\(\int \frac {x^{5/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [405]

Optimal result

Integrand size = 29, antiderivative size = 185 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(A b-a B) x^{5/2}}{5 b^2 (a+b x)^5}-\frac {(A b-3 a B) x^{3/2}}{8 b^3 (a+b x)^4}-\frac {(3 A b-25 a B) \sqrt {x}}{48 b^4 (a+b x)^3}+\frac {(3 A b-121 a B) \sqrt {x}}{192 a b^4 (a+b x)^2}+\frac {(3 A b+7 a B) \sqrt {x}}{128 a^2 b^4 (a+b x)}+\frac {(3 A b+7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{5/2} b^{9/2}} \] Output:

-1/5*(A*b-B*a)*x^(5/2)/b^2/(b*x+a)^5-1/8*(A*b-3*B*a)*x^(3/2)/b^3/(b*x+a)^4 
-1/48*(3*A*b-25*B*a)*x^(1/2)/b^4/(b*x+a)^3+1/192*(3*A*b-121*B*a)*x^(1/2)/a 
/b^4/(b*x+a)^2+1/128*(3*A*b+7*B*a)*x^(1/2)/a^2/b^4/(b*x+a)+1/128*(3*A*b+7* 
B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(5/2)/b^(9/2)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.78 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-105 a^5 B+45 A b^5 x^4+105 a b^4 x^3 (2 A+B x)-14 a^3 b^2 x (15 A+64 B x)-5 a^4 b (9 A+98 B x)-2 a^2 b^3 x^2 (192 A+395 B x)\right )}{1920 a^2 b^4 (a+b x)^5}+\frac {(3 A b+7 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{5/2} b^{9/2}} \] Input:

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

Output:

(Sqrt[x]*(-105*a^5*B + 45*A*b^5*x^4 + 105*a*b^4*x^3*(2*A + B*x) - 14*a^3*b 
^2*x*(15*A + 64*B*x) - 5*a^4*b*(9*A + 98*B*x) - 2*a^2*b^3*x^2*(192*A + 395 
*B*x)))/(1920*a^2*b^4*(a + b*x)^5) + ((3*A*b + 7*a*B)*ArcTan[(Sqrt[b]*Sqrt 
[x])/Sqrt[a]])/(128*a^(5/2)*b^(9/2))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1184, 27, 87, 51, 51, 51, 52, 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.

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

\(\Big \downarrow \) 1184

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {(7 a B+3 A b) \int \frac {x^{5/2}}{(a+b x)^5}dx}{10 a b}+\frac {x^{7/2} (A b-a B)}{5 a b (a+b x)^5}\)

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

((A*b - a*B)*x^(7/2))/(5*a*b*(a + b*x)^5) + ((3*A*b + 7*a*B)*(-1/4*x^(5/2) 
/(b*(a + b*x)^4) + (5*(-1/3*x^(3/2)/(b*(a + b*x)^3) + (-1/2*Sqrt[x]/(b*(a 
+ b*x)^2) + (Sqrt[x]/(a*(a + b*x)) + ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(a^ 
(3/2)*Sqrt[b]))/(4*b))/(2*b)))/(8*b)))/(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 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 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 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 = 2.03 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {\frac {\left (3 A b +7 B a \right ) x^{\frac {9}{2}}}{128 a^{2}}+\frac {\left (21 A b -79 B a \right ) x^{\frac {7}{2}}}{192 a b}-\frac {\left (3 A b +7 B a \right ) x^{\frac {5}{2}}}{15 b^{2}}-\frac {7 a \left (3 A b +7 B a \right ) x^{\frac {3}{2}}}{192 b^{3}}-\frac {\left (3 A b +7 B a \right ) a^{2} \sqrt {x}}{128 b^{4}}}{\left (b x +a \right )^{5}}+\frac {\left (3 A b +7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 a^{2} b^{4} \sqrt {a b}}\)	\(137\)
default	\(\frac {\frac {\left (3 A b +7 B a \right ) x^{\frac {9}{2}}}{128 a^{2}}+\frac {\left (21 A b -79 B a \right ) x^{\frac {7}{2}}}{192 a b}-\frac {\left (3 A b +7 B a \right ) x^{\frac {5}{2}}}{15 b^{2}}-\frac {7 a \left (3 A b +7 B a \right ) x^{\frac {3}{2}}}{192 b^{3}}-\frac {\left (3 A b +7 B a \right ) a^{2} \sqrt {x}}{128 b^{4}}}{\left (b x +a \right )^{5}}+\frac {\left (3 A b +7 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 a^{2} b^{4} \sqrt {a b}}\)	\(137\)

