\(\int \frac {x^{5/2}}{(a+\frac {b}{x})^2} \, dx\) [125]

Optimal result

Integrand size = 15, antiderivative size = 101 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=-\frac {8 b^3 \sqrt {x}}{a^5}+\frac {2 b^2 x^{3/2}}{a^4}-\frac {4 b x^{5/2}}{5 a^3}+\frac {2 x^{7/2}}{7 a^2}-\frac {b^4 \sqrt {x}}{a^5 (b+a x)}+\frac {9 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{11/2}} \] Output:

-8*b^3*x^(1/2)/a^5+2*b^2*x^(3/2)/a^4-4/5*b*x^(5/2)/a^3+2/7*x^(7/2)/a^2-b^4 
*x^(1/2)/a^5/(a*x+b)+9*b^(7/2)*arctan(a^(1/2)*x^(1/2)/b^(1/2))/a^(11/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {\sqrt {x} \left (-315 b^4-210 a b^3 x+42 a^2 b^2 x^2-18 a^3 b x^3+10 a^4 x^4\right )}{35 a^5 (b+a x)}+\frac {9 b^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{a^{11/2}} \] Input:

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

Output:

(Sqrt[x]*(-315*b^4 - 210*a*b^3*x + 42*a^2*b^2*x^2 - 18*a^3*b*x^3 + 10*a^4* 
x^4))/(35*a^5*(b + a*x)) + (9*b^(7/2)*ArcTan[(Sqrt[a]*Sqrt[x])/Sqrt[b]])/a 
^(11/2)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {795, 51, 60, 60, 60, 60, 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.

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

\(\Big \downarrow \) 795

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 51

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

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[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 = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.77

method	result	size
risch	\(\frac {2 \left (5 a^{3} x^{3}-14 a^{2} b \,x^{2}+35 a \,b^{2} x -140 b^{3}\right ) \sqrt {x}}{35 a^{5}}+\frac {b^{4} \left (-\frac {\sqrt {x}}{a x +b}+\frac {9 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{5}}\)	\(78\)
derivativedivides	\(\frac {\frac {2 a^{3} x^{\frac {7}{2}}}{7}-\frac {4 a^{2} b \,x^{\frac {5}{2}}}{5}+2 a \,b^{2} x^{\frac {3}{2}}-8 b^{3} \sqrt {x}}{a^{5}}+\frac {2 b^{4} \left (-\frac {\sqrt {x}}{2 \left (a x +b \right )}+\frac {9 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{5}}\)	\(80\)
default	\(\frac {\frac {2 a^{3} x^{\frac {7}{2}}}{7}-\frac {4 a^{2} b \,x^{\frac {5}{2}}}{5}+2 a \,b^{2} x^{\frac {3}{2}}-8 b^{3} \sqrt {x}}{a^{5}}+\frac {2 b^{4} \left (-\frac {\sqrt {x}}{2 \left (a x +b \right )}+\frac {9 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{5}}\)	\(80\)

Input:

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

Output:

2/35*(5*a^3*x^3-14*a^2*b*x^2+35*a*b^2*x-140*b^3)*x^(1/2)/a^5+1/a^5*b^4*(-x 
^(1/2)/(a*x+b)+9/(a*b)^(1/2)*arctan(a*x^(1/2)/(a*b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.07 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\left [\frac {315 \, {\left (a b^{3} x + b^{4}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt {x}}{70 \, {\left (a^{6} x + a^{5} b\right )}}, \frac {315 \, {\left (a b^{3} x + b^{4}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) + {\left (10 \, a^{4} x^{4} - 18 \, a^{3} b x^{3} + 42 \, a^{2} b^{2} x^{2} - 210 \, a b^{3} x - 315 \, b^{4}\right )} \sqrt {x}}{35 \, {\left (a^{6} x + a^{5} b\right )}}\right ] \] Input:

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

Output:

[1/70*(315*(a*b^3*x + b^4)*sqrt(-b/a)*log((a*x + 2*a*sqrt(x)*sqrt(-b/a) - 
b)/(a*x + b)) + 2*(10*a^4*x^4 - 18*a^3*b*x^3 + 42*a^2*b^2*x^2 - 210*a*b^3* 
x - 315*b^4)*sqrt(x))/(a^6*x + a^5*b), 1/35*(315*(a*b^3*x + b^4)*sqrt(b/a) 
*arctan(a*sqrt(x)*sqrt(b/a)/b) + (10*a^4*x^4 - 18*a^3*b*x^3 + 42*a^2*b^2*x 
^2 - 210*a*b^3*x - 315*b^4)*sqrt(x))/(a^6*x + a^5*b)]
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (97) = 194\).

