\(\int \frac {A+B x}{x^{3/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\) [409]

Optimal result

Integrand size = 29, antiderivative size = 191 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {2 A}{a^6 \sqrt {x}}-\frac {(A b-a B) \sqrt {x}}{5 a^2 (a+b x)^5}-\frac {(19 A b-9 a B) \sqrt {x}}{40 a^3 (a+b x)^4}-\frac {(71 A b-21 a B) \sqrt {x}}{80 a^4 (a+b x)^3}-\frac {(103 A b-21 a B) \sqrt {x}}{64 a^5 (a+b x)^2}-\frac {(437 A b-63 a B) \sqrt {x}}{128 a^6 (a+b x)}-\frac {63 (11 A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{128 a^{13/2} \sqrt {b}} \] Output:

-2*A/a^6/x^(1/2)-1/5*(A*b-B*a)*x^(1/2)/a^2/(b*x+a)^5-1/40*(19*A*b-9*B*a)*x 
^(1/2)/a^3/(b*x+a)^4-1/80*(71*A*b-21*B*a)*x^(1/2)/a^4/(b*x+a)^3-1/64*(103* 
A*b-21*B*a)*x^(1/2)/a^5/(b*x+a)^2-1/128*(437*A*b-63*B*a)*x^(1/2)/a^6/(b*x+ 
a)-63/128*(11*A*b-B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(13/2)/b^(1/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {\sqrt {a} \left (-3465 A b^5 x^5+105 a b^4 x^4 (-154 A+3 B x)+42 a^2 b^3 x^3 (-704 A+35 B x)+6 a^3 b^2 x^2 (-4345 A+448 B x)+5 a^4 b x (-2123 A+474 B x)+a^5 (-1280 A+965 B x)\right )}{\sqrt {x} (a+b x)^5}+\frac {315 (-11 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {b}}}{640 a^{13/2}} \] Input:

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

Output:

((Sqrt[a]*(-3465*A*b^5*x^5 + 105*a*b^4*x^4*(-154*A + 3*B*x) + 42*a^2*b^3*x 
^3*(-704*A + 35*B*x) + 6*a^3*b^2*x^2*(-4345*A + 448*B*x) + 5*a^4*b*x*(-212 
3*A + 474*B*x) + a^5*(-1280*A + 965*B*x)))/(Sqrt[x]*(a + b*x)^5) + (315*(- 
11*A*b + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/Sqrt[b])/(640*a^(13/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1184, 27, 87, 52, 52, 52, 52, 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.

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

\(\Big \downarrow \) 1184

\(\displaystyle b^6 \int \frac {A+B x}{b^6 x^{3/2} (a+b x)^6}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {A+B x}{x^{3/2} (a+b x)^6}dx\)

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 52

\(\displaystyle \frac {(11 A b-a B) \left (\frac {9 \int \frac {1}{x^{3/2} (a+b x)^4}dx}{8 a}+\frac {1}{4 a \sqrt {x} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(11 A b-a B) \left (\frac {9 \left (\frac {7 \int \frac {1}{x^{3/2} (a+b x)^3}dx}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a \sqrt {x} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(11 A b-a B) \left (\frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^{3/2} (a+b x)^2}dx}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a \sqrt {x} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {(11 A b-a B) \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{x^{3/2} (a+b x)}dx}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a \sqrt {x} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}\)

\(\Big \downarrow \) 61

\(\displaystyle \frac {(11 A b-a B) \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a \sqrt {x} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {(11 A b-a B) \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a \sqrt {x} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(11 A b-a B) \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{2 a}+\frac {1}{a \sqrt {x} (a+b x)}\right )}{4 a}+\frac {1}{2 a \sqrt {x} (a+b x)^2}\right )}{6 a}+\frac {1}{3 a \sqrt {x} (a+b x)^3}\right )}{8 a}+\frac {1}{4 a \sqrt {x} (a+b x)^4}\right )}{10 a b}+\frac {A b-a B}{5 a b \sqrt {x} (a+b x)^5}\)

Input:

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

Output:

