\(\int \frac {(d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [179]

Optimal result

Integrand size = 28, antiderivative size = 201 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {231 e^3 (b d-a e)^2 \sqrt {d+e x}}{8 b^6}+\frac {77 e^3 (b d-a e) (d+e x)^{3/2}}{8 b^5}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}-\frac {231 e^3 (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}} \] Output:

231/8*e^3*(-a*e+b*d)^2*(e*x+d)^(1/2)/b^6+77/8*e^3*(-a*e+b*d)*(e*x+d)^(3/2) 
/b^5+231/40*e^3*(e*x+d)^(5/2)/b^4-33/8*e^2*(e*x+d)^(7/2)/b^3/(b*x+a)-11/12 
*e*(e*x+d)^(9/2)/b^2/(b*x+a)^2-1/3*(e*x+d)^(11/2)/b/(b*x+a)^3-231/8*e^3*(- 
a*e+b*d)^(5/2)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(13/2)
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (3465 a^5 e^5+1155 a^4 b e^4 (-7 d+8 e x)+231 a^3 b^2 e^3 \left (23 d^2-94 d e x+33 e^2 x^2\right )+99 a^2 b^3 e^2 \left (-5 d^3+146 d^2 e x-183 d e^2 x^2+16 e^3 x^3\right )-11 a b^4 e \left (10 d^4+130 d^3 e x-1119 d^2 e^2 x^2+352 d e^3 x^3+16 e^4 x^4\right )+b^5 \left (-40 d^5-310 d^4 e x-1335 d^3 e^2 x^2+2768 d^2 e^3 x^3+416 d e^4 x^4+48 e^5 x^5\right )\right )}{120 b^6 (a+b x)^3}-\frac {231 e^3 (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{13/2}} \] Input:

Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
 

Output:

(Sqrt[d + e*x]*(3465*a^5*e^5 + 1155*a^4*b*e^4*(-7*d + 8*e*x) + 231*a^3*b^2 
*e^3*(23*d^2 - 94*d*e*x + 33*e^2*x^2) + 99*a^2*b^3*e^2*(-5*d^3 + 146*d^2*e 
*x - 183*d*e^2*x^2 + 16*e^3*x^3) - 11*a*b^4*e*(10*d^4 + 130*d^3*e*x - 1119 
*d^2*e^2*x^2 + 352*d*e^3*x^3 + 16*e^4*x^4) + b^5*(-40*d^5 - 310*d^4*e*x - 
1335*d^3*e^2*x^2 + 2768*d^2*e^3*x^3 + 416*d*e^4*x^4 + 48*e^5*x^5)))/(120*b 
^6*(a + b*x)^3) - (231*e^3*(-(b*d) + a*e)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[d + e 
*x])/Sqrt[-(b*d) + a*e]])/(8*b^(13/2))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1098, 27, 51, 51, 51, 60, 60, 60, 73, 221}

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.

\(\displaystyle \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

\(\Big \downarrow \) 1098

\(\displaystyle b^4 \int \frac {(d+e x)^{11/2}}{b^4 (a+b x)^4}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \int \frac {(d+e x)^{11/2}}{(a+b x)^4}dx\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {11 e \int \frac {(d+e x)^{9/2}}{(a+b x)^3}dx}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {11 e \left (\frac {9 e \int \frac {(d+e x)^{7/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {11 e \left (\frac {9 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \int \frac {(d+e x)^{3/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\)

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {11 e \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}\)

Input:

Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^2,x]
 

Output:

-1/3*(d + e*x)^(11/2)/(b*(a + b*x)^3) + (11*e*(-1/2*(d + e*x)^(9/2)/(b*(a 
+ b*x)^2) + (9*e*(-((d + e*x)^(7/2)/(b*(a + b*x))) + (7*e*((2*(d + e*x)^(5 
/2))/(5*b) + ((b*d - a*e)*((2*(d + e*x)^(3/2))/(3*b) + ((b*d - a*e)*((2*Sq 
rt[d + e*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b 
*d - a*e]])/b^(3/2)))/b))/b))/(2*b)))/(4*b)))/(6*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 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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1098

