\(\int \frac {A+B x}{x^{5/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [410]

Optimal result

Integrand size = 29, antiderivative size = 217 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {2 A}{3 a^6 x^{3/2}}+\frac {2 (6 A b-a B)}{a^7 \sqrt {x}}+\frac {b (A b-a B) \sqrt {x}}{5 a^3 (a+b x)^5}+\frac {b (29 A b-19 a B) \sqrt {x}}{40 a^4 (a+b x)^4}+\frac {b (443 A b-213 a B) \sqrt {x}}{240 a^5 (a+b x)^3}+\frac {b (827 A b-309 a B) \sqrt {x}}{192 a^6 (a+b x)^2}+\frac {b (1467 A b-437 a B) \sqrt {x}}{128 a^7 (a+b x)}+\frac {231 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{15/2}} \] Output:

-2/3*A/a^6/x^(3/2)+2*(6*A*b-B*a)/a^7/x^(1/2)+1/5*b*(A*b-B*a)*x^(1/2)/a^3/( 
b*x+a)^5+1/40*b*(29*A*b-19*B*a)*x^(1/2)/a^4/(b*x+a)^4+1/240*b*(443*A*b-213 
*B*a)*x^(1/2)/a^5/(b*x+a)^3+1/192*b*(827*A*b-309*B*a)*x^(1/2)/a^6/(b*x+a)^ 
2+1/128*b*(1467*A*b-437*B*a)*x^(1/2)/a^7/(b*x+a)+231/128*b^(1/2)*(13*A*b-3 
*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(15/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-\frac {\sqrt {a} \left (-45045 A b^6 x^6+1280 a^6 (A+3 B x)+1155 a b^5 x^5 (-182 A+9 B x)+462 a^2 b^4 x^4 (-832 A+105 B x)+66 a^3 b^3 x^3 (-5135 A+1344 B x)+55 a^4 b^2 x^2 (-2509 A+1422 B x)+5 a^5 b x (-3328 A+6369 B x)\right )}{x^{3/2} (a+b x)^5}+3465 \sqrt {b} (13 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{1920 a^{15/2}} \] Input:

Integrate[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

(-((Sqrt[a]*(-45045*A*b^6*x^6 + 1280*a^6*(A + 3*B*x) + 1155*a*b^5*x^5*(-18 
2*A + 9*B*x) + 462*a^2*b^4*x^4*(-832*A + 105*B*x) + 66*a^3*b^3*x^3*(-5135* 
A + 1344*B*x) + 55*a^4*b^2*x^2*(-2509*A + 1422*B*x) + 5*a^5*b*x*(-3328*A + 
 6369*B*x)))/(x^(3/2)*(a + b*x)^5)) + 3465*Sqrt[b]*(13*A*b - 3*a*B)*ArcTan 
[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(1920*a^(15/2))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1184, 27, 87, 52, 52, 52, 52, 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.

\(\displaystyle \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {A+B x}{b^6 x^{5/2} (a+b x)^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {A+B x}{x^{5/2} (a+b x)^6}dx\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(13 A b-3 a B) \int \frac {1}{x^{5/2} (a+b x)^5}dx}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \int \frac {1}{x^{5/2} (a+b x)^4}dx}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \int \frac {1}{x^{5/2} (a+b x)^3}dx}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \int \frac {1}{x^{5/2} (a+b x)^2}dx}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^{5/2} (a+b x)}dx}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \int \frac {1}{x^{3/2} (a+b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(13 A b-3 a B) \left (\frac {11 \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (-\frac {b \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{3/2} (a+b x)^2}\right )}{2 a}+\frac {1}{3 a x^{3/2} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a x^{3/2} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b x^{3/2} (a+b x)^5}\)

Input:

Int[(A + B*x)/(x^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]
 

Output:

(A*b - a*B)/(5*a*b*x^(3/2)*(a + b*x)^5) + ((13*A*b - 3*a*B)*(1/(4*a*x^(3/2 
)*(a + b*x)^4) + (11*(1/(3*a*x^(3/2)*(a + b*x)^3) + (3*(1/(2*a*x^(3/2)*(a 
+ b*x)^2) + (7*(1/(a*x^(3/2)*(a + b*x)) + (5*(-2/(3*a*x^(3/2)) - (b*(-2/(a 
*Sqrt[x]) - (2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)))/a))/(2 
*a)))/(4*a)))/(2*a)))/(8*a)))/(10*a*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

