\(\int \frac {x^{9/2} (A+B x)}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [456]

Optimal result

Integrand size = 31, antiderivative size = 334 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {(315 A b-1027 a B) x^{3/2}}{192 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) x^{9/2}}{4 b^2 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(9 A b-17 a B) x^{7/2}}{24 b^3 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(63 A b-167 a B) x^{5/2}}{96 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 (3 A b-11 a B) \sqrt {x} (a+b x)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 B x^{3/2} (a+b x)}{3 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 \sqrt {a} (3 A b-11 a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:

-1/192*(315*A*b-1027*B*a)*x^(3/2)/b^5/((b*x+a)^2)^(1/2)-1/4*(A*b-B*a)*x^(9 
/2)/b^2/(b*x+a)^3/((b*x+a)^2)^(1/2)-1/24*(9*A*b-17*B*a)*x^(7/2)/b^3/(b*x+a 
)^2/((b*x+a)^2)^(1/2)-1/96*(63*A*b-167*B*a)*x^(5/2)/b^4/(b*x+a)/((b*x+a)^2 
)^(1/2)+105/64*(3*A*b-11*B*a)*x^(1/2)*(b*x+a)/b^6/((b*x+a)^2)^(1/2)+2/3*B* 
x^(3/2)*(b*x+a)/b^5/((b*x+a)^2)^(1/2)-105/64*a^(1/2)*(3*A*b-11*B*a)*(b*x+a 
)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/b^(13/2)/((b*x+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.50 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {b} \sqrt {x} \left (-3465 a^5 B+a b^4 x^3 (2511 A-1408 B x)+9 a^2 b^3 x^2 (511 A-1023 B x)+105 a^4 b (9 A-121 B x)+231 a^3 b^2 x (15 A-73 B x)+128 b^5 x^4 (3 A+B x)\right )+315 \sqrt {a} (-3 A b+11 a B) (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{192 b^{13/2} (a+b x)^3 \sqrt {(a+b x)^2}} \] Input:

Integrate[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

(Sqrt[b]*Sqrt[x]*(-3465*a^5*B + a*b^4*x^3*(2511*A - 1408*B*x) + 9*a^2*b^3* 
x^2*(511*A - 1023*B*x) + 105*a^4*b*(9*A - 121*B*x) + 231*a^3*b^2*x*(15*A - 
 73*B*x) + 128*b^5*x^4*(3*A + B*x)) + 315*Sqrt[a]*(-3*A*b + 11*a*B)*(a + b 
*x)^4*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(192*b^(13/2)*(a + b*x)^3*Sqrt[(a 
 + b*x)^2])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.64, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {1187, 27, 87, 51, 51, 51, 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.

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

\(\Big \downarrow \) 1187

\(\displaystyle \frac {b^5 (a+b x) \int \frac {x^{9/2} (A+B x)}{b^5 (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(a+b x) \int \frac {x^{9/2} (A+B x)}{(a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 87

\(\displaystyle \frac {(a+b x) \left (\frac {x^{11/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(3 A b-11 a B) \int \frac {x^{9/2}}{(a+b x)^4}dx}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(a+b x) \left (\frac {x^{11/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(3 A b-11 a B) \left (\frac {3 \int \frac {x^{7/2}}{(a+b x)^3}dx}{2 b}-\frac {x^{9/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(a+b x) \left (\frac {x^{11/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(3 A b-11 a B) \left (\frac {3 \left (\frac {7 \int \frac {x^{5/2}}{(a+b x)^2}dx}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {x^{9/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {(a+b x) \left (\frac {x^{11/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(3 A b-11 a B) \left (\frac {3 \left (\frac {7 \left (\frac {5 \int \frac {x^{3/2}}{a+b x}dx}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {x^{9/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x) \left (\frac {x^{11/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(3 A b-11 a B) \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 b}-\frac {a \int \frac {\sqrt {x}}{a+b x}dx}{b}\right )}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {x^{9/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {(a+b x) \left (\frac {x^{11/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(3 A b-11 a B) \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {a \int \frac {1}{\sqrt {x} (a+b x)}dx}{b}\right )}{b}\right )}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {x^{9/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(a+b x) \left (\frac {x^{11/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(3 A b-11 a B) \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \int \frac {1}{a+b x}d\sqrt {x}}{b}\right )}{b}\right )}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {x^{9/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(a+b x) \left (\frac {x^{11/2} (A b-a B)}{4 a b (a+b x)^4}-\frac {(3 A b-11 a B) \left (\frac {3 \left (\frac {7 \left (\frac {5 \left (\frac {2 x^{3/2}}{3 b}-\frac {a \left (\frac {2 \sqrt {x}}{b}-\frac {2 \sqrt {a} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}}\right )}{b}\right )}{2 b}-\frac {x^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {x^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {x^{9/2}}{3 b (a+b x)^3}\right )}{8 a b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

