\(\int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx\) [1759]

Optimal result

Integrand size = 17, antiderivative size = 106 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^{5/2}} \]

[Out]

-1/8*a^3*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))/b^(5/2)-1/3*(a+b/x)^(1/2)/x^(5/2)-1/12*a*(a+b/x)^(1/2)/b/x^(3/
2)+1/8*a^2*(a+b/x)^(1/2)/b^2/x^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {344, 285, 327, 223, 212} \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=-\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 b^{5/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}-\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}} \]

[In]

Int[Sqrt[a + b/x]/x^(7/2),x]

[Out]

-1/3*Sqrt[a + b/x]/x^(5/2) - (a*Sqrt[a + b/x])/(12*b*x^(3/2)) + (a^2*Sqrt[a + b/x])/(8*b^2*Sqrt[x]) - (a^3*Arc
Tanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(8*b^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 344

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[-k/c, Subst[
Int[(a + b/(c^n*x^(k*n)))^p/x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && ILtQ[n,
0] && FractionQ[m]

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int x^4 \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {1}{3} a \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 b} \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a^3 \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{8 b^2} \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a^3 \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^2} \\ & = -\frac {\sqrt {a+\frac {b}{x}}}{3 x^{5/2}}-\frac {a \sqrt {a+\frac {b}{x}}}{12 b x^{3/2}}+\frac {a^2 \sqrt {a+\frac {b}{x}}}{8 b^2 \sqrt {x}}-\frac {a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 b^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\frac {\sqrt {a+\frac {b}{x}} \left (\frac {\sqrt {b} \left (-8 b^2-2 a b x+3 a^2 x^2\right )}{x^{5/2}}-\frac {3 a^{5/2} \text {arcsinh}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{\sqrt {1+\frac {b}{a x}}}\right )}{24 b^{5/2}} \]

[In]

Integrate[Sqrt[a + b/x]/x^(7/2),x]

[Out]

(Sqrt[a + b/x]*((Sqrt[b]*(-8*b^2 - 2*a*b*x + 3*a^2*x^2))/x^(5/2) - (3*a^(5/2)*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x]
)])/Sqrt[1 + b/(a*x)]))/(24*b^(5/2))

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {\left (3 a^{2} x^{2}-2 a b x -8 b^{2}\right ) \sqrt {\frac {a x +b}{x}}}{24 x^{\frac {5}{2}} b^{2}}-\frac {a^{3} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{8 b^{\frac {5}{2}} \sqrt {a x +b}}\)	\(81\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{3} x^{3}-3 a^{2} x^{2} \sqrt {b}\, \sqrt {a x +b}+2 a \,b^{\frac {3}{2}} x \sqrt {a x +b}+8 b^{\frac {5}{2}} \sqrt {a x +b}\right )}{24 x^{\frac {5}{2}} b^{\frac {5}{2}} \sqrt {a x +b}}\)	\(92\)

[In]

int((a+b/x)^(1/2)/x^(7/2),x,method=_RETURNVERBOSE)

[Out]

1/24*(3*a^2*x^2-2*a*b*x-8*b^2)/x^(5/2)/b^2*((a*x+b)/x)^(1/2)-1/8*a^3/b^(5/2)*arctanh((a*x+b)^(1/2)/b^(1/2))*((
a*x+b)/x)^(1/2)/(a*x+b)^(1/2)*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\left [\frac {3 \, a^{3} \sqrt {b} x^{3} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (3 \, a^{2} b x^{2} - 2 \, a b^{2} x - 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{48 \, b^{3} x^{3}}, \frac {3 \, a^{3} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a^{2} b x^{2} - 2 \, a b^{2} x - 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, b^{3} x^{3}}\right ] \]

[In]

integrate((a+b/x)^(1/2)/x^(7/2),x, algorithm="fricas")

[Out]

[1/48*(3*a^3*sqrt(b)*x^3*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2*(3*a^2*b*x^2 - 2*a*b^2*x
- 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b^3*x^3), 1/24*(3*a^3*sqrt(-b)*x^3*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b
)/x)/b) + (3*a^2*b*x^2 - 2*a*b^2*x - 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b^3*x^3)]

Sympy [A] (verification not implemented)

Time = 10.16 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\frac {a^{\frac {5}{2}}}{8 b^{2} \sqrt {x} \sqrt {1 + \frac {b}{a x}}} + \frac {a^{\frac {3}{2}}}{24 b x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {5 \sqrt {a}}{12 x^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}} - \frac {a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{8 b^{\frac {5}{2}}} - \frac {b}{3 \sqrt {a} x^{\frac {7}{2}} \sqrt {1 + \frac {b}{a x}}} \]

[In]

integrate((a+b/x)**(1/2)/x**(7/2),x)

[Out]

a**(5/2)/(8*b**2*sqrt(x)*sqrt(1 + b/(a*x))) + a**(3/2)/(24*b*x**(3/2)*sqrt(1 + b/(a*x))) - 5*sqrt(a)/(12*x**(5
/2)*sqrt(1 + b/(a*x))) - a**3*asinh(sqrt(b)/(sqrt(a)*sqrt(x)))/(8*b**(5/2)) - b/(3*sqrt(a)*x**(7/2)*sqrt(1 + b
/(a*x)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (78) = 156\).

Time = 0.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.52 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\frac {a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{16 \, b^{\frac {5}{2}}} + \frac {3 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} x^{\frac {5}{2}} - 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3} b x^{\frac {3}{2}} - 3 \, \sqrt {a + \frac {b}{x}} a^{3} b^{2} \sqrt {x}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} b^{2} x^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} b^{3} x^{2} + 3 \, {\left (a + \frac {b}{x}\right )} b^{4} x - b^{5}\right )}} \]

[In]

integrate((a+b/x)^(1/2)/x^(7/2),x, algorithm="maxima")

[Out]

1/16*a^3*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b)))/b^(5/2) + 1/24*(3*(a + b/x)^
(5/2)*a^3*x^(5/2) - 8*(a + b/x)^(3/2)*a^3*b*x^(3/2) - 3*sqrt(a + b/x)*a^3*b^2*sqrt(x))/((a + b/x)^3*b^2*x^3 -
3*(a + b/x)^2*b^3*x^2 + 3*(a + b/x)*b^4*x - b^5)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\frac {{\left (\frac {3 \, a^{4} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {3 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{4} - 8 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{4} b - 3 \, \sqrt {a x + b} a^{4} b^{2}}{a^{3} b^{2} x^{3}}\right )} \mathrm {sgn}\left (x\right )}{24 \, a} \]

[In]

integrate((a+b/x)^(1/2)/x^(7/2),x, algorithm="giac")

[Out]

1/24*(3*a^4*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (3*(a*x + b)^(5/2)*a^4 - 8*(a*x + b)^(3/2)*a^4*b -
3*sqrt(a*x + b)*a^4*b^2)/(a^3*b^2*x^3))*sgn(x)/a

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x}}}{x^{7/2}} \,d x \]

[In]

int((a + b/x)^(1/2)/x^(7/2),x)

[Out]

int((a + b/x)^(1/2)/x^(7/2), x)