\(\int \frac {1}{(a+\frac {b}{x})^{5/2} x^{7/2}} \, dx\) [253]

Optimal result

Integrand size = 17, antiderivative size = 75 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 10.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {2 \left (\sqrt {b} \sqrt {x} (4 b+3 a x)-3 \sqrt {a} \sqrt {1+\frac {b}{a x}} x (b+a x) \text {arcsinh}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )\right )}{3 b^{5/2} \sqrt {a+\frac {b}{x}} x (b+a x)} \] Input:

Integrate[1/((a + b/x)^(5/2)*x^(7/2)),x]
 

Output:

(2*(Sqrt[b]*Sqrt[x]*(4*b + 3*a*x) - 3*Sqrt[a]*Sqrt[1 + b/(a*x)]*x*(b + a*x 
)*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x])]))/(3*b^(5/2)*Sqrt[a + b/x]*x*(b + a*x 
))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {860, 252, 252, 224, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x^{7/2} \left (a+\frac {b}{x}\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 860

\(\displaystyle -2 \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^2}d\frac {1}{\sqrt {x}}\)

\(\Big \downarrow \) 252

\(\displaystyle -2 \left (\frac {\int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x}d\frac {1}{\sqrt {x}}}{b}-\frac {1}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\)

\(\Big \downarrow \) 252

\(\displaystyle -2 \left (\frac {\frac {\int \frac {1}{\sqrt {a+\frac {b}{x}}}d\frac {1}{\sqrt {x}}}{b}-\frac {1}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}}{b}-\frac {1}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle -2 \left (\frac {\frac {\int \frac {1}{1-\frac {b}{x}}d\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}}{b}-\frac {1}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}}{b}-\frac {1}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle -2 \left (\frac {\frac {\text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{3/2}}-\frac {1}{b \sqrt {x} \sqrt {a+\frac {b}{x}}}}{b}-\frac {1}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}}\right )\)

Input:

Int[1/((a + b/x)^(5/2)*x^(7/2)),x]
 

Output:

-2*(-1/3*1/(b*(a + b/x)^(3/2)*x^(3/2)) + (-(1/(b*Sqrt[a + b/x]*Sqrt[x])) + 
 ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])]/b^(3/2))/b)
 

Defintions of rubi rules used

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 860

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[-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]
 

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (-3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}\, a x +4 b^{\frac {3}{2}}+3 x a \sqrt {b}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) b \sqrt {a x +b}\right )}{3 b^{\frac {5}{2}} \left (a x +b \right )^{2}}\)	\(85\)

Input:

int(1/(a+b/x)^(5/2)/x^(7/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.79 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {b} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{a x + b}\right ) + {\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}\right )}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}\right ] \] Input:

integrate(1/(a+b/x)^(5/2)/x^(7/2),x, algorithm="fricas")
 

Output:

[1/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(b)*log((a*x - 2*sqrt(b)*sqrt(x)*sqr 
t((a*x + b)/x) + 2*b)/x) + 2*(3*a*b*x + 4*b^2)*sqrt(x)*sqrt((a*x + b)/x))/ 
(a^2*b^3*x^2 + 2*a*b^4*x + b^5), 2/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(-b) 
*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/(a*x + b)) + (3*a*b*x + 4*b^2)* 
sqrt(x)*sqrt((a*x + b)/x))/(a^2*b^3*x^2 + 2*a*b^4*x + b^5)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (63) = 126\).

Time = 30.97 (sec) , antiderivative size = 697, normalized size of antiderivative = 9.29 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {3 a^{3} b^{4} x^{3} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {6 a^{3} b^{4} x^{3} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {6 a^{2} b^{5} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {9 a^{2} b^{5} x^{2} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {18 a^{2} b^{5} x^{2} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {14 a b^{6} x \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {9 a b^{6} x \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {18 a b^{6} x \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {8 b^{7} \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {3 b^{7} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {6 b^{7} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} \] Input:

integrate(1/(a+b/x)**(5/2)/x**(7/2),x)
 

Output:

3*a**3*b**4*x**3*log(a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 
 + 9*a*b**(17/2)*x + 3*b**(19/2)) - 6*a**3*b**4*x**3*log(sqrt(a*x/b + 1) + 
 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b 
**(19/2)) + 6*a**2*b**5*x**2*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 + 9*a* 
*2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) + 9*a**2*b**5*x**2*log( 
a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 
3*b**(19/2)) - 18*a**2*b**5*x**2*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2 
)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) + 14*a*b** 
6*x*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b 
**(17/2)*x + 3*b**(19/2)) + 9*a*b**6*x*log(a*x/b)/(3*a**3*b**(13/2)*x**3 + 
 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) - 18*a*b**6*x*log( 
sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a* 
b**(17/2)*x + 3*b**(19/2)) + 8*b**7*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 
 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) + 3*b**7*log(a*x 
/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b 
**(19/2)) - 6*b**7*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a** 
2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2))
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{b^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} x + b\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}}} \] Input:

integrate(1/(a+b/x)^(5/2)/x^(7/2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b}\right )} \mathrm {sgn}\left (x\right )}{3 \, \sqrt {-b} b^{\frac {5}{2}}} + \frac {2 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2} \mathrm {sgn}\left (x\right )} + \frac {2 \, {\left (3 \, a x + 4 \, b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (x\right )} \] Input:

integrate(1/(a+b/x)^(5/2)/x^(7/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\int \frac {1}{x^{7/2}\,{\left (a+\frac {b}{x}\right )}^{5/2}} \,d x \] Input:

int(1/(x^(7/2)*(a + b/x)^(5/2)),x)
 

Output:

int(1/(x^(7/2)*(a + b/x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx=\frac {3 \sqrt {b}\, \sqrt {a x +b}\, \mathrm {log}\left (\sqrt {a x +b}-\sqrt {b}\right ) a x +3 \sqrt {b}\, \sqrt {a x +b}\, \mathrm {log}\left (\sqrt {a x +b}-\sqrt {b}\right ) b -3 \sqrt {b}\, \sqrt {a x +b}\, \mathrm {log}\left (\sqrt {a x +b}+\sqrt {b}\right ) a x -3 \sqrt {b}\, \sqrt {a x +b}\, \mathrm {log}\left (\sqrt {a x +b}+\sqrt {b}\right ) b +6 a b x +8 b^{2}}{3 \sqrt {a x +b}\, b^{3} \left (a x +b \right )} \] Input:

int(1/(a+b/x)^(5/2)/x^(7/2),x)
 

Output:

(3*sqrt(b)*sqrt(a*x + b)*log(sqrt(a*x + b) - sqrt(b))*a*x + 3*sqrt(b)*sqrt 
(a*x + b)*log(sqrt(a*x + b) - sqrt(b))*b - 3*sqrt(b)*sqrt(a*x + b)*log(sqr 
t(a*x + b) + sqrt(b))*a*x - 3*sqrt(b)*sqrt(a*x + b)*log(sqrt(a*x + b) + sq 
rt(b))*b + 6*a*b*x + 8*b**2)/(3*sqrt(a*x + b)*b**3*(a*x + b))