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[1/c^p   Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ 
{a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.14

method	result	size
risch	\(\frac {2 e^{3} \left (3 b^{2} e^{2} x^{2}-20 x a b \,e^{2}+26 b^{2} d e x +150 e^{2} a^{2}-320 a b d e +173 b^{2} d^{2}\right ) \sqrt {e x +d}}{15 b^{6}}-\frac {\left (2 e^{3} a^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) e^{3} \left (\frac {-\frac {89 \left (e x +d \right )^{\frac {5}{2}} b^{2}}{16}-\frac {59 b \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {71}{16} e^{2} a^{2}+\frac {71}{8} a b d e -\frac {71}{16} b^{2} d^{2}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {231 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 \sqrt {b \left (a e -b d \right )}}\right )}{b^{6}}\)	\(229\)
pseudoelliptic	\(-\frac {231 \left (e^{3} \left (b x +a \right )^{3} \left (a e -b d \right )^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )-\sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}\, \left (\left (\frac {16}{1155} e^{5} x^{5}+\frac {416}{3465} d \,e^{4} x^{4}+\frac {2768}{3465} d^{2} e^{3} x^{3}-\frac {89}{231} d^{3} e^{2} x^{2}-\frac {62}{693} d^{4} e x -\frac {8}{693} d^{5}\right ) b^{5}-\frac {2 e \left (\frac {8}{5} e^{4} x^{4}+\frac {176}{5} d \,e^{3} x^{3}-\frac {1119}{10} d^{2} e^{2} x^{2}+13 d^{3} e x +d^{4}\right ) a \,b^{4}}{63}-\frac {e^{2} a^{2} \left (-\frac {16}{5} e^{3} x^{3}+\frac {183}{5} d \,e^{2} x^{2}-\frac {146}{5} d^{2} e x +d^{3}\right ) b^{3}}{7}+\frac {23 e^{3} \left (\frac {33}{23} e^{2} x^{2}-\frac {94}{23} d e x +d^{2}\right ) a^{3} b^{2}}{15}-\frac {7 e^{4} \left (-\frac {8 e x}{7}+d \right ) a^{4} b}{3}+e^{5} a^{5}\right )\right )}{8 \sqrt {b \left (a e -b d \right )}\, b^{6} \left (b x +a \right )^{3}}\)	\(282\)
derivativedivides	\(2 e^{3} \left (\frac {\frac {\left (e x +d \right )^{\frac {5}{2}} b^{2}}{5}-\frac {4 a b e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 b^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+10 a^{2} e^{2} \sqrt {e x +d}-20 a b d e \sqrt {e x +d}+10 b^{2} d^{2} \sqrt {e x +d}}{b^{6}}-\frac {\frac {\left (-\frac {89}{16} a^{3} b^{2} e^{3}+\frac {267}{16} a^{2} b^{3} d \,e^{2}-\frac {267}{16} a \,b^{4} d^{2} e +\frac {89}{16} b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {59 b \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {71}{16} e^{5} a^{5}+\frac {355}{16} a^{4} b d \,e^{4}-\frac {355}{8} a^{3} b^{2} d^{2} e^{3}+\frac {355}{8} a^{2} b^{3} d^{3} e^{2}-\frac {355}{16} a \,b^{4} d^{4} e +\frac {71}{16} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {231 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 \sqrt {b \left (a e -b d \right )}}}{b^{6}}\right )\)	\(371\)
default	\(2 e^{3} \left (\frac {\frac {\left (e x +d \right )^{\frac {5}{2}} b^{2}}{5}-\frac {4 a b e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 b^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+10 a^{2} e^{2} \sqrt {e x +d}-20 a b d e \sqrt {e x +d}+10 b^{2} d^{2} \sqrt {e x +d}}{b^{6}}-\frac {\frac {\left (-\frac {89}{16} a^{3} b^{2} e^{3}+\frac {267}{16} a^{2} b^{3} d \,e^{2}-\frac {267}{16} a \,b^{4} d^{2} e +\frac {89}{16} b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {59 b \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {71}{16} e^{5} a^{5}+\frac {355}{16} a^{4} b d \,e^{4}-\frac {355}{8} a^{3} b^{2} d^{2} e^{3}+\frac {355}{8} a^{2} b^{3} d^{3} e^{2}-\frac {355}{16} a \,b^{4} d^{4} e +\frac {71}{16} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {231 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 \sqrt {b \left (a e -b d \right )}}}{b^{6}}\right )\)	\(371\)

