3.1729 \(\int \frac {1}{\sqrt {a+\frac {b}{x}} x^4} \, dx\)

Optimal. Leaf size=57 \[ -\frac {2 a^2 \sqrt {a+\frac {b}{x}}}{b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3} \]

[Out]

4/3*a*(a+b/x)^(3/2)/b^3-2/5*(a+b/x)^(5/2)/b^3-2*a^2*(a+b/x)^(1/2)/b^3

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Rubi [A]  time = 0.02, antiderivative size = 57, 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} \[ -\frac {2 a^2 \sqrt {a+\frac {b}{x}}}{b^3}-\frac {2 \left (a+\frac {b}{x}\right )^{5/2}}{5 b^3}+\frac {4 a \left (a+\frac {b}{x}\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*x^4),x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 40, normalized size = 0.70 \[ -\frac {2 \sqrt {a+\frac {b}{x}} \left (8 a^2 x^2-4 a b x+3 b^2\right )}{15 b^3 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b/x]*x^4),x]

[Out]

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

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fricas [A]  time = 0.98, size = 38, normalized size = 0.67 \[ -\frac {2 \, {\left (8 \, a^{2} x^{2} - 4 \, a b x + 3 \, b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{15 \, b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.17, size = 67, normalized size = 1.18 \[ -\frac {2 \, {\left (15 \, a^{2} \sqrt {\frac {a x + b}{x}} - \frac {10 \, {\left (a x + b\right )} a \sqrt {\frac {a x + b}{x}}}{x} + \frac {3 \, {\left (a x + b\right )}^{2} \sqrt {\frac {a x + b}{x}}}{x^{2}}\right )}}{15 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 44, normalized size = 0.77 \[ -\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}-4 a b x +3 b^{2}\right )}{15 \sqrt {\frac {a x +b}{x}}\, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, size = 47, normalized size = 0.82 \[ -\frac {2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{5 \, b^{3}} + \frac {4 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a}{3 \, b^{3}} - \frac {2 \, \sqrt {a + \frac {b}{x}} a^{2}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.30, size = 54, normalized size = 0.95 \[ -\frac {6\,b^2\,\sqrt {a+\frac {b}{x}}+16\,a^2\,x^2\,\sqrt {a+\frac {b}{x}}-8\,a\,b\,x\,\sqrt {a+\frac {b}{x}}}{15\,b^3\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^4*(a + b/x)^(1/2)),x)

[Out]

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

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sympy [B]  time = 2.25, size = 813, normalized size = 14.26 \[ - \frac {16 a^{\frac {15}{2}} b^{\frac {9}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {40 a^{\frac {13}{2}} b^{\frac {11}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {30 a^{\frac {11}{2}} b^{\frac {13}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {10 a^{\frac {9}{2}} b^{\frac {15}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {10 a^{\frac {7}{2}} b^{\frac {17}{2}} x \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} - \frac {6 a^{\frac {5}{2}} b^{\frac {19}{2}} \sqrt {\frac {a x}{b} + 1}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} + \frac {16 a^{8} b^{4} x^{\frac {11}{2}}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} + \frac {48 a^{7} b^{5} x^{\frac {9}{2}}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} + \frac {48 a^{6} b^{6} x^{\frac {7}{2}}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} + \frac {16 a^{5} b^{7} x^{\frac {5}{2}}}{15 a^{\frac {11}{2}} b^{7} x^{\frac {11}{2}} + 45 a^{\frac {9}{2}} b^{8} x^{\frac {9}{2}} + 45 a^{\frac {7}{2}} b^{9} x^{\frac {7}{2}} + 15 a^{\frac {5}{2}} b^{10} x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*a**(15/2)*b**(9/2)*x**5*sqrt(a*x/b + 1)/(15*a**(11/2)*b**7*x**(11/2) + 45*a**(9/2)*b**8*x**(9/2) + 45*a**(
7/2)*b**9*x**(7/2) + 15*a**(5/2)*b**10*x**(5/2)) - 40*a**(13/2)*b**(11/2)*x**4*sqrt(a*x/b + 1)/(15*a**(11/2)*b
**7*x**(11/2) + 45*a**(9/2)*b**8*x**(9/2) + 45*a**(7/2)*b**9*x**(7/2) + 15*a**(5/2)*b**10*x**(5/2)) - 30*a**(1
1/2)*b**(13/2)*x**3*sqrt(a*x/b + 1)/(15*a**(11/2)*b**7*x**(11/2) + 45*a**(9/2)*b**8*x**(9/2) + 45*a**(7/2)*b**
9*x**(7/2) + 15*a**(5/2)*b**10*x**(5/2)) - 10*a**(9/2)*b**(15/2)*x**2*sqrt(a*x/b + 1)/(15*a**(11/2)*b**7*x**(1
1/2) + 45*a**(9/2)*b**8*x**(9/2) + 45*a**(7/2)*b**9*x**(7/2) + 15*a**(5/2)*b**10*x**(5/2)) - 10*a**(7/2)*b**(1
7/2)*x*sqrt(a*x/b + 1)/(15*a**(11/2)*b**7*x**(11/2) + 45*a**(9/2)*b**8*x**(9/2) + 45*a**(7/2)*b**9*x**(7/2) +
15*a**(5/2)*b**10*x**(5/2)) - 6*a**(5/2)*b**(19/2)*sqrt(a*x/b + 1)/(15*a**(11/2)*b**7*x**(11/2) + 45*a**(9/2)*
b**8*x**(9/2) + 45*a**(7/2)*b**9*x**(7/2) + 15*a**(5/2)*b**10*x**(5/2)) + 16*a**8*b**4*x**(11/2)/(15*a**(11/2)
*b**7*x**(11/2) + 45*a**(9/2)*b**8*x**(9/2) + 45*a**(7/2)*b**9*x**(7/2) + 15*a**(5/2)*b**10*x**(5/2)) + 48*a**
7*b**5*x**(9/2)/(15*a**(11/2)*b**7*x**(11/2) + 45*a**(9/2)*b**8*x**(9/2) + 45*a**(7/2)*b**9*x**(7/2) + 15*a**(
5/2)*b**10*x**(5/2)) + 48*a**6*b**6*x**(7/2)/(15*a**(11/2)*b**7*x**(11/2) + 45*a**(9/2)*b**8*x**(9/2) + 45*a**
(7/2)*b**9*x**(7/2) + 15*a**(5/2)*b**10*x**(5/2)) + 16*a**5*b**7*x**(5/2)/(15*a**(11/2)*b**7*x**(11/2) + 45*a*
*(9/2)*b**8*x**(9/2) + 45*a**(7/2)*b**9*x**(7/2) + 15*a**(5/2)*b**10*x**(5/2))

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