Time = 27.78 (sec) , antiderivative size = 495, normalized size of antiderivative = 4.90 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\begin {cases} \tilde {\infty } x^{\frac {11}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {11}{2}}}{11 b^{2}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {7}{2}}}{7 a^{2}} & \text {for}\: b = 0 \\\frac {20 a^{5} x^{\frac {9}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {36 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {84 a^{3} b^{2} x^{\frac {5}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {420 a^{2} b^{3} x^{\frac {3}{2}} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {630 a b^{4} \sqrt {x} \sqrt {- \frac {b}{a}}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {315 a b^{4} x \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {315 a b^{4} x \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} + \frac {315 b^{5} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} - \frac {315 b^{5} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{a}} \right )}}{70 a^{7} x \sqrt {- \frac {b}{a}} + 70 a^{6} b \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((zoo*x**(11/2), Eq(a, 0) & Eq(b, 0)), (2*x**(11/2)/(11*b**2), Eq 
(a, 0)), (2*x**(7/2)/(7*a**2), Eq(b, 0)), (20*a**5*x**(9/2)*sqrt(-b/a)/(70 
*a**7*x*sqrt(-b/a) + 70*a**6*b*sqrt(-b/a)) - 36*a**4*b*x**(7/2)*sqrt(-b/a) 
/(70*a**7*x*sqrt(-b/a) + 70*a**6*b*sqrt(-b/a)) + 84*a**3*b**2*x**(5/2)*sqr 
t(-b/a)/(70*a**7*x*sqrt(-b/a) + 70*a**6*b*sqrt(-b/a)) - 420*a**2*b**3*x**( 
3/2)*sqrt(-b/a)/(70*a**7*x*sqrt(-b/a) + 70*a**6*b*sqrt(-b/a)) - 630*a*b**4 
*sqrt(x)*sqrt(-b/a)/(70*a**7*x*sqrt(-b/a) + 70*a**6*b*sqrt(-b/a)) + 315*a* 
b**4*x*log(sqrt(x) - sqrt(-b/a))/(70*a**7*x*sqrt(-b/a) + 70*a**6*b*sqrt(-b 
/a)) - 315*a*b**4*x*log(sqrt(x) + sqrt(-b/a))/(70*a**7*x*sqrt(-b/a) + 70*a 
**6*b*sqrt(-b/a)) + 315*b**5*log(sqrt(x) - sqrt(-b/a))/(70*a**7*x*sqrt(-b/ 
a) + 70*a**6*b*sqrt(-b/a)) - 315*b**5*log(sqrt(x) + sqrt(-b/a))/(70*a**7*x 
*sqrt(-b/a) + 70*a**6*b*sqrt(-b/a)), True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {10 \, a^{4} - \frac {18 \, a^{3} b}{x} + \frac {42 \, a^{2} b^{2}}{x^{2}} - \frac {210 \, a b^{3}}{x^{3}} - \frac {315 \, b^{4}}{x^{4}}}{35 \, {\left (\frac {a^{6}}{x^{\frac {7}{2}}} + \frac {a^{5} b}{x^{\frac {9}{2}}}\right )}} - \frac {9 \, b^{4} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} a^{5}} \] Input:

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

Output:

1/35*(10*a^4 - 18*a^3*b/x + 42*a^2*b^2/x^2 - 210*a*b^3/x^3 - 315*b^4/x^4)/ 
(a^6/x^(7/2) + a^5*b/x^(9/2)) - 9*b^4*arctan(b/(sqrt(a*b)*sqrt(x)))/(sqrt( 
a*b)*a^5)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {9 \, b^{4} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{5}} - \frac {b^{4} \sqrt {x}}{{\left (a x + b\right )} a^{5}} + \frac {2 \, {\left (5 \, a^{12} x^{\frac {7}{2}} - 14 \, a^{11} b x^{\frac {5}{2}} + 35 \, a^{10} b^{2} x^{\frac {3}{2}} - 140 \, a^{9} b^{3} \sqrt {x}\right )}}{35 \, a^{14}} \] Input:

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

Output:

9*b^4*arctan(a*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) - b^4*sqrt(x)/((a*x + b) 
*a^5) + 2/35*(5*a^12*x^(7/2) - 14*a^11*b*x^(5/2) + 35*a^10*b^2*x^(3/2) - 1 
40*a^9*b^3*sqrt(x))/a^14
 

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {2\,x^{7/2}}{7\,a^2}-\frac {4\,b\,x^{5/2}}{5\,a^3}+\frac {2\,b^2\,x^{3/2}}{a^4}-\frac {8\,b^3\,\sqrt {x}}{a^5}-\frac {b^4\,\sqrt {x}}{x\,a^6+b\,a^5}+\frac {9\,b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{a^{11/2}} \] Input:

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

Output:

(2*x^(7/2))/(7*a^2) - (4*b*x^(5/2))/(5*a^3) + (2*b^2*x^(3/2))/a^4 - (8*b^3 
*x^(1/2))/a^5 - (b^4*x^(1/2))/(a^5*b + a^6*x) + (9*b^(7/2)*atan((a^(1/2)*x 
^(1/2))/b^(1/2)))/a^(11/2)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.11 \[ \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} x +315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, a}{\sqrt {b}\, \sqrt {a}}\right ) b^{4}+10 \sqrt {x}\, a^{5} x^{4}-18 \sqrt {x}\, a^{4} b \,x^{3}+42 \sqrt {x}\, a^{3} b^{2} x^{2}-210 \sqrt {x}\, a^{2} b^{3} x -315 \sqrt {x}\, a \,b^{4}}{35 a^{6} \left (a x +b \right )} \] Input:

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

Output:

(315*sqrt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*a*b**3*x + 315*sq 
rt(b)*sqrt(a)*atan((sqrt(x)*a)/(sqrt(b)*sqrt(a)))*b**4 + 10*sqrt(x)*a**5*x 
**4 - 18*sqrt(x)*a**4*b*x**3 + 42*sqrt(x)*a**3*b**2*x**2 - 210*sqrt(x)*a** 
2*b**3*x - 315*sqrt(x)*a*b**4)/(35*a**6*(a*x + b))