Input:

Int[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
 

Output:

((a + b*x)*(((A*b - a*B)*x^(11/2))/(4*a*b*(a + b*x)^4) - ((3*A*b - 11*a*B) 
*(-1/3*x^(9/2)/(b*(a + b*x)^3) + (3*(-1/2*x^(7/2)/(b*(a + b*x)^2) + (7*(-( 
x^(5/2)/(b*(a + b*x))) + (5*((2*x^(3/2))/(3*b) - (a*((2*Sqrt[x])/b - (2*Sq 
rt[a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2)))/b))/(2*b)))/(4*b)))/(2* 
b)))/(8*a*b)))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
 

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 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 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 1187

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ 
IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 
 - 4*a*c, 0] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.52

method	result	size
risch	\(\frac {2 \left (B b x +3 A b -15 B a \right ) \sqrt {x}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{6} \left (b x +a \right )}-\frac {a \left (\frac {2 \left (-\frac {325}{128} A \,b^{4}+\frac {765}{128} B a \,b^{3}\right ) x^{\frac {7}{2}}+2 \left (-\frac {765}{128} A a \,b^{3}+\frac {5855}{384} B \,a^{2} b^{2}\right ) x^{\frac {5}{2}}-\frac {a^{2} b \left (1929 A b -5153 B a \right ) x^{\frac {3}{2}}}{192}+2 \left (-\frac {187}{128} A \,a^{3} b +\frac {515}{128} a^{4} B \right ) \sqrt {x}}{\left (b x +a \right )^{4}}+\frac {105 \left (3 A b -11 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \sqrt {a b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{6} \left (b x +a \right )}\)	\(174\)
default	\(\frac {\left (3465 A \sqrt {a b}\, x^{\frac {3}{2}} a^{3} b^{2}-5670 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{3} b^{3} x^{2}-12705 B \sqrt {a b}\, x^{\frac {3}{2}} a^{4} b -9207 B \sqrt {a b}\, x^{\frac {7}{2}} a^{2} b^{3}+20790 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{4} b^{2} x^{2}-945 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{5} b -3465 B \sqrt {a b}\, \sqrt {x}\, a^{5}-945 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a \,b^{5} x^{4}-16863 B \sqrt {a b}\, x^{\frac {5}{2}} a^{3} b^{2}+3465 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{2} b^{4} x^{4}+128 B \sqrt {a b}\, x^{\frac {11}{2}} b^{5}-3780 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{4} b^{2} x +13860 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{5} b x +945 A \sqrt {a b}\, \sqrt {x}\, a^{4} b -1408 B \sqrt {a b}\, x^{\frac {9}{2}} a \,b^{4}+384 A \sqrt {a b}\, x^{\frac {9}{2}} b^{5}-3780 A \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{2} b^{4} x^{3}+4599 A \sqrt {a b}\, x^{\frac {5}{2}} a^{2} b^{3}+2511 A \sqrt {a b}\, x^{\frac {7}{2}} a \,b^{4}+13860 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{3} b^{3} x^{3}+3465 B \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{6}\right ) \left (b x +a \right )}{192 \sqrt {a b}\, b^{6} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(407\)

