\(\int \frac {A+B x}{x^{7/2} (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\) [464]

Optimal result

Integrand size = 31, antiderivative size = 391 \[ \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {b^2 (1083 A b-515 a B) \sqrt {x}}{64 a^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (A b-a B) \sqrt {x}}{4 a^4 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (31 A b-23 a B) \sqrt {x}}{24 a^5 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (443 A b-259 a B) \sqrt {x}}{96 a^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 A (a+b x)}{5 a^5 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (5 A b-a B) (a+b x)}{3 a^6 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 b (3 A b-a B) (a+b x)}{a^7 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {231 b^{3/2} (13 A b-5 a B) (a+b x) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{64 a^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:

-1/64*b^2*(1083*A*b-515*B*a)*x^(1/2)/a^7/((b*x+a)^2)^(1/2)-1/4*b^2*(A*b-B* 
a)*x^(1/2)/a^4/(b*x+a)^3/((b*x+a)^2)^(1/2)-1/24*b^2*(31*A*b-23*B*a)*x^(1/2 
)/a^5/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/96*b^2*(443*A*b-259*B*a)*x^(1/2)/a^6/( 
b*x+a)/((b*x+a)^2)^(1/2)-2/5*A*(b*x+a)/a^5/x^(5/2)/((b*x+a)^2)^(1/2)+2/3*( 
5*A*b-B*a)*(b*x+a)/a^6/x^(3/2)/((b*x+a)^2)^(1/2)-10*b*(3*A*b-B*a)*(b*x+a)/ 
a^7/x^(1/2)/((b*x+a)^2)^(1/2)-231/64*b^(3/2)*(13*A*b-5*B*a)*(b*x+a)*arctan 
(b^(1/2)*x^(1/2)/a^(1/2))/a^(15/2)/((b*x+a)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.49 \[ \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {a} \left (-45045 A b^6 x^6-128 a^6 (3 A+5 B x)+1155 a b^5 x^5 (-143 A+15 B x)+128 a^5 b x (13 A+55 B x)+231 a^2 b^4 x^4 (-949 A+275 B x)+33 a^3 b^3 x^3 (-3627 A+2555 B x)+11 a^4 b^2 x^2 (-1664 A+4185 B x)\right )}{x^{5/2}}+3465 b^{3/2} (-13 A b+5 a B) (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{960 a^{15/2} (a+b x)^3 \sqrt {(a+b x)^2}} \] Input:

Integrate[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
 

Output:

((Sqrt[a]*(-45045*A*b^6*x^6 - 128*a^6*(3*A + 5*B*x) + 1155*a*b^5*x^5*(-143 
*A + 15*B*x) + 128*a^5*b*x*(13*A + 55*B*x) + 231*a^2*b^4*x^4*(-949*A + 275 
*B*x) + 33*a^3*b^3*x^3*(-3627*A + 2555*B*x) + 11*a^4*b^2*x^2*(-1664*A + 41 
85*B*x)))/x^(5/2) + 3465*b^(3/2)*(-13*A*b + 5*a*B)*(a + b*x)^4*ArcTan[(Sqr 
t[b]*Sqrt[x])/Sqrt[a]])/(960*a^(15/2)*(a + b*x)^3*Sqrt[(a + b*x)^2])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.59, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {1187, 27, 87, 52, 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.

