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*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)

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Rubi [A]  time = 0.0251826, antiderivative size = 59, normalized size of antiderivative = 1., 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 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]]

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])

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 = 0.0203619, 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)

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Maple [A]  time = 0.004, size = 44, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 8\,{a}^{2}{x}^{2}-12\,xab+15\,{b}^{2} \right ) }{105\,{b}^{3}{x}^{3}}\sqrt{{\frac{ax+b}{x}}}} \end{align*}

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

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Maxima [A]  time = 0.997731, size = 63, normalized size = 1.07 \begin{align*} -\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}} \end{align*}

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

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Fricas [A]  time = 1.80158, size = 112, normalized size = 1.9 \begin{align*} -\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}} \end{align*}

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)

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Sympy [B]  time = 1.90633, size = 899, normalized size = 15.24 \begin{align*} \text{result too large to display} \end{align*}

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))

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Giac [B]  time = 1.16006, size = 197, normalized size = 3.34 \begin{align*} \frac{2 \,{\left (140 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{4} a^{2} \mathrm{sgn}\left (x\right ) + 315 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}} b \mathrm{sgn}\left (x\right ) + 273 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b^{2} \mathrm{sgn}\left (x\right ) + 105 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{3} \mathrm{sgn}\left (x\right ) + 15 \, b^{4} \mathrm{sgn}\left (x\right )\right )}}{105 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{7}} \end{align*}

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