Input:

int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
 

Output:

2/15*e^3*(3*b^2*e^2*x^2-20*a*b*e^2*x+26*b^2*d*e*x+150*a^2*e^2-320*a*b*d*e+ 
173*b^2*d^2)*(e*x+d)^(1/2)/b^6-1/b^6*(2*a^3*e^3-6*a^2*b*d*e^2+6*a*b^2*d^2* 
e-2*b^3*d^3)*e^3*((-89/16*(e*x+d)^(5/2)*b^2-59/6*b*(a*e-b*d)*(e*x+d)^(3/2) 
+(-71/16*e^2*a^2+71/8*a*b*d*e-71/16*b^2*d^2)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e 
-b*d)^3+231/16/(b*(a*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1 
/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (165) = 330\).

Time = 0.12 (sec) , antiderivative size = 994, normalized size of antiderivative = 4.95 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fricas")
 

Output:

[1/240*(3465*(a^3*b^2*d^2*e^3 - 2*a^4*b*d*e^4 + a^5*e^5 + (b^5*d^2*e^3 - 2 
*a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + 3*(a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 + a^3 
*b^2*e^5)*x^2 + 3*(a^2*b^3*d^2*e^3 - 2*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*sqrt( 
(b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d - a* 
e)/b))/(b*x + a)) + 2*(48*b^5*e^5*x^5 - 40*b^5*d^5 - 110*a*b^4*d^4*e - 495 
*a^2*b^3*d^3*e^2 + 5313*a^3*b^2*d^2*e^3 - 8085*a^4*b*d*e^4 + 3465*a^5*e^5 
+ 16*(26*b^5*d*e^4 - 11*a*b^4*e^5)*x^4 + 16*(173*b^5*d^2*e^3 - 242*a*b^4*d 
*e^4 + 99*a^2*b^3*e^5)*x^3 - 3*(445*b^5*d^3*e^2 - 4103*a*b^4*d^2*e^3 + 603 
9*a^2*b^3*d*e^4 - 2541*a^3*b^2*e^5)*x^2 - 2*(155*b^5*d^4*e + 715*a*b^4*d^3 
*e^2 - 7227*a^2*b^3*d^2*e^3 + 10857*a^3*b^2*d*e^4 - 4620*a^4*b*e^5)*x)*sqr 
t(e*x + d))/(b^9*x^3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6), -1/120*(3465* 
(a^3*b^2*d^2*e^3 - 2*a^4*b*d*e^4 + a^5*e^5 + (b^5*d^2*e^3 - 2*a*b^4*d*e^4 
+ a^2*b^3*e^5)*x^3 + 3*(a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 
 + 3*(a^2*b^3*d^2*e^3 - 2*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*sqrt(-(b*d - a*e)/ 
b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (48*b^5*e^5 
*x^5 - 40*b^5*d^5 - 110*a*b^4*d^4*e - 495*a^2*b^3*d^3*e^2 + 5313*a^3*b^2*d 
^2*e^3 - 8085*a^4*b*d*e^4 + 3465*a^5*e^5 + 16*(26*b^5*d*e^4 - 11*a*b^4*e^5 
)*x^4 + 16*(173*b^5*d^2*e^3 - 242*a*b^4*d*e^4 + 99*a^2*b^3*e^5)*x^3 - 3*(4 
45*b^5*d^3*e^2 - 4103*a*b^4*d^2*e^3 + 6039*a^2*b^3*d*e^4 - 2541*a^3*b^2*e^ 
5)*x^2 - 2*(155*b^5*d^4*e + 715*a*b^4*d^3*e^2 - 7227*a^2*b^3*d^2*e^3 + ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (165) = 330\).