Input:

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

Output:

2*(1/256*(3*A*b+7*B*a)/a^2*x^(9/2)+1/384*(21*A*b-79*B*a)/a/b*x^(7/2)-1/30/ 
b^2*(3*A*b+7*B*a)*x^(5/2)-7/384*a/b^3*(3*A*b+7*B*a)*x^(3/2)-1/256*(3*A*b+7 
*B*a)*a^2/b^4*x^(1/2))/(b*x+a)^5+1/128*(3*A*b+7*B*a)/a^2/b^4/(a*b)^(1/2)*a 
rctan(b*x^(1/2)/(a*b)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (154) = 308\).

Time = 0.11 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.55 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {15 \, {\left (7 \, B a^{6} + 3 \, A a^{5} b + {\left (7 \, B a b^{5} + 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (7 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (7 \, B a^{5} b + 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a - 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) + 2 \, {\left (105 \, B a^{6} b + 45 \, A a^{5} b^{2} - 15 \, {\left (7 \, B a^{2} b^{5} + 3 \, A a b^{6}\right )} x^{4} + 10 \, {\left (79 \, B a^{3} b^{4} - 21 \, A a^{2} b^{5}\right )} x^{3} + 128 \, {\left (7 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4}\right )} x^{2} + 70 \, {\left (7 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{3840 \, {\left (a^{3} b^{10} x^{5} + 5 \, a^{4} b^{9} x^{4} + 10 \, a^{5} b^{8} x^{3} + 10 \, a^{6} b^{7} x^{2} + 5 \, a^{7} b^{6} x + a^{8} b^{5}\right )}}, -\frac {15 \, {\left (7 \, B a^{6} + 3 \, A a^{5} b + {\left (7 \, B a b^{5} + 3 \, A b^{6}\right )} x^{5} + 5 \, {\left (7 \, B a^{2} b^{4} + 3 \, A a b^{5}\right )} x^{4} + 10 \, {\left (7 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (7 \, B a^{4} b^{2} + 3 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (7 \, B a^{5} b + 3 \, A a^{4} b^{2}\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (105 \, B a^{6} b + 45 \, A a^{5} b^{2} - 15 \, {\left (7 \, B a^{2} b^{5} + 3 \, A a b^{6}\right )} x^{4} + 10 \, {\left (79 \, B a^{3} b^{4} - 21 \, A a^{2} b^{5}\right )} x^{3} + 128 \, {\left (7 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4}\right )} x^{2} + 70 \, {\left (7 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3}\right )} x\right )} \sqrt {x}}{1920 \, {\left (a^{3} b^{10} x^{5} + 5 \, a^{4} b^{9} x^{4} + 10 \, a^{5} b^{8} x^{3} + 10 \, a^{6} b^{7} x^{2} + 5 \, a^{7} b^{6} x + a^{8} b^{5}\right )}}\right ] \] Input:

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

Output:

[-1/3840*(15*(7*B*a^6 + 3*A*a^5*b + (7*B*a*b^5 + 3*A*b^6)*x^5 + 5*(7*B*a^2 
*b^4 + 3*A*a*b^5)*x^4 + 10*(7*B*a^3*b^3 + 3*A*a^2*b^4)*x^3 + 10*(7*B*a^4*b 
^2 + 3*A*a^3*b^3)*x^2 + 5*(7*B*a^5*b + 3*A*a^4*b^2)*x)*sqrt(-a*b)*log((b*x 
 - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(105*B*a^6*b + 45*A*a^5*b^2 - 
15*(7*B*a^2*b^5 + 3*A*a*b^6)*x^4 + 10*(79*B*a^3*b^4 - 21*A*a^2*b^5)*x^3 + 
128*(7*B*a^4*b^3 + 3*A*a^3*b^4)*x^2 + 70*(7*B*a^5*b^2 + 3*A*a^4*b^3)*x)*sq 
rt(x))/(a^3*b^10*x^5 + 5*a^4*b^9*x^4 + 10*a^5*b^8*x^3 + 10*a^6*b^7*x^2 + 5 
*a^7*b^6*x + a^8*b^5), -1/1920*(15*(7*B*a^6 + 3*A*a^5*b + (7*B*a*b^5 + 3*A 
*b^6)*x^5 + 5*(7*B*a^2*b^4 + 3*A*a*b^5)*x^4 + 10*(7*B*a^3*b^3 + 3*A*a^2*b^ 
4)*x^3 + 10*(7*B*a^4*b^2 + 3*A*a^3*b^3)*x^2 + 5*(7*B*a^5*b + 3*A*a^4*b^2)* 
x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (105*B*a^6*b + 45*A*a^5*b^2 - 
 15*(7*B*a^2*b^5 + 3*A*a*b^6)*x^4 + 10*(79*B*a^3*b^4 - 21*A*a^2*b^5)*x^3 + 
 128*(7*B*a^4*b^3 + 3*A*a^3*b^4)*x^2 + 70*(7*B*a^5*b^2 + 3*A*a^4*b^3)*x)*s 
qrt(x))/(a^3*b^10*x^5 + 5*a^4*b^9*x^4 + 10*a^5*b^8*x^3 + 10*a^6*b^7*x^2 + 
5*a^7*b^6*x + a^8*b^5)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.11 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {15 \, {\left (7 \, B a b^{4} + 3 \, A b^{5}\right )} x^{\frac {9}{2}} - 10 \, {\left (79 \, B a^{2} b^{3} - 21 \, A a b^{4}\right )} x^{\frac {7}{2}} - 128 \, {\left (7 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3}\right )} x^{\frac {5}{2}} - 70 \, {\left (7 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{\frac {3}{2}} - 15 \, {\left (7 \, B a^{5} + 3 \, A a^{4} b\right )} \sqrt {x}}{1920 \, {\left (a^{2} b^{9} x^{5} + 5 \, a^{3} b^{8} x^{4} + 10 \, a^{4} b^{7} x^{3} + 10 \, a^{5} b^{6} x^{2} + 5 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} + \frac {{\left (7 \, B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{2} b^{4}} \] Input:

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

Output:

1/1920*(15*(7*B*a*b^4 + 3*A*b^5)*x^(9/2) - 10*(79*B*a^2*b^3 - 21*A*a*b^4)* 
x^(7/2) - 128*(7*B*a^3*b^2 + 3*A*a^2*b^3)*x^(5/2) - 70*(7*B*a^4*b + 3*A*a^ 
3*b^2)*x^(3/2) - 15*(7*B*a^5 + 3*A*a^4*b)*sqrt(x))/(a^2*b^9*x^5 + 5*a^3*b^ 
8*x^4 + 10*a^4*b^7*x^3 + 10*a^5*b^6*x^2 + 5*a^6*b^5*x + a^7*b^4) + 1/128*( 
7*B*a + 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b^4)
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.84 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {{\left (7 \, B a + 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{2} b^{4}} + \frac {105 \, B a b^{4} x^{\frac {9}{2}} + 45 \, A b^{5} x^{\frac {9}{2}} - 790 \, B a^{2} b^{3} x^{\frac {7}{2}} + 210 \, A a b^{4} x^{\frac {7}{2}} - 896 \, B a^{3} b^{2} x^{\frac {5}{2}} - 384 \, A a^{2} b^{3} x^{\frac {5}{2}} - 490 \, B a^{4} b x^{\frac {3}{2}} - 210 \, A a^{3} b^{2} x^{\frac {3}{2}} - 105 \, B a^{5} \sqrt {x} - 45 \, A a^{4} b \sqrt {x}}{1920 \, {\left (b x + a\right )}^{5} a^{2} b^{4}} \] Input:

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

Output:

1/128*(7*B*a + 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2*b^4) + 1/ 
1920*(105*B*a*b^4*x^(9/2) + 45*A*b^5*x^(9/2) - 790*B*a^2*b^3*x^(7/2) + 210 
*A*a*b^4*x^(7/2) - 896*B*a^3*b^2*x^(5/2) - 384*A*a^2*b^3*x^(5/2) - 490*B*a 
^4*b*x^(3/2) - 210*A*a^3*b^2*x^(3/2) - 105*B*a^5*sqrt(x) - 45*A*a^4*b*sqrt 
(x))/((b*x + a)^5*a^2*b^4)
 

Mupad [B] (verification not implemented)

Time = 10.75 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.95 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b+7\,B\,a\right )}{128\,a^{5/2}\,b^{9/2}}-\frac {\frac {x^{5/2}\,\left (3\,A\,b+7\,B\,a\right )}{15\,b^2}-\frac {x^{9/2}\,\left (3\,A\,b+7\,B\,a\right )}{128\,a^2}+\frac {a^2\,\sqrt {x}\,\left (3\,A\,b+7\,B\,a\right )}{128\,b^4}-\frac {x^{7/2}\,\left (21\,A\,b-79\,B\,a\right )}{192\,a\,b}+\frac {7\,a\,x^{3/2}\,\left (3\,A\,b+7\,B\,a\right )}{192\,b^3}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \] Input:

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

Output:

(atan((b^(1/2)*x^(1/2))/a^(1/2))*(3*A*b + 7*B*a))/(128*a^(5/2)*b^(9/2)) - 
((x^(5/2)*(3*A*b + 7*B*a))/(15*b^2) - (x^(9/2)*(3*A*b + 7*B*a))/(128*a^2) 
+ (a^2*x^(1/2)*(3*A*b + 7*B*a))/(128*b^4) - (x^(7/2)*(21*A*b - 79*B*a))/(1 
92*a*b) + (7*a*x^(3/2)*(3*A*b + 7*B*a))/(192*b^3))/(a^5 + b^5*x^5 + 5*a*b^ 
4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.17 \[ \int \frac {x^{5/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{4}+60 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b x +90 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} x^{2}+60 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} x^{3}+15 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{4} x^{4}-15 \sqrt {x}\, a^{4} b -55 \sqrt {x}\, a^{3} b^{2} x -73 \sqrt {x}\, a^{2} b^{3} x^{2}+15 \sqrt {x}\, a \,b^{4} x^{3}}{192 a^{2} b^{4} \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )} \] Input:

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

Output:

(15*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**4 + 60*sqrt(b)* 
sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**3*b*x + 90*sqrt(b)*sqrt(a)* 
atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**2*b**2*x**2 + 60*sqrt(b)*sqrt(a)*at 
an((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a*b**3*x**3 + 15*sqrt(b)*sqrt(a)*atan((s 
qrt(x)*b)/(sqrt(b)*sqrt(a)))*b**4*x**4 - 15*sqrt(x)*a**4*b - 55*sqrt(x)*a* 
*3*b**2*x - 73*sqrt(x)*a**2*b**3*x**2 + 15*sqrt(x)*a*b**4*x**3)/(192*a**2* 
b**4*(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4))