3.1753 \(\int \sqrt {a+\frac {b}{x}} x^{5/2} \, dx\)

Optimal. Leaf size=74 \[ \frac {16 b^2 x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}{105 a^3}-\frac {8 b x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}{35 a^2}+\frac {2 x^{7/2} \left (a+\frac {b}{x}\right )^{3/2}}{7 a} \]

[Out]

16/105*b^2*(a+b/x)^(3/2)*x^(3/2)/a^3-8/35*b*(a+b/x)^(3/2)*x^(5/2)/a^2+2/7*(a+b/x)^(3/2)*x^(7/2)/a

________________________________________________________________________________________

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} \[ \frac {16 b^2 x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}{105 a^3}-\frac {8 b x^{5/2} \left (a+\frac {b}{x}\right )^{3/2}}{35 a^2}+\frac {2 x^{7/2} \left (a+\frac {b}{x}\right )^{3/2}}{7 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]*x^(5/2),x]

[Out]

(16*b^2*(a + b/x)^(3/2)*x^(3/2))/(105*a^3) - (8*b*(a + b/x)^(3/2)*x^(5/2))/(35*a^2) + (2*(a + b/x)^(3/2)*x^(7/
2))/(7*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 \sqrt {a+\frac {b}{x}} x^{5/2} \, dx &=\frac {2 \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}{7 a}-\frac {(4 b) \int \sqrt {a+\frac {b}{x}} x^{3/2} \, dx}{7 a}\\ &=-\frac {8 b \left (a+\frac {b}{x}\right )^{3/2} x^{5/2}}{35 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}{7 a}+\frac {\left (8 b^2\right ) \int \sqrt {a+\frac {b}{x}} \sqrt {x} \, dx}{35 a^2}\\ &=\frac {16 b^2 \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}{105 a^3}-\frac {8 b \left (a+\frac {b}{x}\right )^{3/2} x^{5/2}}{35 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}{7 a}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 47, normalized size = 0.64 \[ \frac {2 \sqrt {x} \sqrt {a+\frac {b}{x}} (a x+b) \left (15 a^2 x^2-12 a b x+8 b^2\right )}{105 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]*x^(5/2),x]

[Out]

(2*Sqrt[a + b/x]*Sqrt[x]*(b + a*x)*(8*b^2 - 12*a*b*x + 15*a^2*x^2))/(105*a^3)

________________________________________________________________________________________

fricas [A]  time = 0.78, size = 49, normalized size = 0.66 \[ \frac {2 \, {\left (15 \, a^{3} x^{3} + 3 \, a^{2} b x^{2} - 4 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{105 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)*x^(5/2),x, algorithm="fricas")

[Out]

2/105*(15*a^3*x^3 + 3*a^2*b*x^2 - 4*a*b^2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x)/a^3

________________________________________________________________________________________

giac [A]  time = 0.16, size = 50, normalized size = 0.68 \[ -\frac {2}{105} \, {\left (\frac {8 \, b^{\frac {7}{2}}}{a^{3}} - \frac {15 \, {\left (a x + b\right )}^{\frac {7}{2}} - 42 \, {\left (a x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}}{a^{3}}\right )} \mathrm {sgn}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)*x^(5/2),x, algorithm="giac")

[Out]

-2/105*(8*b^(7/2)/a^3 - (15*(a*x + b)^(7/2) - 42*(a*x + b)^(5/2)*b + 35*(a*x + b)^(3/2)*b^2)/a^3)*sgn(x)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 44, normalized size = 0.59 \[ \frac {2 \left (a x +b \right ) \left (15 a^{2} x^{2}-12 a b x +8 b^{2}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{105 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(1/2)*x^(5/2),x)

[Out]

2/105*(a*x+b)*(15*a^2*x^2-12*a*b*x+8*b^2)*x^(1/2)*((a*x+b)/x)^(1/2)/a^3

________________________________________________________________________________________

maxima [A]  time = 1.11, size = 52, normalized size = 0.70 \[ \frac {2 \, {\left (15 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} x^{\frac {7}{2}} - 42 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b x^{\frac {5}{2}} + 35 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}}\right )}}{105 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)*x^(5/2),x, algorithm="maxima")

[Out]

2/105*(15*(a + b/x)^(7/2)*x^(7/2) - 42*(a + b/x)^(5/2)*b*x^(5/2) + 35*(a + b/x)^(3/2)*b^2*x^(3/2))/a^3

________________________________________________________________________________________

mupad [B]  time = 1.40, size = 47, normalized size = 0.64 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,x^{7/2}}{7}+\frac {2\,b\,x^{5/2}}{35\,a}-\frac {8\,b^2\,x^{3/2}}{105\,a^2}+\frac {16\,b^3\,\sqrt {x}}{105\,a^3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(a + b/x)^(1/2),x)

[Out]

(a + b/x)^(1/2)*((2*x^(7/2))/7 + (2*b*x^(5/2))/(35*a) - (8*b^2*x^(3/2))/(105*a^2) + (16*b^3*x^(1/2))/(105*a^3)
)

________________________________________________________________________________________

sympy [B]  time = 16.44, size = 314, normalized size = 4.24 \[ \frac {30 a^{5} b^{\frac {9}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac {66 a^{4} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac {34 a^{3} b^{\frac {13}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac {6 a^{2} b^{\frac {15}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac {24 a b^{\frac {17}{2}} x \sqrt {\frac {a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} + \frac {16 b^{\frac {19}{2}} \sqrt {\frac {a x}{b} + 1}}{105 a^{5} b^{4} x^{2} + 210 a^{4} b^{5} x + 105 a^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(1/2)*x**(5/2),x)

[Out]

30*a**5*b**(9/2)*x**5*sqrt(a*x/b + 1)/(105*a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*b**6) + 66*a**4*b**(11/
2)*x**4*sqrt(a*x/b + 1)/(105*a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*b**6) + 34*a**3*b**(13/2)*x**3*sqrt(a
*x/b + 1)/(105*a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*b**6) + 6*a**2*b**(15/2)*x**2*sqrt(a*x/b + 1)/(105*
a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*b**6) + 24*a*b**(17/2)*x*sqrt(a*x/b + 1)/(105*a**5*b**4*x**2 + 210
*a**4*b**5*x + 105*a**3*b**6) + 16*b**(19/2)*sqrt(a*x/b + 1)/(105*a**5*b**4*x**2 + 210*a**4*b**5*x + 105*a**3*
b**6)

________________________________________________________________________________________