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 1184

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(f + g*x 
)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E 
qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {2 \left (-18 A b x +3 B a x +A a \right )}{3 a^{7} x^{\frac {3}{2}}}+\frac {b \left (\frac {2 \left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {a \,b^{3} \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{192}+2 \left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+2 \left (\frac {12131}{384} a^{3} A \,b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+2 \left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} B \,a^{5}\right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}}\right )}{a^{7}}\)	\(165\)
derivativedivides	\(\frac {2 b \left (\frac {\left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {a \,b^{3} \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {12131}{384} a^{3} A \,b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} B \,a^{5}\right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{a^{7}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}}-\frac {2 \left (-6 A b +B a \right )}{a^{7} \sqrt {x}}\)	\(168\)
default	\(\frac {2 b \left (\frac {\left (\frac {1467}{256} A \,b^{5}-\frac {437}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {a \,b^{3} \left (9629 A b -2931 B a \right ) x^{\frac {7}{2}}}{384}+\left (\frac {1253}{30} A \,a^{2} b^{3}-\frac {131}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {12131}{384} a^{3} A \,b^{2}-\frac {1327}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {2373}{256} A \,a^{4} b -\frac {843}{256} B \,a^{5}\right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {231 \left (13 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{a^{7}}-\frac {2 A}{3 a^{6} x^{\frac {3}{2}}}-\frac {2 \left (-6 A b +B a \right )}{a^{7} \sqrt {x}}\)	\(168\)

Input:

int((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)
 

Output:

-2/3*(-18*A*b*x+3*B*a*x+A*a)/a^7/x^(3/2)+1/a^7*b*(2*((1467/256*A*b^5-437/2 
56*B*a*b^4)*x^(9/2)+1/384*a*b^3*(9629*A*b-2931*B*a)*x^(7/2)+(1253/30*A*a^2 
*b^3-131/10*B*a^3*b^2)*x^(5/2)+(12131/384*a^3*A*b^2-1327/128*B*a^4*b)*x^(3 
/2)+(2373/256*A*a^4*b-843/256*B*a^5)*x^(1/2))/(b*x+a)^5+231/128*(13*A*b-3* 
B*a)/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.36 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
 

Output:

[-1/3840*(3465*((3*B*a*b^5 - 13*A*b^6)*x^7 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)* 
x^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^5 + 10*(3*B*a^4*b^2 - 13*A*a^3*b^3 
)*x^4 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^3 + (3*B*a^6 - 13*A*a^5*b)*x^2)*sqr 
t(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(1280*A*a^6 
+ 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x^5 + 
 29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^3*b^ 
3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*(3*B*a^6 - 13*A*a^5*b 
)*x)*sqrt(x))/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2* 
x^4 + 5*a^11*b*x^3 + a^12*x^2), -1/1920*(3465*((3*B*a*b^5 - 13*A*b^6)*x^7 
+ 5*(3*B*a^2*b^4 - 13*A*a*b^5)*x^6 + 10*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^5 + 
 10*(3*B*a^4*b^2 - 13*A*a^3*b^3)*x^4 + 5*(3*B*a^5*b - 13*A*a^4*b^2)*x^3 + 
(3*B*a^6 - 13*A*a^5*b)*x^2)*sqrt(b/a)*arctan(sqrt(x)*sqrt(b/a)) + (1280*A* 
a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 - 13*A*a*b^5)*x 
^5 + 29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a^4*b^2 - 13*A*a^ 
3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*(3*B*a^6 - 13*A*a 
^5*b)*x)*sqrt(x))/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10* 
b^2*x^4 + 5*a^11*b*x^3 + a^12*x^2)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/x**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.07 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1280 \, A a^{6} + 3465 \, {\left (3 \, B a b^{5} - 13 \, A b^{6}\right )} x^{6} + 16170 \, {\left (3 \, B a^{2} b^{4} - 13 \, A a b^{5}\right )} x^{5} + 29568 \, {\left (3 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{4} + 26070 \, {\left (3 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{3} + 10615 \, {\left (3 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x^{2} + 1280 \, {\left (3 \, B a^{6} - 13 \, A a^{5} b\right )} x}{1920 \, {\left (a^{7} b^{5} x^{\frac {13}{2}} + 5 \, a^{8} b^{4} x^{\frac {11}{2}} + 10 \, a^{9} b^{3} x^{\frac {9}{2}} + 10 \, a^{10} b^{2} x^{\frac {7}{2}} + 5 \, a^{11} b x^{\frac {5}{2}} + a^{12} x^{\frac {3}{2}}\right )}} - \frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} \] Input:

integrate((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
 

Output:

