} \, dx\)

Optimal. Leaf size=74 \[ \frac {16 b^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}{15 a^3}-\frac {8 b x^{3/2} \sqrt {a+\frac {b}{x}}}{15 a^2}+\frac {2 x^{5/2} \sqrt {a+\frac {b}{x}}}{5 a} \]

[Out]

-8/15*b*x^(3/2)*(a+b/x)^(1/2)/a^2+2/5*x^(5/2)*(a+b/x)^(1/2)/a+16/15*b^2*(a+b/x)^(1/2)*x^(1/2)/a^3

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Rubi [A]  time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {16 b^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}{15 a^3}-\frac {8 b x^{3/2} \sqrt {a+\frac {b}{x}}}{15 a^2}+\frac {2 x^{5/2} \sqrt {a+\frac {b}{x}}}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)/Sqrt[a + b/x],x]

[Out]

(16*b^2*Sqrt[a + b/x]*Sqrt[x])/(15*a^3) - (8*b*Sqrt[a + b/x]*x^(3/2))/(15*a^2) + (2*Sqrt[a + b/x]*x^(5/2))/(5*
a)

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

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Mathematica [A]  time = 0.02, size = 42, normalized size = 0.57 \[ \frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}} \left (3 a^2 x^2-4 a b x+8 b^2\right )}{15 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)/Sqrt[a + b/x],x]

[Out]

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

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fricas [A]  time = 0.78, size = 38, normalized size = 0.51 \[ \frac {2 \, {\left (3 \, a^{2} x^{2} - 4 \, a b x + 8 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{15 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b/x)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*a^2*x^2 - 4*a*b*x + 8*b^2)*sqrt(x)*sqrt((a*x + b)/x)/a^3

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giac [A]  time = 0.16, size = 49, normalized size = 0.66 \[ \frac {2 \, \sqrt {a x + b} b^{2}}{a^{3}} - \frac {16 \, b^{\frac {5}{2}}}{15 \, a^{3}} + \frac {2 \, {\left (3 \, {\left (a x + b\right )}^{\frac {5}{2}} - 10 \, {\left (a x + b\right )}^{\frac {3}{2}} b\right )}}{15 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b/x)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(a*x + b)*b^2/a^3 - 16/15*b^(5/2)/a^3 + 2/15*(3*(a*x + b)^(5/2) - 10*(a*x + b)^(3/2)*b)/a^3

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maple [A]  time = 0.00, size = 44, normalized size = 0.59 \[ \frac {2 \left (a x +b \right ) \left (3 a^{2} x^{2}-4 a b x +8 b^{2}\right )}{15 \sqrt {\frac {a x +b}{x}}\, a^{3} \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(a+b/x)^(1/2),x)

[Out]

2/15*(a*x+b)*(3*a^2*x^2-4*a*b*x+8*b^2)/a^3/x^(1/2)/((a*x+b)/x)^(1/2)

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maxima [A]  time = 1.05, size = 52, normalized size = 0.70 \[ \frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} x^{\frac {5}{2}} - 10 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b x^{\frac {3}{2}} + 15 \, \sqrt {a + \frac {b}{x}} b^{2} \sqrt {x}\right )}}{15 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)/(a+b/x)^(1/2),x, algorithm="maxima")

[Out]

2/15*(3*(a + b/x)^(5/2)*x^(5/2) - 10*(a + b/x)^(3/2)*b*x^(3/2) + 15*sqrt(a + b/x)*b^2*sqrt(x))/a^3

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mupad [B]  time = 1.44, size = 39, normalized size = 0.53 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{5/2}}{5\,a}-\frac {8\,b\,x^{3/2}}{15\,a^2}+\frac {16\,b^2\,\sqrt {x}}{15\,a^3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)/(a + b/x)^(1/2),x)

[Out]

(a + b/x)^(1/2)*((2*x^(5/2))/(5*a) - (8*b*x^(3/2))/(15*a^2) + (16*b^2*x^(1/2))/(15*a^3))

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sympy [B]  time = 3.83, size = 260, normalized size = 3.51 \[ \frac {6 a^{4} b^{\frac {9}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac {4 a^{3} b^{\frac {11}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac {6 a^{2} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac {24 a b^{\frac {15}{2}} x \sqrt {\frac {a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} + \frac {16 b^{\frac {17}{2}} \sqrt {\frac {a x}{b} + 1}}{15 a^{5} b^{4} x^{2} + 30 a^{4} b^{5} x + 15 a^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)/(a+b/x)**(1/2),x)

[Out]

6*a**4*b**(9/2)*x**4*sqrt(a*x/b + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x + 15*a**3*b**6) + 4*a**3*b**(11/2)*x*
*3*sqrt(a*x/b + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x + 15*a**3*b**6) + 6*a**2*b**(13/2)*x**2*sqrt(a*x/b + 1)
/(15*a**5*b**4*x**2 + 30*a**4*b**5*x + 15*a**3*b**6) + 24*a*b**(15/2)*x*sqrt(a*x/b + 1)/(15*a**5*b**4*x**2 + 3
0*a**4*b**5*x + 15*a**3*b**6) + 16*b**(17/2)*sqrt(a*x/b + 1)/(15*a**5*b**4*x**2 + 30*a**4*b**5*x + 15*a**3*b**
6)

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