(A*b - a*B)/(5*a*b*Sqrt[x]*(a + b*x)^5) + ((11*A*b - a*B)*(1/(4*a*Sqrt[x]* 
(a + b*x)^4) + (9*(1/(3*a*Sqrt[x]*(a + b*x)^3) + (7*(1/(2*a*Sqrt[x]*(a + b 
*x)^2) + (5*(1/(a*Sqrt[x]*(a + b*x)) + (3*(-2/(a*Sqrt[x]) - (2*Sqrt[b]*Arc 
Tan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)))/(2*a)))/(4*a)))/(6*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 = 1.18 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {\left (\frac {437}{256} A \,b^{5}-\frac {63}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {a \,b^{3} \left (977 A b -147 B a \right ) x^{\frac {7}{2}}}{128}+\left (\frac {131}{10} A \,a^{2} b^{3}-\frac {21}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {1327}{128} a^{3} A \,b^{2}-\frac {237}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {843}{256} A \,a^{4} b -\frac {193}{256} B \,a^{5}\right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {63 \left (11 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{a^{6}}-\frac {2 A}{a^{6} \sqrt {x}}\)	\(151\)
default	\(-\frac {2 \left (\frac {\left (\frac {437}{256} A \,b^{5}-\frac {63}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {a \,b^{3} \left (977 A b -147 B a \right ) x^{\frac {7}{2}}}{128}+\left (\frac {131}{10} A \,a^{2} b^{3}-\frac {21}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+\left (\frac {1327}{128} a^{3} A \,b^{2}-\frac {237}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+\left (\frac {843}{256} A \,a^{4} b -\frac {193}{256} B \,a^{5}\right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {63 \left (11 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{a^{6}}-\frac {2 A}{a^{6} \sqrt {x}}\)	\(151\)
risch	\(-\frac {2 A}{a^{6} \sqrt {x}}-\frac {\frac {2 \left (\frac {437}{256} A \,b^{5}-\frac {63}{256} B a \,b^{4}\right ) x^{\frac {9}{2}}+\frac {a \,b^{3} \left (977 A b -147 B a \right ) x^{\frac {7}{2}}}{64}+2 \left (\frac {131}{10} A \,a^{2} b^{3}-\frac {21}{10} B \,a^{3} b^{2}\right ) x^{\frac {5}{2}}+2 \left (\frac {1327}{128} a^{3} A \,b^{2}-\frac {237}{128} B \,a^{4} b \right ) x^{\frac {3}{2}}+2 \left (\frac {843}{256} A \,a^{4} b -\frac {193}{256} B \,a^{5}\right ) \sqrt {x}}{\left (b x +a \right )^{5}}+\frac {63 \left (11 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \sqrt {a b}}}{a^{6}}\)	\(152\)

Input:

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

Output:

-2/a^6*(((437/256*A*b^5-63/256*B*a*b^4)*x^(9/2)+1/128*a*b^3*(977*A*b-147*B 
*a)*x^(7/2)+(131/10*A*a^2*b^3-21/10*B*a^3*b^2)*x^(5/2)+(1327/128*a^3*A*b^2 
-237/128*B*a^4*b)*x^(3/2)+(843/256*A*a^4*b-193/256*B*a^5)*x^(1/2))/(b*x+a) 
^5+63/256*(11*A*b-B*a)/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))-2*A/a^6/ 
x^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (156) = 312\).

Time = 0.10 (sec) , antiderivative size = 673, normalized size of antiderivative = 3.52 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [\frac {315 \, {\left ({\left (B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 5 \, {\left (B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 10 \, {\left (B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 10 \, {\left (B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} + 5 \, {\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (1280 \, A a^{6} b - 315 \, {\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x^{5} - 1470 \, {\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x^{4} - 2688 \, {\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x^{3} - 2370 \, {\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x^{2} - 965 \, {\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt {x}}{1280 \, {\left (a^{7} b^{6} x^{6} + 5 \, a^{8} b^{5} x^{5} + 10 \, a^{9} b^{4} x^{4} + 10 \, a^{10} b^{3} x^{3} + 5 \, a^{11} b^{2} x^{2} + a^{12} b x\right )}}, -\frac {315 \, {\left ({\left (B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 5 \, {\left (B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 10 \, {\left (B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 10 \, {\left (B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} + 5 \, {\left (B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} + {\left (B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (1280 \, A a^{6} b - 315 \, {\left (B a^{2} b^{5} - 11 \, A a b^{6}\right )} x^{5} - 1470 \, {\left (B a^{3} b^{4} - 11 \, A a^{2} b^{5}\right )} x^{4} - 2688 \, {\left (B a^{4} b^{3} - 11 \, A a^{3} b^{4}\right )} x^{3} - 2370 \, {\left (B a^{5} b^{2} - 11 \, A a^{4} b^{3}\right )} x^{2} - 965 \, {\left (B a^{6} b - 11 \, A a^{5} b^{2}\right )} x\right )} \sqrt {x}}{640 \, {\left (a^{7} b^{6} x^{6} + 5 \, a^{8} b^{5} x^{5} + 10 \, a^{9} b^{4} x^{4} + 10 \, a^{10} b^{3} x^{3} + 5 \, a^{11} b^{2} x^{2} + a^{12} b x\right )}}\right ] \] Input:

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

Output:

[1/1280*(315*((B*a*b^5 - 11*A*b^6)*x^6 + 5*(B*a^2*b^4 - 11*A*a*b^5)*x^5 + 
10*(B*a^3*b^3 - 11*A*a^2*b^4)*x^4 + 10*(B*a^4*b^2 - 11*A*a^3*b^3)*x^3 + 5* 
(B*a^5*b - 11*A*a^4*b^2)*x^2 + (B*a^6 - 11*A*a^5*b)*x)*sqrt(-a*b)*log((b*x 
 - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2*(1280*A*a^6*b - 315*(B*a^2*b^5 
 - 11*A*a*b^6)*x^5 - 1470*(B*a^3*b^4 - 11*A*a^2*b^5)*x^4 - 2688*(B*a^4*b^3 
 - 11*A*a^3*b^4)*x^3 - 2370*(B*a^5*b^2 - 11*A*a^4*b^3)*x^2 - 965*(B*a^6*b 
- 11*A*a^5*b^2)*x)*sqrt(x))/(a^7*b^6*x^6 + 5*a^8*b^5*x^5 + 10*a^9*b^4*x^4 
+ 10*a^10*b^3*x^3 + 5*a^11*b^2*x^2 + a^12*b*x), -1/640*(315*((B*a*b^5 - 11 
*A*b^6)*x^6 + 5*(B*a^2*b^4 - 11*A*a*b^5)*x^5 + 10*(B*a^3*b^3 - 11*A*a^2*b^ 
4)*x^4 + 10*(B*a^4*b^2 - 11*A*a^3*b^3)*x^3 + 5*(B*a^5*b - 11*A*a^4*b^2)*x^ 
2 + (B*a^6 - 11*A*a^5*b)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (128 
0*A*a^6*b - 315*(B*a^2*b^5 - 11*A*a*b^6)*x^5 - 1470*(B*a^3*b^4 - 11*A*a^2* 
b^5)*x^4 - 2688*(B*a^4*b^3 - 11*A*a^3*b^4)*x^3 - 2370*(B*a^5*b^2 - 11*A*a^ 
4*b^3)*x^2 - 965*(B*a^6*b - 11*A*a^5*b^2)*x)*sqrt(x))/(a^7*b^6*x^6 + 5*a^8 
*b^5*x^5 + 10*a^9*b^4*x^4 + 10*a^10*b^3*x^3 + 5*a^11*b^2*x^2 + a^12*b*x)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {1280 \, A a^{5} - 315 \, {\left (B a b^{4} - 11 \, A b^{5}\right )} x^{5} - 1470 \, {\left (B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{4} - 2688 \, {\left (B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{3} - 2370 \, {\left (B a^{4} b - 11 \, A a^{3} b^{2}\right )} x^{2} - 965 \, {\left (B a^{5} - 11 \, A a^{4} b\right )} x}{640 \, {\left (a^{6} b^{5} x^{\frac {11}{2}} + 5 \, a^{7} b^{4} x^{\frac {9}{2}} + 10 \, a^{8} b^{3} x^{\frac {7}{2}} + 10 \, a^{9} b^{2} x^{\frac {5}{2}} + 5 \, a^{10} b x^{\frac {3}{2}} + a^{11} \sqrt {x}\right )}} + \frac {63 \, {\left (B a - 11 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{6}} \] Input:

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

Output:

-1/640*(1280*A*a^5 - 315*(B*a*b^4 - 11*A*b^5)*x^5 - 1470*(B*a^2*b^3 - 11*A 
*a*b^4)*x^4 - 2688*(B*a^3*b^2 - 11*A*a^2*b^3)*x^3 - 2370*(B*a^4*b - 11*A*a 
^3*b^2)*x^2 - 965*(B*a^5 - 11*A*a^4*b)*x)/(a^6*b^5*x^(11/2) + 5*a^7*b^4*x^ 
(9/2) + 10*a^8*b^3*x^(7/2) + 10*a^9*b^2*x^(5/2) + 5*a^10*b*x^(3/2) + a^11* 
sqrt(x)) + 63/128*(B*a - 11*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^ 
6)
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63 \, {\left (B a - 11 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{6}} - \frac {2 \, A}{a^{6} \sqrt {x}} + \frac {315 \, B a b^{4} x^{\frac {9}{2}} - 2185 \, A b^{5} x^{\frac {9}{2}} + 1470 \, B a^{2} b^{3} x^{\frac {7}{2}} - 9770 \, A a b^{4} x^{\frac {7}{2}} + 2688 \, B a^{3} b^{2} x^{\frac {5}{2}} - 16768 \, A a^{2} b^{3} x^{\frac {5}{2}} + 2370 \, B a^{4} b x^{\frac {3}{2}} - 13270 \, A a^{3} b^{2} x^{\frac {3}{2}} + 965 \, B a^{5} \sqrt {x} - 4215 \, A a^{4} b \sqrt {x}}{640 \, {\left (b x + a\right )}^{5} a^{6}} \] Input:

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

Output:

63/128*(B*a - 11*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^6) - 2*A/(a 
^6*sqrt(x)) + 1/640*(315*B*a*b^4*x^(9/2) - 2185*A*b^5*x^(9/2) + 1470*B*a^2 
*b^3*x^(7/2) - 9770*A*a*b^4*x^(7/2) + 2688*B*a^3*b^2*x^(5/2) - 16768*A*a^2 
*b^3*x^(5/2) + 2370*B*a^4*b*x^(3/2) - 13270*A*a^3*b^2*x^(3/2) + 965*B*a^5* 
sqrt(x) - 4215*A*a^4*b*sqrt(x))/((b*x + a)^5*a^6)
 

Mupad [B] (verification not implemented)

Time = 11.09 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\frac {2\,A}{a}+\frac {193\,x\,\left (11\,A\,b-B\,a\right )}{128\,a^2}+\frac {21\,b^2\,x^3\,\left (11\,A\,b-B\,a\right )}{5\,a^4}+\frac {147\,b^3\,x^4\,\left (11\,A\,b-B\,a\right )}{64\,a^5}+\frac {63\,b^4\,x^5\,\left (11\,A\,b-B\,a\right )}{128\,a^6}+\frac {237\,b\,x^2\,\left (11\,A\,b-B\,a\right )}{64\,a^3}}{a^5\,\sqrt {x}+b^5\,x^{11/2}+5\,a^4\,b\,x^{3/2}+5\,a\,b^4\,x^{9/2}+10\,a^3\,b^2\,x^{5/2}+10\,a^2\,b^3\,x^{7/2}}-\frac {63\,\mathrm {atan}\left (\frac {63\,\sqrt {b}\,\sqrt {x}\,\left (11\,A\,b-B\,a\right )}{\sqrt {a}\,\left (693\,A\,b-63\,B\,a\right )}\right )\,\left (11\,A\,b-B\,a\right )}{128\,a^{13/2}\,\sqrt {b}} \] Input:

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

Output:

- ((2*A)/a + (193*x*(11*A*b - B*a))/(128*a^2) + (21*b^2*x^3*(11*A*b - B*a) 
)/(5*a^4) + (147*b^3*x^4*(11*A*b - B*a))/(64*a^5) + (63*b^4*x^5*(11*A*b - 
B*a))/(128*a^6) + (237*b*x^2*(11*A*b - B*a))/(64*a^3))/(a^5*x^(1/2) + b^5* 
x^(11/2) + 5*a^4*b*x^(3/2) + 5*a*b^4*x^(9/2) + 10*a^3*b^2*x^(5/2) + 10*a^2 
*b^3*x^(7/2)) - (63*atan((63*b^(1/2)*x^(1/2)*(11*A*b - B*a))/(a^(1/2)*(693 
*A*b - 63*B*a)))*(11*A*b - B*a))/(128*a^(13/2)*b^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {-315 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{4}-1260 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{3} b x -1890 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a^{2} b^{2} x^{2}-1260 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) a \,b^{3} x^{3}-315 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {x}\, b}{\sqrt {b}\, \sqrt {a}}\right ) b^{4} x^{4}-128 a^{5}-837 a^{4} b x -1533 a^{3} b^{2} x^{2}-1155 a^{2} b^{3} x^{3}-315 a \,b^{4} x^{4}}{64 \sqrt {x}\, a^{6} \left (b^{4} x^{4}+4 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )} \] Input:

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

Output:

( - 315*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**4 - 
 1260*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**3*b*x 
 - 1890*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)))*a**2*b 
**2*x**2 - 1260*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt(a)) 
)*a*b**3*x**3 - 315*sqrt(x)*sqrt(b)*sqrt(a)*atan((sqrt(x)*b)/(sqrt(b)*sqrt 
(a)))*b**4*x**4 - 128*a**5 - 837*a**4*b*x - 1533*a**3*b**2*x**2 - 1155*a** 
2*b**3*x**3 - 315*a*b**4*x**4)/(64*sqrt(x)*a**6*(a**4 + 4*a**3*b*x + 6*a** 
2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4))