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

\(\Big \downarrow \) 1187

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a+b x) \left (\frac {(13 A b-5 a B) \left (\frac {11 \int \frac {1}{x^{7/2} (a+b x)^3}dx}{6 a}+\frac {1}{3 a x^{5/2} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a+b x) \left (\frac {(13 A b-5 a B) \left (\frac {11 \left (\frac {9 \int \frac {1}{x^{7/2} (a+b x)^2}dx}{4 a}+\frac {1}{2 a x^{5/2} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a x^{5/2} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(a+b x) \left (\frac {(13 A b-5 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \int \frac {1}{x^{7/2} (a+b x)}dx}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{5/2} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a x^{5/2} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a+b x) \left (\frac {(13 A b-5 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \int \frac {1}{x^{5/2} (a+b x)}dx}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{5/2} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a x^{5/2} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a+b x) \left (\frac {(13 A b-5 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \int \frac {1}{x^{3/2} (a+b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{5/2} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a x^{5/2} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(a+b x) \left (\frac {(13 A b-5 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{5/2} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a x^{5/2} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(a+b x) \left (\frac {(13 A b-5 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{5/2} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a x^{5/2} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(a+b x) \left (\frac {(13 A b-5 a B) \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (-\frac {b \left (-\frac {b \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a}+\frac {1}{a x^{5/2} (a+b x)}\right )}{4 a}+\frac {1}{2 a x^{5/2} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a x^{5/2} (a+b x)^3}\right )}{8 a b}+\frac {A b-a B}{4 a b x^{5/2} (a+b x)^4}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\)

Input:

Int[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
 

Output:

((a + b*x)*((A*b - a*B)/(4*a*b*x^(5/2)*(a + b*x)^4) + ((13*A*b - 5*a*B)*(1 
/(3*a*x^(5/2)*(a + b*x)^3) + (11*(1/(2*a*x^(5/2)*(a + b*x)^2) + (9*(1/(a*x 
^(5/2)*(a + b*x)) + (7*(-2/(5*a*x^(5/2)) - (b*(-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))/a 
))/(2*a)))/(4*a)))/(6*a)))/(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 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 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 = 1.15 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.51

method	result	size
risch	\(-\frac {2 \left (225 x^{2} b^{2} A -75 B a \,x^{2} b -25 a b A x +5 a^{2} B x +3 a^{2} A \right ) \sqrt {\left (b x +a \right )^{2}}}{15 a^{7} x^{\frac {5}{2}} \left (b x +a \right )}-\frac {b^{2} \left (\frac {2 \left (\frac {1083}{128} A \,b^{4}-\frac {515}{128} B a \,b^{3}\right ) x^{\frac {7}{2}}+\frac {a \,b^{2} \left (10633 A b -5153 B a \right ) x^{\frac {5}{2}}}{192}+2 \left (\frac {11767}{384} a^{2} A \,b^{2}-\frac {5855}{384} B \,a^{3} b \right ) x^{\frac {3}{2}}+2 \left (\frac {1477}{128} A \,a^{3} b -\frac {765}{128} a^{4} B \right ) \sqrt {x}}{\left (b x +a \right )^{4}}+\frac {231 \left (13 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \sqrt {a b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{a^{7} \left (b x +a \right )}\)	\(200\)
default	\(-\frac {\left (-7040 B \,x^{2} \sqrt {a b}\, a^{5} b -1664 A x \sqrt {a b}\, a^{5} b +219219 A \,x^{4} \sqrt {a b}\, a^{2} b^{4}-84315 B \,x^{4} \sqrt {a b}\, a^{3} b^{3}+119691 A \,x^{3} \sqrt {a b}\, a^{3} b^{3}+45045 A \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{4} b^{3}-46035 B \,x^{3} \sqrt {a b}\, a^{4} b^{2}-17325 B \,x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{5} b^{2}-17325 B \,x^{\frac {13}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a \,b^{6}+180180 A \,x^{\frac {11}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a \,b^{6}+18304 A \,x^{2} \sqrt {a b}\, a^{4} b^{2}-17325 B \,x^{6} \sqrt {a b}\, a \,b^{5}+165165 A \,x^{5} \sqrt {a b}\, a \,b^{5}+45045 A \,x^{\frac {13}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) b^{7}-63525 B \,x^{5} \sqrt {a b}\, a^{2} b^{4}+180180 A \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{3} b^{4}-69300 B \,x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{4} b^{3}+640 B x \sqrt {a b}\, a^{6}-69300 B \,x^{\frac {11}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{2} b^{5}+270270 A \,x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{2} b^{5}-103950 B \,x^{\frac {9}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) a^{3} b^{4}+45045 A \,x^{6} \sqrt {a b}\, b^{6}+384 A \sqrt {a b}\, a^{6}\right ) \left (b x +a \right )}{960 \sqrt {a b}\, x^{\frac {5}{2}} a^{7} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\)	\(449\)