-1/1920*(1280*A*a^6 + 3465*(3*B*a*b^5 - 13*A*b^6)*x^6 + 16170*(3*B*a^2*b^4 
 - 13*A*a*b^5)*x^5 + 29568*(3*B*a^3*b^3 - 13*A*a^2*b^4)*x^4 + 26070*(3*B*a 
^4*b^2 - 13*A*a^3*b^3)*x^3 + 10615*(3*B*a^5*b - 13*A*a^4*b^2)*x^2 + 1280*( 
3*B*a^6 - 13*A*a^5*b)*x)/(a^7*b^5*x^(13/2) + 5*a^8*b^4*x^(11/2) + 10*a^9*b 
^3*x^(9/2) + 10*a^10*b^2*x^(7/2) + 5*a^11*b*x^(5/2) + a^12*x^(3/2)) - 231/ 
128*(3*B*a*b - 13*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {231 \, {\left (3 \, B a b - 13 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{7}} - \frac {2 \, {\left (3 \, B a x - 18 \, A b x + A a\right )}}{3 \, a^{7} x^{\frac {3}{2}}} - \frac {6555 \, B a b^{5} x^{\frac {9}{2}} - 22005 \, A b^{6} x^{\frac {9}{2}} + 29310 \, B a^{2} b^{4} x^{\frac {7}{2}} - 96290 \, A a b^{5} x^{\frac {7}{2}} + 50304 \, B a^{3} b^{3} x^{\frac {5}{2}} - 160384 \, A a^{2} b^{4} x^{\frac {5}{2}} + 39810 \, B a^{4} b^{2} x^{\frac {3}{2}} - 121310 \, A a^{3} b^{3} x^{\frac {3}{2}} + 12645 \, B a^{5} b \sqrt {x} - 35595 \, A a^{4} b^{2} \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{7}} \] Input:

integrate((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
 

Output:

-231/128*(3*B*a*b - 13*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7) 
- 2/3*(3*B*a*x - 18*A*b*x + A*a)/(a^7*x^(3/2)) - 1/1920*(6555*B*a*b^5*x^(9 
/2) - 22005*A*b^6*x^(9/2) + 29310*B*a^2*b^4*x^(7/2) - 96290*A*a*b^5*x^(7/2 
) + 50304*B*a^3*b^3*x^(5/2) - 160384*A*a^2*b^4*x^(5/2) + 39810*B*a^4*b^2*x 
^(3/2) - 121310*A*a^3*b^3*x^(3/2) + 12645*B*a^5*b*sqrt(x) - 35595*A*a^4*b^ 
2*sqrt(x))/((b*x + a)^5*a^7)
 

Mupad [B] (verification not implemented)

Time = 11.04 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {2\,x\,\left (13\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {869\,b^2\,x^3\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^4}+\frac {77\,b^3\,x^4\,\left (13\,A\,b-3\,B\,a\right )}{5\,a^5}+\frac {539\,b^4\,x^5\,\left (13\,A\,b-3\,B\,a\right )}{64\,a^6}+\frac {231\,b^5\,x^6\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^7}+\frac {2123\,b\,x^2\,\left (13\,A\,b-3\,B\,a\right )}{384\,a^3}}{a^5\,x^{3/2}+b^5\,x^{13/2}+5\,a^4\,b\,x^{5/2}+5\,a\,b^4\,x^{11/2}+10\,a^3\,b^2\,x^{7/2}+10\,a^2\,b^3\,x^{9/2}}+\frac {231\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (13\,A\,b-3\,B\,a\right )}{128\,a^{15/2}} \] Input:

int((A + B*x)/(x^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)
                                                                                    
                                                                                    
 

Output:

((2*x*(13*A*b - 3*B*a))/(3*a^2) - (2*A)/(3*a) + (869*b^2*x^3*(13*A*b - 3*B 
*a))/(64*a^4) + (77*b^3*x^4*(13*A*b - 3*B*a))/(5*a^5) + (539*b^4*x^5*(13*A 
*b - 3*B*a))/(64*a^6) + (231*b^5*x^6*(13*A*b - 3*B*a))/(128*a^7) + (2123*b 
*x^2*(13*A*b - 3*B*a))/(384*a^3))/(a^5*x^(3/2) + b^5*x^(13/2) + 5*a^4*b*x^ 
(5/2) + 5*a*b^4*x^(11/2) + 10*a^3*b^2*x^(7/2) + 10*a^2*b^3*x^(9/2)) + (231 
*b^(1/2)*atan((b^(1/2)*x^(1/2))/a^(1/2))*(13*A*b - 3*B*a))/(128*a^(15/2))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x}{x^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3465 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{4} b x +13860 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{2}+20790 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{3}+13860 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{4}+3465 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{5}-128 a^{6}+1408 a^{5} b x +9207 a^{4} b^{2} x^{2}+16863 a^{3} b^{3} x^{3}+12705 a^{2} b^{4} x^{4}+3465 a \,b^{5} x^{5}}{192 \sqrt {x}\, a^{7} x \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )} \] Input:

int((B*x+A)/x^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)
 

Output:

(3465*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**4*b*x 
 + 13860*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**3* 
b**2*x**2 + 20790*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a 
)))*a**2*b**3*x**3 + 13860*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt( 
b)*sqrt(a)))*a*b**4*x**4 + 3465*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/( 
sqrt(b)*sqrt(a)))*b**5*x**5 - 128*a**6 + 1408*a**5*b*x + 9207*a**4*b**2*x* 
*2 + 16863*a**3*b**3*x**3 + 12705*a**2*b**4*x**4 + 3465*a*b**5*x**5)/(192* 
sqrt(x)*a**7*x*(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b** 
4*x**4))