3.1698 \(\int \frac {\sqrt {a+\frac {b}{x}}}{x^4} \, dx\)

Optimal. Leaf size=59 \[ -\frac {2 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3} \]

[Out]

-2/3*a^2*(a+b/x)^(3/2)/b^3+4/5*a*(a+b/x)^(5/2)/b^3-2/7*(a+b/x)^(7/2)/b^3

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac {2 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]/x^4,x]

[Out]

(-2*a^2*(a + b/x)^(3/2))/(3*b^3) + (4*a*(a + b/x)^(5/2))/(5*b^3) - (2*(a + b/x)^(7/2))/(7*b^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+\frac {b}{x}}}{x^4} \, dx &=-\operatorname {Subst}\left (\int x^2 \sqrt {a+b x} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a^2 \sqrt {a+b x}}{b^2}-\frac {2 a (a+b x)^{3/2}}{b^2}+\frac {(a+b x)^{5/2}}{b^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {2 a^2 \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{7/2}}{7 b^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 45, normalized size = 0.76 \[ -\frac {2 \sqrt {a+\frac {b}{x}} (a x+b) \left (8 a^2 x^2-12 a b x+15 b^2\right )}{105 b^3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]/x^4,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.06, size = 49, normalized size = 0.83 \[ -\frac {2 \, {\left (8 \, a^{3} x^{3} - 4 \, a^{2} b x^{2} + 3 \, a b^{2} x + 15 \, b^{3}\right )} \sqrt {\frac {a x + b}{x}}}{105 \, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.25, size = 146, normalized size = 2.47 \[ \frac {2 \, {\left (140 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{4} a^{2} \mathrm {sgn}\relax (x) + 315 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{3} a^{\frac {3}{2}} b \mathrm {sgn}\relax (x) + 273 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{2} a b^{2} \mathrm {sgn}\relax (x) + 105 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )} \sqrt {a} b^{3} \mathrm {sgn}\relax (x) + 15 \, b^{4} \mathrm {sgn}\relax (x)\right )}}{105 \, {\left (\sqrt {a} x - \sqrt {a x^{2} + b x}\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)/x^4,x, algorithm="giac")

[Out]

2/105*(140*(sqrt(a)*x - sqrt(a*x^2 + b*x))^4*a^2*sgn(x) + 315*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b*sgn(
x) + 273*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a*b^2*sgn(x) + 105*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3*sgn(
x) + 15*b^4*sgn(x))/(sqrt(a)*x - sqrt(a*x^2 + b*x))^7

________________________________________________________________________________________

maple [A]  time = 0.00, size = 44, normalized size = 0.75 \[ -\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}-12 a b x +15 b^{2}\right ) \sqrt {\frac {a x +b}{x}}}{105 b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(1/2)/x^4,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.06, size = 47, normalized size = 0.80 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}}}{7 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a}{5 \, b^{3}} - \frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{2}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(1/2)/x^4,x, algorithm="maxima")

[Out]

-2/7*(a + b/x)^(7/2)/b^3 + 4/5*(a + b/x)^(5/2)*a/b^3 - 2/3*(a + b/x)^(3/2)*a^2/b^3

________________________________________________________________________________________

mupad [B]  time = 1.53, size = 70, normalized size = 1.19 \[ \frac {8\,a^2\,\sqrt {a+\frac {b}{x}}}{105\,b^2\,x}-\frac {16\,a^3\,\sqrt {a+\frac {b}{x}}}{105\,b^3}-\frac {2\,a\,\sqrt {a+\frac {b}{x}}}{35\,b\,x^2}-\frac {2\,\sqrt {a+\frac {b}{x}}}{7\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(1/2)/x^4,x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 2.19, size = 899, normalized size = 15.24 \[ - \frac {16 a^{\frac {19}{2}} b^{\frac {9}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} - \frac {40 a^{\frac {17}{2}} b^{\frac {11}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} - \frac {30 a^{\frac {15}{2}} b^{\frac {13}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} - \frac {40 a^{\frac {13}{2}} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} - \frac {100 a^{\frac {11}{2}} b^{\frac {17}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} - \frac {96 a^{\frac {9}{2}} b^{\frac {19}{2}} x \sqrt {\frac {a x}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} - \frac {30 a^{\frac {7}{2}} b^{\frac {21}{2}} \sqrt {\frac {a x}{b} + 1}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} + \frac {16 a^{10} b^{4} x^{\frac {13}{2}}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} + \frac {48 a^{9} b^{5} x^{\frac {11}{2}}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} + \frac {48 a^{8} b^{6} x^{\frac {9}{2}}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} + \frac {16 a^{7} b^{7} x^{\frac {7}{2}}}{105 a^{\frac {13}{2}} b^{7} x^{\frac {13}{2}} + 315 a^{\frac {11}{2}} b^{8} x^{\frac {11}{2}} + 315 a^{\frac {9}{2}} b^{9} x^{\frac {9}{2}} + 105 a^{\frac {7}{2}} b^{10} x^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(1/2)/x**4,x)

[Out]

-16*a**(19/2)*b**(9/2)*x**6*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315
*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 40*a**(17/2)*b**(11/2)*x**5*sqrt(a*x/b + 1)/(105*a**(
13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)
) - 30*a**(15/2)*b**(13/2)*x**4*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) +
315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 40*a**(13/2)*b**(15/2)*x**3*sqrt(a*x/b + 1)/(105*
a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(
7/2)) - 100*a**(11/2)*b**(17/2)*x**2*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11
/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) - 96*a**(9/2)*b**(19/2)*x*sqrt(a*x/b + 1)/(105
*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**
(7/2)) - 30*a**(7/2)*b**(21/2)*sqrt(a*x/b + 1)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) +
315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 16*a**10*b**4*x**(13/2)/(105*a**(13/2)*b**7*x**(13
/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 48*a**9*b**5*
x**(11/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(
7/2)*b**10*x**(7/2)) + 48*a**8*b**6*x**(9/2)/(105*a**(13/2)*b**7*x**(13/2) + 315*a**(11/2)*b**8*x**(11/2) + 31
5*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2)) + 16*a**7*b**7*x**(7/2)/(105*a**(13/2)*b**7*x**(13/2)
+ 315*a**(11/2)*b**8*x**(11/2) + 315*a**(9/2)*b**9*x**(9/2) + 105*a**(7/2)*b**10*x**(7/2))

________________________________________________________________________________________