Input:

int((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-2/15*(225*A*b^2*x^2-75*B*a*b*x^2-25*A*a*b*x+5*B*a^2*x+3*A*a^2)/a^7/x^(5/2 
)*((b*x+a)^2)^(1/2)/(b*x+a)-1/a^7*b^2*(2*((1083/128*A*b^4-515/128*B*a*b^3) 
*x^(7/2)+1/384*a*b^2*(10633*A*b-5153*B*a)*x^(5/2)+(11767/384*a^2*A*b^2-585 
5/384*B*a^3*b)*x^(3/2)+(1477/128*A*a^3*b-765/128*a^4*B)*x^(1/2))/(b*x+a)^4 
+231/64*(13*A*b-5*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 = 0.11 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.71 \[ \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas 
")
 

Output:

[-1/1920*(3465*((5*B*a*b^5 - 13*A*b^6)*x^7 + 4*(5*B*a^2*b^4 - 13*A*a*b^5)* 
x^6 + 6*(5*B*a^3*b^3 - 13*A*a^2*b^4)*x^5 + 4*(5*B*a^4*b^2 - 13*A*a^3*b^3)* 
x^4 + (5*B*a^5*b - 13*A*a^4*b^2)*x^3)*sqrt(-b/a)*log((b*x - 2*a*sqrt(x)*sq 
rt(-b/a) - a)/(b*x + a)) + 2*(384*A*a^6 - 3465*(5*B*a*b^5 - 13*A*b^6)*x^6 
- 12705*(5*B*a^2*b^4 - 13*A*a*b^5)*x^5 - 16863*(5*B*a^3*b^3 - 13*A*a^2*b^4 
)*x^4 - 9207*(5*B*a^4*b^2 - 13*A*a^3*b^3)*x^3 - 1408*(5*B*a^5*b - 13*A*a^4 
*b^2)*x^2 + 128*(5*B*a^6 - 13*A*a^5*b)*x)*sqrt(x))/(a^7*b^4*x^7 + 4*a^8*b^ 
3*x^6 + 6*a^9*b^2*x^5 + 4*a^10*b*x^4 + a^11*x^3), 1/960*(3465*((5*B*a*b^5 
- 13*A*b^6)*x^7 + 4*(5*B*a^2*b^4 - 13*A*a*b^5)*x^6 + 6*(5*B*a^3*b^3 - 13*A 
*a^2*b^4)*x^5 + 4*(5*B*a^4*b^2 - 13*A*a^3*b^3)*x^4 + (5*B*a^5*b - 13*A*a^4 
*b^2)*x^3)*sqrt(b/a)*arctan(sqrt(x)*sqrt(b/a)) - (384*A*a^6 - 3465*(5*B*a* 
b^5 - 13*A*b^6)*x^6 - 12705*(5*B*a^2*b^4 - 13*A*a*b^5)*x^5 - 16863*(5*B*a^ 
3*b^3 - 13*A*a^2*b^4)*x^4 - 9207*(5*B*a^4*b^2 - 13*A*a^3*b^3)*x^3 - 1408*( 
5*B*a^5*b - 13*A*a^4*b^2)*x^2 + 128*(5*B*a^6 - 13*A*a^5*b)*x)*sqrt(x))/(a^ 
7*b^4*x^7 + 4*a^8*b^3*x^6 + 6*a^9*b^2*x^5 + 4*a^10*b*x^4 + a^11*x^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/x**(7/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (263) = 526\).