Input:

int(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/3*(B*b*x+3*A*b-15*B*a)*x^(1/2)/b^6*((b*x+a)^2)^(1/2)/(b*x+a)-a/b^6*(2*(( 
-325/128*A*b^4+765/128*B*a*b^3)*x^(7/2)+(-765/128*A*a*b^3+5855/384*B*a^2*b 
^2)*x^(5/2)-1/384*a^2*b*(1929*A*b-5153*B*a)*x^(3/2)+(-187/128*A*a^3*b+515/ 
128*a^4*B)*x^(1/2))/(b*x+a)^4+105/64*(3*A*b-11*B*a)/(a*b)^(1/2)*arctan(b*x 
^(1/2)/(a*b)^(1/2)))*((b*x+a)^2)^(1/2)/(b*x+a)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.75 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [-\frac {315 \, {\left (11 \, B a^{5} - 3 \, A a^{4} b + {\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \, {\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 6 \, {\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (128 \, B b^{5} x^{5} - 3465 \, B a^{5} + 945 \, A a^{4} b - 128 \, {\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} - 837 \, {\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} - 1533 \, {\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 1155 \, {\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt {x}}{384 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}, \frac {315 \, {\left (11 \, B a^{5} - 3 \, A a^{4} b + {\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \, {\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 6 \, {\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 4 \, {\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (128 \, B b^{5} x^{5} - 3465 \, B a^{5} + 945 \, A a^{4} b - 128 \, {\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} - 837 \, {\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} - 1533 \, {\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 1155 \, {\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt {x}}{192 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}\right ] \] Input:

integrate(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas 
")
 

Output:

[-1/384*(315*(11*B*a^5 - 3*A*a^4*b + (11*B*a*b^4 - 3*A*b^5)*x^4 + 4*(11*B* 
a^2*b^3 - 3*A*a*b^4)*x^3 + 6*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 4*(11*B*a^ 
4*b - 3*A*a^3*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b) - a)/(b 
*x + a)) - 2*(128*B*b^5*x^5 - 3465*B*a^5 + 945*A*a^4*b - 128*(11*B*a*b^4 - 
 3*A*b^5)*x^4 - 837*(11*B*a^2*b^3 - 3*A*a*b^4)*x^3 - 1533*(11*B*a^3*b^2 - 
3*A*a^2*b^3)*x^2 - 1155*(11*B*a^4*b - 3*A*a^3*b^2)*x)*sqrt(x))/(b^10*x^4 + 
 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6), 1/192*(315*(11*B*a^ 
5 - 3*A*a^4*b + (11*B*a*b^4 - 3*A*b^5)*x^4 + 4*(11*B*a^2*b^3 - 3*A*a*b^4)* 
x^3 + 6*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 + 4*(11*B*a^4*b - 3*A*a^3*b^2)*x) 
*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (128*B*b^5*x^5 - 3465*B*a^5 + 9 
45*A*a^4*b - 128*(11*B*a*b^4 - 3*A*b^5)*x^4 - 837*(11*B*a^2*b^3 - 3*A*a*b^ 
4)*x^3 - 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*x^2 - 1155*(11*B*a^4*b - 3*A*a^ 
3*b^2)*x)*sqrt(x))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + 
 a^4*b^6)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(x**(9/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.14 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {5 \, {\left ({\left (2747 \, B a b^{5} - 693 \, A b^{6}\right )} x^{2} + 3 \, {\left (437 \, B a^{2} b^{4} - 63 \, A a b^{5}\right )} x\right )} x^{\frac {9}{2}} + 10 \, {\left (359 \, {\left (13 \, B a^{2} b^{4} - 3 \, A a b^{5}\right )} x^{2} + 183 \, {\left (11 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x\right )} x^{\frac {7}{2}} + 20 \, {\left (242 \, {\left (13 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{2} + 117 \, {\left (11 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x\right )} x^{\frac {5}{2}} + 198 \, {\left (15 \, {\left (13 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} + 7 \, {\left (11 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} x^{\frac {3}{2}} + 63 \, {\left (11 \, {\left (13 \, B a^{5} b - 3 \, A a^{4} b^{2}\right )} x^{2} + 5 \, {\left (11 \, B a^{6} - A a^{5} b\right )} x\right )} \sqrt {x}}{1920 \, {\left (a b^{10} x^{5} + 5 \, a^{2} b^{9} x^{4} + 10 \, a^{3} b^{8} x^{3} + 10 \, a^{4} b^{7} x^{2} + 5 \, a^{5} b^{6} x + a^{6} b^{5}\right )}} + \frac {105 \, {\left (11 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{6}} + \frac {7 \, {\left (11 \, {\left (13 \, B a b - 3 \, A b^{2}\right )} x^{\frac {3}{2}} - 30 \, {\left (11 \, B a^{2} - 3 \, A a b\right )} \sqrt {x}\right )}}{128 \, a b^{6}} \] Input:

integrate(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima 
")
 