Time = 0.21 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.44 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {231 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {267 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{3} - 472 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{3} + 213 \, \sqrt {e x + d} b^{5} d^{5} e^{3} - 801 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{4} + 1888 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{4} - 1065 \, \sqrt {e x + d} a b^{4} d^{4} e^{4} + 801 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{5} - 2832 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{5} + 2130 \, \sqrt {e x + d} a^{2} b^{3} d^{3} e^{5} - 267 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{6} + 1888 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{6} - 2130 \, \sqrt {e x + d} a^{3} b^{2} d^{2} e^{6} - 472 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b e^{7} + 1065 \, \sqrt {e x + d} a^{4} b d e^{7} - 213 \, \sqrt {e x + d} a^{5} e^{8}}{24 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3} b^{6}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{16} e^{3} + 20 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{16} d e^{3} + 150 \, \sqrt {e x + d} b^{16} d^{2} e^{3} - 20 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{15} e^{4} - 300 \, \sqrt {e x + d} a b^{15} d e^{4} + 150 \, \sqrt {e x + d} a^{2} b^{14} e^{5}\right )}}{15 \, b^{20}} \] Input:

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac")
 

Output:

231/8*(b^3*d^3*e^3 - 3*a*b^2*d^2*e^4 + 3*a^2*b*d*e^5 - a^3*e^6)*arctan(sqr 
t(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6) - 1/24*(267* 
(e*x + d)^(5/2)*b^5*d^3*e^3 - 472*(e*x + d)^(3/2)*b^5*d^4*e^3 + 213*sqrt(e 
*x + d)*b^5*d^5*e^3 - 801*(e*x + d)^(5/2)*a*b^4*d^2*e^4 + 1888*(e*x + d)^( 
3/2)*a*b^4*d^3*e^4 - 1065*sqrt(e*x + d)*a*b^4*d^4*e^4 + 801*(e*x + d)^(5/2 
)*a^2*b^3*d*e^5 - 2832*(e*x + d)^(3/2)*a^2*b^3*d^2*e^5 + 2130*sqrt(e*x + d 
)*a^2*b^3*d^3*e^5 - 267*(e*x + d)^(5/2)*a^3*b^2*e^6 + 1888*(e*x + d)^(3/2) 
*a^3*b^2*d*e^6 - 2130*sqrt(e*x + d)*a^3*b^2*d^2*e^6 - 472*(e*x + d)^(3/2)* 
a^4*b*e^7 + 1065*sqrt(e*x + d)*a^4*b*d*e^7 - 213*sqrt(e*x + d)*a^5*e^8)/(( 
(e*x + d)*b - b*d + a*e)^3*b^6) + 2/15*(3*(e*x + d)^(5/2)*b^16*e^3 + 20*(e 
*x + d)^(3/2)*b^16*d*e^3 + 150*sqrt(e*x + d)*b^16*d^2*e^3 - 20*(e*x + d)^( 
3/2)*a*b^15*e^4 - 300*sqrt(e*x + d)*a*b^15*d*e^4 + 150*sqrt(e*x + d)*a^2*b 
^14*e^5)/b^20
 

Mupad [B] (verification not implemented)

Time = 9.27 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.46 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left (\frac {2\,e^3\,{\left (4\,b^4\,d-4\,a\,b^3\,e\right )}^2}{b^{12}}-\frac {12\,e^3\,{\left (a\,e-b\,d\right )}^2}{b^6}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (\frac {71\,a^5\,e^8}{8}-\frac {355\,a^4\,b\,d\,e^7}{8}+\frac {355\,a^3\,b^2\,d^2\,e^6}{4}-\frac {355\,a^2\,b^3\,d^3\,e^5}{4}+\frac {355\,a\,b^4\,d^4\,e^4}{8}-\frac {71\,b^5\,d^5\,e^3}{8}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {89\,a^3\,b^2\,e^6}{8}-\frac {267\,a^2\,b^3\,d\,e^5}{8}+\frac {267\,a\,b^4\,d^2\,e^4}{8}-\frac {89\,b^5\,d^3\,e^3}{8}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {59\,a^4\,b\,e^7}{3}-\frac {236\,a^3\,b^2\,d\,e^6}{3}+118\,a^2\,b^3\,d^2\,e^5-\frac {236\,a\,b^4\,d^3\,e^4}{3}+\frac {59\,b^5\,d^4\,e^3}{3}\right )}{b^9\,{\left (d+e\,x\right )}^3-\left (3\,b^9\,d-3\,a\,b^8\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^7\,e^2-6\,a\,b^8\,d\,e+3\,b^9\,d^2\right )-b^9\,d^3+a^3\,b^6\,e^3-3\,a^2\,b^7\,d\,e^2+3\,a\,b^8\,d^2\,e}+\frac {2\,e^3\,{\left (d+e\,x\right )}^{5/2}}{5\,b^4}+\frac {2\,e^3\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^8}-\frac {231\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^3\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a\,b^2\,d^2\,e^4-b^3\,d^3\,e^3}\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{8\,b^{13/2}} \] Input:

int((d + e*x)^(11/2)/(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
 

Output:

((2*e^3*(4*b^4*d - 4*a*b^3*e)^2)/b^12 - (12*e^3*(a*e - b*d)^2)/b^6)*(d + e 
*x)^(1/2) + ((d + e*x)^(1/2)*((71*a^5*e^8)/8 - (71*b^5*d^5*e^3)/8 + (355*a 
*b^4*d^4*e^4)/8 - (355*a^2*b^3*d^3*e^5)/4 + (355*a^3*b^2*d^2*e^6)/4 - (355 
*a^4*b*d*e^7)/8) + (d + e*x)^(5/2)*((89*a^3*b^2*e^6)/8 - (89*b^5*d^3*e^3)/ 
8 + (267*a*b^4*d^2*e^4)/8 - (267*a^2*b^3*d*e^5)/8) + (d + e*x)^(3/2)*((59* 
a^4*b*e^7)/3 + (59*b^5*d^4*e^3)/3 - (236*a*b^4*d^3*e^4)/3 - (236*a^3*b^2*d 
*e^6)/3 + 118*a^2*b^3*d^2*e^5))/(b^9*(d + e*x)^3 - (3*b^9*d - 3*a*b^8*e)*( 
d + e*x)^2 + (d + e*x)*(3*b^9*d^2 + 3*a^2*b^7*e^2 - 6*a*b^8*d*e) - b^9*d^3 
 + a^3*b^6*e^3 - 3*a^2*b^7*d*e^2 + 3*a*b^8*d^2*e) + (2*e^3*(d + e*x)^(5/2) 
)/(5*b^4) + (2*e^3*(4*b^4*d - 4*a*b^3*e)*(d + e*x)^(3/2))/(3*b^8) - (231*e 
^3*atan((b^(1/2)*e^3*(a*e - b*d)^(5/2)*(d + e*x)^(1/2))/(a^3*e^6 - b^3*d^3 
*e^3 + 3*a*b^2*d^2*e^4 - 3*a^2*b*d*e^5))*(a*e - b*d)^(5/2))/(8*b^(13/2))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1000, normalized size of antiderivative = 4.98 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^2,x)
 

Output:

( - 3465*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e 
- b*d)))*a**5*e**5 + 6930*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/( 
sqrt(b)*sqrt(a*e - b*d)))*a**4*b*d*e**4 - 10395*sqrt(b)*sqrt(a*e - b*d)*at 
an((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**4*b*e**5*x - 3465*sqrt( 
b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**3* 
b**2*d**2*e**3 + 20790*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqr 
t(b)*sqrt(a*e - b*d)))*a**3*b**2*d*e**4*x - 10395*sqrt(b)*sqrt(a*e - b*d)* 
atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**3*b**2*e**5*x**2 - 10 
395*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d 
)))*a**2*b**3*d**2*e**3*x + 20790*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e 
*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2*b**3*d*e**4*x**2 - 3465*sqrt(b)*sqr 
t(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a**2*b**3*e 
**5*x**3 - 10395*sqrt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*s 
qrt(a*e - b*d)))*a*b**4*d**2*e**3*x**2 + 6930*sqrt(b)*sqrt(a*e - b*d)*atan 
((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*a*b**4*d*e**4*x**3 - 3465*sq 
rt(b)*sqrt(a*e - b*d)*atan((sqrt(d + e*x)*b)/(sqrt(b)*sqrt(a*e - b*d)))*b* 
*5*d**2*e**3*x**3 + 3465*sqrt(d + e*x)*a**5*b*e**5 - 8085*sqrt(d + e*x)*a* 
*4*b**2*d*e**4 + 9240*sqrt(d + e*x)*a**4*b**2*e**5*x + 5313*sqrt(d + e*x)* 
a**3*b**3*d**2*e**3 - 21714*sqrt(d + e*x)*a**3*b**3*d*e**4*x + 7623*sqrt(d 
 + e*x)*a**3*b**3*e**5*x**2 - 495*sqrt(d + e*x)*a**2*b**4*d**3*e**2 + 1...