Time = 0.21 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.41 \[ \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {1155 \, {\left ({\left (3 \, B a b^{8} - 13 \, A b^{9}\right )} x^{2} + 39 \, {\left (B a^{2} b^{7} - 3 \, A a b^{8}\right )} x\right )} x^{\frac {9}{2}} + 2310 \, {\left ({\left (3 \, B a^{2} b^{7} - 13 \, A a b^{8}\right )} x^{2} + 117 \, {\left (B a^{3} b^{6} - 3 \, A a^{2} b^{7}\right )} x\right )} x^{\frac {7}{2}} - 4620 \, {\left (2 \, {\left (3 \, B a^{3} b^{6} - 13 \, A a^{2} b^{7}\right )} x^{2} - 143 \, {\left (B a^{4} b^{5} - 3 \, A a^{3} b^{6}\right )} x\right )} x^{\frac {5}{2}} - 462 \, {\left (85 \, {\left (3 \, B a^{4} b^{5} - 13 \, A a^{3} b^{6}\right )} x^{2} - 1807 \, {\left (B a^{5} b^{4} - 3 \, A a^{4} b^{5}\right )} x\right )} x^{\frac {3}{2}} - 33 \, {\left (1771 \, {\left (3 \, B a^{5} b^{4} - 13 \, A a^{4} b^{5}\right )} x^{2} - 17095 \, {\left (B a^{6} b^{3} - 3 \, A a^{5} b^{4}\right )} x\right )} \sqrt {x} - \frac {14080 \, {\left (3 \, {\left (3 \, B a^{6} b^{3} - 13 \, A a^{5} b^{4}\right )} x^{2} - 13 \, {\left (B a^{7} b^{2} - 3 \, A a^{6} b^{3}\right )} x\right )}}{\sqrt {x}} - \frac {1280 \, {\left (11 \, {\left (3 \, B a^{7} b^{2} - 13 \, A a^{6} b^{3}\right )} x^{2} - 13 \, {\left (B a^{8} b - 3 \, A a^{7} b^{2}\right )} x\right )}}{x^{\frac {3}{2}}} - \frac {1280 \, {\left ({\left (3 \, B a^{8} b - 13 \, A a^{7} b^{2}\right )} x^{2} + {\left (B a^{9} - 3 \, A a^{8} b\right )} x\right )}}{x^{\frac {5}{2}}} - \frac {256 \, {\left (5 \, A a^{8} b x^{2} + 3 \, A a^{9} x\right )}}{x^{\frac {7}{2}}}}{1920 \, {\left (a^{9} b^{5} x^{5} + 5 \, a^{10} b^{4} x^{4} + 10 \, a^{11} b^{3} x^{3} + 10 \, a^{12} b^{2} x^{2} + 5 \, a^{13} b x + a^{14}\right )}} + \frac {231 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{7}} - \frac {77 \, {\left ({\left (3 \, B a b^{3} - 13 \, A b^{4}\right )} x^{\frac {3}{2}} + 6 \, {\left (5 \, B a^{2} b^{2} - 13 \, A a b^{3}\right )} \sqrt {x}\right )}}{128 \, a^{9}} \] Input:

integrate((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima 
")
 

Output:

1/1920*(1155*((3*B*a*b^8 - 13*A*b^9)*x^2 + 39*(B*a^2*b^7 - 3*A*a*b^8)*x)*x 
^(9/2) + 2310*((3*B*a^2*b^7 - 13*A*a*b^8)*x^2 + 117*(B*a^3*b^6 - 3*A*a^2*b 
^7)*x)*x^(7/2) - 4620*(2*(3*B*a^3*b^6 - 13*A*a^2*b^7)*x^2 - 143*(B*a^4*b^5 
 - 3*A*a^3*b^6)*x)*x^(5/2) - 462*(85*(3*B*a^4*b^5 - 13*A*a^3*b^6)*x^2 - 18 
07*(B*a^5*b^4 - 3*A*a^4*b^5)*x)*x^(3/2) - 33*(1771*(3*B*a^5*b^4 - 13*A*a^4 
*b^5)*x^2 - 17095*(B*a^6*b^3 - 3*A*a^5*b^4)*x)*sqrt(x) - 14080*(3*(3*B*a^6 
*b^3 - 13*A*a^5*b^4)*x^2 - 13*(B*a^7*b^2 - 3*A*a^6*b^3)*x)/sqrt(x) - 1280* 
(11*(3*B*a^7*b^2 - 13*A*a^6*b^3)*x^2 - 13*(B*a^8*b - 3*A*a^7*b^2)*x)/x^(3/ 
2) - 1280*((3*B*a^8*b - 13*A*a^7*b^2)*x^2 + (B*a^9 - 3*A*a^8*b)*x)/x^(5/2) 
 - 256*(5*A*a^8*b*x^2 + 3*A*a^9*x)/x^(7/2))/(a^9*b^5*x^5 + 5*a^10*b^4*x^4 
+ 10*a^11*b^3*x^3 + 10*a^12*b^2*x^2 + 5*a^13*b*x + a^14) + 231/64*(5*B*a*b 
^2 - 13*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7) - 77/128*((3*B* 
a*b^3 - 13*A*b^4)*x^(3/2) + 6*(5*B*a^2*b^2 - 13*A*a*b^3)*sqrt(x))/a^9
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.53 \[ \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {231 \, {\left (5 \, B a b^{2} - 13 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{64 \, \sqrt {a b} a^{7} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left (75 \, B a b x^{2} - 225 \, A b^{2} x^{2} - 5 \, B a^{2} x + 25 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{7} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right )} + \frac {1545 \, B a b^{5} x^{\frac {7}{2}} - 3249 \, A b^{6} x^{\frac {7}{2}} + 5153 \, B a^{2} b^{4} x^{\frac {5}{2}} - 10633 \, A a b^{5} x^{\frac {5}{2}} + 5855 \, B a^{3} b^{3} x^{\frac {3}{2}} - 11767 \, A a^{2} b^{4} x^{\frac {3}{2}} + 2295 \, B a^{4} b^{2} \sqrt {x} - 4431 \, A a^{3} b^{3} \sqrt {x}}{192 \, {\left (b x + a\right )}^{4} a^{7} \mathrm {sgn}\left (b x + a\right )} \] Input:

integrate((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
 

Output:

231/64*(5*B*a*b^2 - 13*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^7*s 
gn(b*x + a)) + 2/15*(75*B*a*b*x^2 - 225*A*b^2*x^2 - 5*B*a^2*x + 25*A*a*b*x 
 - 3*A*a^2)/(a^7*x^(5/2)*sgn(b*x + a)) + 1/192*(1545*B*a*b^5*x^(7/2) - 324 
9*A*b^6*x^(7/2) + 5153*B*a^2*b^4*x^(5/2) - 10633*A*a*b^5*x^(5/2) + 5855*B* 
a^3*b^3*x^(3/2) - 11767*A*a^2*b^4*x^(3/2) + 2295*B*a^4*b^2*sqrt(x) - 4431* 
A*a^3*b^3*sqrt(x))/((b*x + a)^4*a^7*sgn(b*x + a))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{x^{7/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \] Input:

int((A + B*x)/(x^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
 

Output:

int((A + B*x)/(x^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.54 \[ \int \frac {A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-3465 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b^{2} x^{2}-10395 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{3} x^{3}-10395 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{4} x^{4}-3465 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{5} x^{5}-48 a^{6}+176 a^{5} b x -1584 a^{4} b^{2} x^{2}-7623 a^{3} b^{3} x^{3}-9240 a^{2} b^{4} x^{4}-3465 a \,b^{5} x^{5}}{120 \sqrt {x}\, a^{7} x^{2} \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right )} \] Input:

int((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
 

Output:

( - 3465*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**3* 
b**2*x**2 - 10395*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a 
)))*a**2*b**3*x**3 - 10395*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 - 48*a**6 + 176*a**5*b*x - 1584*a**4*b**2*x**2 
 - 7623*a**3*b**3*x**3 - 9240*a**2*b**4*x**4 - 3465*a*b**5*x**5)/(120*sqrt 
(x)*a**7*x**2*(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3))