\(\int \frac {(d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^2} \, dx\) [1654]

Optimal result

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

[Out]

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)+231/8*e^3*(-a*e+b*d)^2*(e*x+d)^(1/2)/b^6

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {231 e^3 (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}}+\frac {231 e^3 \sqrt {d+e x} (b d-a e)^2}{8 b^6}+\frac {77 e^3 (d+e x)^{3/2} (b d-a e)}{8 b^5}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4} \]

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

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] - Dist[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] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[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/(b*(
m + n + 1))), x] + Dist[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] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{11/2}}{(a+b x)^4} \, dx \\ & = -\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}+\frac {(11 e) \int \frac {(d+e x)^{9/2}}{(a+b x)^3} \, dx}{6 b} \\ & = -\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 {\left (33 e^2\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{8 b^2} \\ & = -\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 {\left (231 e^3\right ) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{16 b^3} \\ & = \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 {\left (231 e^3 (b d-a e)\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{16 b^4} \\ & = \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 {\left (231 e^3 (b d-a e)^2\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^5} \\ & = \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 {\left (231 e^3 (b d-a e)^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^6} \\ & = \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 {\left (231 e^2 (b d-a e)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b^6} \\ & = \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} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.97 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (3465 a^5 e^5+1155 a^4 b e^4 (-7 d+8 e x)+231 a^3 b^2 e^3 \left (23 d^2-94 d e x+33 e^2 x^2\right )+99 a^2 b^3 e^2 \left (-5 d^3+146 d^2 e x-183 d e^2 x^2+16 e^3 x^3\right )-11 a b^4 e \left (10 d^4+130 d^3 e x-1119 d^2 e^2 x^2+352 d e^3 x^3+16 e^4 x^4\right )+b^5 \left (-40 d^5-310 d^4 e x-1335 d^3 e^2 x^2+2768 d^2 e^3 x^3+416 d e^4 x^4+48 e^5 x^5\right )\right )}{120 b^6 (a+b x)^3}-\frac {231 e^3 (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{13/2}} \]

[In]

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

[Out]

(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 - 11
19*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))

Maple [A] (verified)

Time = 2.55 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.14

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

[In]

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

[Out]

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*b^2*(e*x+d)^(5/2)-59/6*(a*e-b*d)*b*(e*x+d)^(3/
2)+(-71/16*a^2*e^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/((a*e-b*d)*b)^(1/2)
*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(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 = 0.39 (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=\left [\frac {3465 \, {\left (a^{3} b^{2} d^{2} e^{3} - 2 \, a^{4} b d e^{4} + a^{5} e^{5} + {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (a^{2} b^{3} d^{2} e^{3} - 2 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (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 \, {\left (26 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 16 \, {\left (173 \, b^{5} d^{2} e^{3} - 242 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 3 \, {\left (445 \, 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}\right )} x^{2} - 2 \, {\left (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}\right )} x\right )} \sqrt {e x + d}}{240 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}}, -\frac {3465 \, {\left (a^{3} b^{2} d^{2} e^{3} - 2 \, a^{4} b d e^{4} + a^{5} e^{5} + {\left (b^{5} d^{2} e^{3} - 2 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 3 \, {\left (a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} + 3 \, {\left (a^{2} b^{3} d^{2} e^{3} - 2 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (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 \, {\left (26 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} + 16 \, {\left (173 \, b^{5} d^{2} e^{3} - 242 \, a b^{4} d e^{4} + 99 \, a^{2} b^{3} e^{5}\right )} x^{3} - 3 \, {\left (445 \, 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}\right )} x^{2} - 2 \, {\left (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}\right )} x\right )} \sqrt {e x + d}}{120 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}}\right ] \]

[In]

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

[Out]

[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)*sqr
t((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 + 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 + 10857*a^3*b^2*d*e^4 - 4620*a^4*b*e^5)*x)*sqrt(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*(445*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 + 10857*a^3*b^2*d*e^4 - 4620*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^9*x^
3 + 3*a*b^8*x^2 + 3*a^2*b^7*x + a^3*b^6)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more 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 = 0.29 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.44 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {231 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {267 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{3} - 472 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{3} + 213 \, \sqrt {e x + d} b^{5} d^{5} e^{3} - 801 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{4} + 1888 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{4} - 1065 \, \sqrt {e x + d} a b^{4} d^{4} e^{4} + 801 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{5} - 2832 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{5} + 2130 \, \sqrt {e x + d} a^{2} b^{3} d^{3} e^{5} - 267 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{6} + 1888 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{6} - 2130 \, \sqrt {e x + d} a^{3} b^{2} d^{2} e^{6} - 472 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b e^{7} + 1065 \, \sqrt {e x + d} a^{4} b d e^{7} - 213 \, \sqrt {e x + d} a^{5} e^{8}}{24 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3} b^{6}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{16} e^{3} + 20 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{16} d e^{3} + 150 \, \sqrt {e x + d} b^{16} d^{2} e^{3} - 20 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{15} e^{4} - 300 \, \sqrt {e x + d} a b^{15} d e^{4} + 150 \, \sqrt {e x + d} a^{2} b^{14} e^{5}\right )}}{15 \, b^{20}} \]

[In]

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

[Out]

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(sqrt(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 = 9.49 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.46 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left (\frac {2\,e^3\,{\left (4\,b^4\,d-4\,a\,b^3\,e\right )}^2}{b^{12}}-\frac {12\,e^3\,{\left (a\,e-b\,d\right )}^2}{b^6}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (\frac {71\,a^5\,e^8}{8}-\frac {355\,a^4\,b\,d\,e^7}{8}+\frac {355\,a^3\,b^2\,d^2\,e^6}{4}-\frac {355\,a^2\,b^3\,d^3\,e^5}{4}+\frac {355\,a\,b^4\,d^4\,e^4}{8}-\frac {71\,b^5\,d^5\,e^3}{8}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {89\,a^3\,b^2\,e^6}{8}-\frac {267\,a^2\,b^3\,d\,e^5}{8}+\frac {267\,a\,b^4\,d^2\,e^4}{8}-\frac {89\,b^5\,d^3\,e^3}{8}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {59\,a^4\,b\,e^7}{3}-\frac {236\,a^3\,b^2\,d\,e^6}{3}+118\,a^2\,b^3\,d^2\,e^5-\frac {236\,a\,b^4\,d^3\,e^4}{3}+\frac {59\,b^5\,d^4\,e^3}{3}\right )}{b^9\,{\left (d+e\,x\right )}^3-\left (3\,b^9\,d-3\,a\,b^8\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^7\,e^2-6\,a\,b^8\,d\,e+3\,b^9\,d^2\right )-b^9\,d^3+a^3\,b^6\,e^3-3\,a^2\,b^7\,d\,e^2+3\,a\,b^8\,d^2\,e}+\frac {2\,e^3\,{\left (d+e\,x\right )}^{5/2}}{5\,b^4}+\frac {2\,e^3\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^8}-\frac {231\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^3\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^6-3\,a^2\,b\,d\,e^5+3\,a\,b^2\,d^2\,e^4-b^3\,d^3\,e^3}\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{8\,b^{13/2}} \]

[In]

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

[Out]

((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 - (3
55*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))