\(\int \frac {1}{(a+\frac {b}{x})^3 x^{13/2}} \, dx\) [142]

Optimal result

Integrand size = 15, antiderivative size = 116 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{13/2}} \, dx=-\frac {2}{5 b^3 x^{5/2}}+\frac {2 a}{b^4 x^{3/2}}-\frac {12 a^2}{b^5 \sqrt {x}}+\frac {a^4}{2 b^5 \left (a+\frac {b}{x}\right )^2 \sqrt {x}}-\frac {17 a^3}{4 b^5 \left (a+\frac {b}{x}\right ) \sqrt {x}}+\frac {63 a^{5/2} \arctan \left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{4 b^{11/2}} \] Output:

-2/5/b^3/x^(5/2)+2*a/b^4/x^(3/2)-12*a^2/b^5/x^(1/2)+1/2*a^4/b^5/(a+b/x)^2/ 
x^(1/2)-17/4*a^3/b^5/(a+b/x)/x^(1/2)+63/4*a^(5/2)*arctan(1/a^(1/2)/x^(1/2) 
*b^(1/2))/b^(11/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{13/2}} \, dx=\frac {-8 b^4+24 a b^3 x-168 a^2 b^2 x^2-525 a^3 b x^3-315 a^4 x^4}{20 b^5 x^{5/2} (b+a x)^2}-\frac {63 a^{5/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{11/2}} \] Input:

Integrate[1/((a + b/x)^3*x^(13/2)),x]
 

Output:

(-8*b^4 + 24*a*b^3*x - 168*a^2*b^2*x^2 - 525*a^3*b*x^3 - 315*a^4*x^4)/(20* 
b^5*x^(5/2)*(b + a*x)^2) - (63*a^(5/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/ 
(4*b^(11/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {795, 52, 52, 61, 61, 61, 73, 218}

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.

\(\displaystyle \int \frac {1}{x^{13/2} \left (a+\frac {b}{x}\right )^3} \, dx\)

\(\Big \downarrow \) 795

\(\displaystyle \int \frac {1}{x^{7/2} (a x+b)^3}dx\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {9 \int \frac {1}{x^{7/2} (b+a x)^2}dx}{4 b}+\frac {1}{2 b x^{5/2} (a x+b)^2}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{x^{7/2} (b+a x)}dx}{2 b}+\frac {1}{b x^{5/2} (a x+b)}\right )}{4 b}+\frac {1}{2 b x^{5/2} (a x+b)^2}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {a \int \frac {1}{x^{5/2} (b+a x)}dx}{b}-\frac {2}{5 b x^{5/2}}\right )}{2 b}+\frac {1}{b x^{5/2} (a x+b)}\right )}{4 b}+\frac {1}{2 b x^{5/2} (a x+b)^2}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {a \left (-\frac {a \int \frac {1}{x^{3/2} (b+a x)}dx}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{2 b}+\frac {1}{b x^{5/2} (a x+b)}\right )}{4 b}+\frac {1}{2 b x^{5/2} (a x+b)^2}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {a \left (-\frac {a \left (-\frac {a \int \frac {1}{\sqrt {x} (b+a x)}dx}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{2 b}+\frac {1}{b x^{5/2} (a x+b)}\right )}{4 b}+\frac {1}{2 b x^{5/2} (a x+b)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {a \left (-\frac {a \left (-\frac {2 a \int \frac {1}{b+a x}d\sqrt {x}}{b}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{2 b}+\frac {1}{b x^{5/2} (a x+b)}\right )}{4 b}+\frac {1}{2 b x^{5/2} (a x+b)^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {9 \left (\frac {7 \left (-\frac {a \left (-\frac {a \left (-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{3/2}}-\frac {2}{b \sqrt {x}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{5 b x^{5/2}}\right )}{2 b}+\frac {1}{b x^{5/2} (a x+b)}\right )}{4 b}+\frac {1}{2 b x^{5/2} (a x+b)^2}\)

Input:

Int[1/((a + b/x)^3*x^(13/2)),x]
 

Output:

1/(2*b*x^(5/2)*(b + a*x)^2) + (9*(1/(b*x^(5/2)*(b + a*x)) + (7*(-2/(5*b*x^ 
(5/2)) - (a*(-2/(3*b*x^(3/2)) - (a*(-2/(b*Sqrt[x]) - (2*Sqrt[a]*ArcTan[(Sq 
rt[a]*Sqrt[x])/Sqrt[b]])/b^(3/2)))/b))/b))/(2*b)))/(4*b)
 

Defintions of rubi rules used

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

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

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 795

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

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {2 \left (30 a^{2} x^{2}-5 a b x +b^{2}\right )}{5 b^{5} x^{\frac {5}{2}}}-\frac {a^{3} \left (\frac {\frac {15 a \,x^{\frac {3}{2}}}{4}+\frac {17 b \sqrt {x}}{4}}{\left (a x +b \right )^{2}}+\frac {63 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{5}}\)	\(76\)
derivativedivides	\(-\frac {2}{5 b^{3} x^{\frac {5}{2}}}-\frac {12 a^{2}}{b^{5} \sqrt {x}}+\frac {2 a}{b^{4} x^{\frac {3}{2}}}-\frac {2 a^{3} \left (\frac {\frac {15 a \,x^{\frac {3}{2}}}{8}+\frac {17 b \sqrt {x}}{8}}{\left (a x +b \right )^{2}}+\frac {63 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{5}}\)	\(78\)
default	\(-\frac {2}{5 b^{3} x^{\frac {5}{2}}}-\frac {12 a^{2}}{b^{5} \sqrt {x}}+\frac {2 a}{b^{4} x^{\frac {3}{2}}}-\frac {2 a^{3} \left (\frac {\frac {15 a \,x^{\frac {3}{2}}}{8}+\frac {17 b \sqrt {x}}{8}}{\left (a x +b \right )^{2}}+\frac {63 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{5}}\)	\(78\)

Input:

int(1/(a+b/x)^3/x^(13/2),x,method=_RETURNVERBOSE)
 

Output:

-2/5*(30*a^2*x^2-5*a*b*x+b^2)/b^5/x^(5/2)-1/b^5*a^3*(2*(15/8*a*x^(3/2)+17/ 
8*b*x^(1/2))/(a*x+b)^2+63/4/(a*b)^(1/2)*arctan(a*x^(1/2)/(a*b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.34 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{13/2}} \, dx=\left [\frac {315 \, {\left (a^{4} x^{5} + 2 \, a^{3} b x^{4} + a^{2} b^{2} x^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {a x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) - 2 \, {\left (315 \, a^{4} x^{4} + 525 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} - 24 \, a b^{3} x + 8 \, b^{4}\right )} \sqrt {x}}{40 \, {\left (a^{2} b^{5} x^{5} + 2 \, a b^{6} x^{4} + b^{7} x^{3}\right )}}, -\frac {315 \, {\left (a^{4} x^{5} + 2 \, a^{3} b x^{4} + a^{2} b^{2} x^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\sqrt {x} \sqrt {\frac {a}{b}}\right ) + {\left (315 \, a^{4} x^{4} + 525 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} - 24 \, a b^{3} x + 8 \, b^{4}\right )} \sqrt {x}}{20 \, {\left (a^{2} b^{5} x^{5} + 2 \, a b^{6} x^{4} + b^{7} x^{3}\right )}}\right ] \] Input:

integrate(1/(a+b/x)^3/x^(13/2),x, algorithm="fricas")
 

Output:

[1/40*(315*(a^4*x^5 + 2*a^3*b*x^4 + a^2*b^2*x^3)*sqrt(-a/b)*log((a*x - 2*b 
*sqrt(x)*sqrt(-a/b) - b)/(a*x + b)) - 2*(315*a^4*x^4 + 525*a^3*b*x^3 + 168 
*a^2*b^2*x^2 - 24*a*b^3*x + 8*b^4)*sqrt(x))/(a^2*b^5*x^5 + 2*a*b^6*x^4 + b 
^7*x^3), -1/20*(315*(a^4*x^5 + 2*a^3*b*x^4 + a^2*b^2*x^3)*sqrt(a/b)*arctan 
(sqrt(x)*sqrt(a/b)) + (315*a^4*x^4 + 525*a^3*b*x^3 + 168*a^2*b^2*x^2 - 24* 
a*b^3*x + 8*b^4)*sqrt(x))/(a^2*b^5*x^5 + 2*a*b^6*x^4 + b^7*x^3)]
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{13/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b/x)**3/x**(13/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{13/2}} \, dx=-\frac {\frac {15 \, a^{4}}{\sqrt {x}} + \frac {17 \, a^{3} b}{x^{\frac {3}{2}}}}{4 \, {\left (a^{2} b^{5} + \frac {2 \, a b^{6}}{x} + \frac {b^{7}}{x^{2}}\right )}} + \frac {63 \, a^{3} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{4 \, \sqrt {a b} b^{5}} - \frac {2 \, {\left (\frac {30 \, a^{2}}{\sqrt {x}} - \frac {5 \, a b}{x^{\frac {3}{2}}} + \frac {b^{2}}{x^{\frac {5}{2}}}\right )}}{5 \, b^{5}} \] Input:

integrate(1/(a+b/x)^3/x^(13/2),x, algorithm="maxima")
 

Output:

-1/4*(15*a^4/sqrt(x) + 17*a^3*b/x^(3/2))/(a^2*b^5 + 2*a*b^6/x + b^7/x^2) + 
 63/4*a^3*arctan(b/(sqrt(a*b)*sqrt(x)))/(sqrt(a*b)*b^5) - 2/5*(30*a^2/sqrt 
(x) - 5*a*b/x^(3/2) + b^2/x^(5/2))/b^5
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{13/2}} \, dx=-\frac {63 \, a^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{5}} - \frac {15 \, a^{4} x^{\frac {3}{2}} + 17 \, a^{3} b \sqrt {x}}{4 \, {\left (a x + b\right )}^{2} b^{5}} - \frac {2 \, {\left (30 \, a^{2} x^{2} - 5 \, a b x + b^{2}\right )}}{5 \, b^{5} x^{\frac {5}{2}}} \] Input:

integrate(1/(a+b/x)^3/x^(13/2),x, algorithm="giac")
 

Output:

-63/4*a^3*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/4*(15*a^4*x^(3/2 
) + 17*a^3*b*sqrt(x))/((a*x + b)^2*b^5) - 2/5*(30*a^2*x^2 - 5*a*b*x + b^2) 
/(b^5*x^(5/2))
 

Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{13/2}} \, dx=-\frac {\frac {2}{5\,b}+\frac {42\,a^2\,x^2}{5\,b^3}+\frac {105\,a^3\,x^3}{4\,b^4}+\frac {63\,a^4\,x^4}{4\,b^5}-\frac {6\,a\,x}{5\,b^2}}{a^2\,x^{9/2}+b^2\,x^{5/2}+2\,a\,b\,x^{7/2}}-\frac {63\,a^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{4\,b^{11/2}} \] Input:

int(1/(x^(13/2)*(a + b/x)^3),x)
 

Output:

- (2/(5*b) + (42*a^2*x^2)/(5*b^3) + (105*a^3*x^3)/(4*b^4) + (63*a^4*x^4)/( 
4*b^5) - (6*a*x)/(5*b^2))/(a^2*x^(9/2) + b^2*x^(5/2) + 2*a*b*x^(7/2)) - (6 
3*a^(5/2)*atan((a^(1/2)*x^(1/2))/b^(1/2)))/(4*b^(11/2))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{13/2}} \, dx=\frac {-315 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} x^{4}-630 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b \,x^{3}-315 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} x^{2}-315 a^{4} b \,x^{4}-525 a^{3} b^{2} x^{3}-168 a^{2} b^{3} x^{2}+24 a \,b^{4} x -8 b^{5}}{20 \sqrt {x}\, b^{6} x^{2} \left (a^{2} x^{2}+2 a b x +b^{2}\right )} \] Input:

int(1/(a+b/x)^3/x^(13/2),x)
 

Output:

( - 315*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*a**4*x 
**4 - 630*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*a**3 
*b*x**3 - 315*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))* 
a**2*b**2*x**2 - 315*a**4*b*x**4 - 525*a**3*b**2*x**3 - 168*a**2*b**3*x**2 
 + 24*a*b**4*x - 8*b**5)/(20*sqrt(x)*b**6*x**2*(a**2*x**2 + 2*a*b*x + b**2 
))