Output:

-1/1920*(5*((2747*B*a*b^5 - 693*A*b^6)*x^2 + 3*(437*B*a^2*b^4 - 63*A*a*b^5 
)*x)*x^(9/2) + 10*(359*(13*B*a^2*b^4 - 3*A*a*b^5)*x^2 + 183*(11*B*a^3*b^3 
- A*a^2*b^4)*x)*x^(7/2) + 20*(242*(13*B*a^3*b^3 - 3*A*a^2*b^4)*x^2 + 117*( 
11*B*a^4*b^2 - A*a^3*b^3)*x)*x^(5/2) + 198*(15*(13*B*a^4*b^2 - 3*A*a^3*b^3 
)*x^2 + 7*(11*B*a^5*b - A*a^4*b^2)*x)*x^(3/2) + 63*(11*(13*B*a^5*b - 3*A*a 
^4*b^2)*x^2 + 5*(11*B*a^6 - A*a^5*b)*x)*sqrt(x))/(a*b^10*x^5 + 5*a^2*b^9*x 
^4 + 10*a^3*b^8*x^3 + 10*a^4*b^7*x^2 + 5*a^5*b^6*x + a^6*b^5) + 105/64*(11 
*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^6) + 7/128*(11* 
(13*B*a*b - 3*A*b^2)*x^(3/2) - 30*(11*B*a^2 - 3*A*a*b)*sqrt(x))/(a*b^6)
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.57 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {105 \, {\left (11 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {2295 \, B a^{2} b^{3} x^{\frac {7}{2}} - 975 \, A a b^{4} x^{\frac {7}{2}} + 5855 \, B a^{3} b^{2} x^{\frac {5}{2}} - 2295 \, A a^{2} b^{3} x^{\frac {5}{2}} + 5153 \, B a^{4} b x^{\frac {3}{2}} - 1929 \, A a^{3} b^{2} x^{\frac {3}{2}} + 1545 \, B a^{5} \sqrt {x} - 561 \, A a^{4} b \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (B b^{10} x^{\frac {3}{2}} - 15 \, B a b^{9} \sqrt {x} + 3 \, A b^{10} \sqrt {x}\right )}}{3 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \] Input:

integrate(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
 

Output:

105/64*(11*B*a^2 - 3*A*a*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^6*sgn 
(b*x + a)) - 1/192*(2295*B*a^2*b^3*x^(7/2) - 975*A*a*b^4*x^(7/2) + 5855*B* 
a^3*b^2*x^(5/2) - 2295*A*a^2*b^3*x^(5/2) + 5153*B*a^4*b*x^(3/2) - 1929*A*a 
^3*b^2*x^(3/2) + 1545*B*a^5*sqrt(x) - 561*A*a^4*b*sqrt(x))/((b*x + a)^4*b^ 
6*sgn(b*x + a)) + 2/3*(B*b^10*x^(3/2) - 15*B*a*b^9*sqrt(x) + 3*A*b^10*sqrt 
(x))/(b^15*sgn(b*x + a))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {x^{9/2}\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \] Input:

int((x^(9/2)*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
 

Output:

int((x^(9/2)*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.56 \[ \int \frac {x^{9/2} (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{4}+945 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b x +945 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} x^{2}+315 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} x^{3}-315 \sqrt {x}\, a^{4} b -840 \sqrt {x}\, a^{3} b^{2} x -693 \sqrt {x}\, a^{2} b^{3} x^{2}-144 \sqrt {x}\, a \,b^{4} x^{3}+16 \sqrt {x}\, b^{5} x^{4}}{24 b^{6} \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )} \] Input:

int(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
 

Output:

(315*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**4 + 945*sqrt(b 
)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**3*b*x + 945*sqrt(b)*sqrt( 
a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**2*b**2*x**2 + 315*sqrt(b)*sqrt(a 
)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a*b**3*x**3 - 315*sqrt(x)*a**4*b - 8 
40*sqrt(x)*a**3*b**2*x - 693*sqrt(x)*a**2*b**3*x**2 - 144*sqrt(x)*a*b**4*x 
**3 + 16*sqrt(x)*b**5*x**4)/(24*b**6*(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + 
b**3*x**3))