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

Optimal. Leaf size=201 \[ \frac {231 e^3 (b d-a e)^2 \sqrt {d+e x}}{8 b^6}+\frac {77 e^3 (b d-a e) (d+e x)^{3/2}}{8 b^5}+\frac {231 e^3 (d+e x)^{5/2}}{40 b^4}-\frac {33 e^2 (d+e x)^{7/2}}{8 b^3 (a+b x)}-\frac {11 e (d+e x)^{9/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{11/2}}{3 b (a+b x)^3}-\frac {231 e^3 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{13/2}} \]

[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]
time = 0.09, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \begin {gather*} -\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}}+\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\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]
time = 1.02, size = 277, normalized size = 1.38 \begin {gather*} \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} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(165)=330\).
time = 0.73, size = 371, normalized size = 1.85

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {\frac {b^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 a b e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 b^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+10 a^{2} e^{2} \sqrt {e x +d}-20 a b d e \sqrt {e x +d}+10 b^{2} d^{2} \sqrt {e x +d}}{b^{6}}-\frac {\frac {\left (-\frac {89}{16} a^{3} b^{2} e^{3}+\frac {267}{16} a^{2} b^{3} d \,e^{2}-\frac {267}{16} a \,b^{4} d^{2} e +\frac {89}{16} b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {59 b \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {71}{16} a^{5} e^{5}+\frac {355}{16} a^{4} b d \,e^{4}-\frac {355}{8} a^{3} b^{2} d^{2} e^{3}+\frac {355}{8} a^{2} b^{3} d^{3} e^{2}-\frac {355}{16} a \,b^{4} d^{4} e +\frac {71}{16} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {231 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 \sqrt {b \left (a e -b d \right )}}}{b^{6}}\right )\)	\(371\)
default	\(2 e^{3} \left (\frac {\frac {b^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 a b e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 b^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+10 a^{2} e^{2} \sqrt {e x +d}-20 a b d e \sqrt {e x +d}+10 b^{2} d^{2} \sqrt {e x +d}}{b^{6}}-\frac {\frac {\left (-\frac {89}{16} a^{3} b^{2} e^{3}+\frac {267}{16} a^{2} b^{3} d \,e^{2}-\frac {267}{16} a \,b^{4} d^{2} e +\frac {89}{16} b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {59 b \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {71}{16} a^{5} e^{5}+\frac {355}{16} a^{4} b d \,e^{4}-\frac {355}{8} a^{3} b^{2} d^{2} e^{3}+\frac {355}{8} a^{2} b^{3} d^{3} e^{2}-\frac {355}{16} a \,b^{4} d^{4} e +\frac {71}{16} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (\left (e x +d \right ) b +a e -b d \right )^{3}}+\frac {231 \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right )}{16 \sqrt {b \left (a e -b d \right )}}}{b^{6}}\right )\)	\(371\)
risch	\(\frac {2 e^{3} \left (3 b^{2} x^{2} e^{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}\right ) \sqrt {e x +d}}{15 b^{6}}-\frac {267 e^{5} \left (e x +d \right )^{\frac {5}{2}} a^{2} d}{8 b^{3} \left (b e x +a e \right )^{3}}+\frac {267 e^{4} \left (e x +d \right )^{\frac {5}{2}} a \,d^{2}}{8 b^{2} \left (b e x +a e \right )^{3}}-\frac {236 e^{6} \left (e x +d \right )^{\frac {3}{2}} a^{3} d}{3 b^{4} \left (b e x +a e \right )^{3}}+\frac {118 e^{5} \left (e x +d \right )^{\frac {3}{2}} a^{2} d^{2}}{b^{3} \left (b e x +a e \right )^{3}}-\frac {236 e^{4} \left (e x +d \right )^{\frac {3}{2}} a \,d^{3}}{3 b^{2} \left (b e x +a e \right )^{3}}+\frac {89 e^{6} \left (e x +d \right )^{\frac {5}{2}} a^{3}}{8 b^{4} \left (b e x +a e \right )^{3}}+\frac {59 e^{7} \left (e x +d \right )^{\frac {3}{2}} a^{4}}{3 b^{5} \left (b e x +a e \right )^{3}}-\frac {89 e^{3} \left (e x +d \right )^{\frac {5}{2}} d^{3}}{8 b \left (b e x +a e \right )^{3}}+\frac {59 e^{3} \left (e x +d \right )^{\frac {3}{2}} d^{4}}{3 b \left (b e x +a e \right )^{3}}+\frac {71 e^{8} \sqrt {e x +d}\, a^{5}}{8 b^{6} \left (b e x +a e \right )^{3}}-\frac {71 e^{3} \sqrt {e x +d}\, d^{5}}{8 b \left (b e x +a e \right )^{3}}-\frac {231 e^{6} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{3}}{8 b^{6} \sqrt {b \left (a e -b d \right )}}+\frac {231 e^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d^{3}}{8 b^{3} \sqrt {b \left (a e -b d \right )}}-\frac {355 e^{7} \sqrt {e x +d}\, a^{4} d}{8 b^{5} \left (b e x +a e \right )^{3}}+\frac {355 e^{6} \sqrt {e x +d}\, a^{3} d^{2}}{4 b^{4} \left (b e x +a e \right )^{3}}-\frac {355 e^{5} \sqrt {e x +d}\, a^{2} d^{3}}{4 b^{3} \left (b e x +a e \right )^{3}}+\frac {355 e^{4} \sqrt {e x +d}\, a \,d^{4}}{8 b^{2} \left (b e x +a e \right )^{3}}+\frac {693 e^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} d}{8 b^{5} \sqrt {b \left (a e -b d \right )}}-\frac {693 e^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a \,d^{2}}{8 b^{4} \sqrt {b \left (a e -b d \right )}}\)	\(684\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^3*(1/b^6*(1/5*b^2*(e*x+d)^(5/2)-4/3*a*b*e*(e*x+d)^(3/2)+4/3*b^2*d*(e*x+d)^(3/2)+10*a^2*e^2*(e*x+d)^(1/2)-2
0*a*b*d*e*(e*x+d)^(1/2)+10*b^2*d^2*(e*x+d)^(1/2))-1/b^6*(((-89/16*a^3*b^2*e^3+267/16*a^2*b^3*d*e^2-267/16*a*b^
4*d^2*e+89/16*b^5*d^3)*(e*x+d)^(5/2)-59/6*b*(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)*(e
*x+d)^(3/2)+(-71/16*a^5*e^5+355/16*a^4*b*d*e^4-355/8*a^3*b^2*d^2*e^3+355/8*a^2*b^3*d^3*e^2-355/16*a*b^4*d^4*e+
71/16*b^5*d^5)*(e*x+d)^(1/2))/((e*x+d)*b+a*e-b*d)^3+231/16*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/(b*(a
*e-b*d))^(1/2)*arctan(b*(e*x+d)^(1/2)/(b*(a*e-b*d))^(1/2))))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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(b*d-%e*a>0)', see `assume?` fo
r more detai

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (172) = 344\).
time = 2.97, size = 930, normalized size = 4.63 \begin {gather*} \left [\frac {3465 \, {\left ({\left (a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} + 3 \, a^{4} b x + a^{5}\right )} e^{5} - 2 \, {\left (a b^{4} d x^{3} + 3 \, a^{2} b^{3} d x^{2} + 3 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4} + {\left (b^{5} d^{2} x^{3} + 3 \, a b^{4} d^{2} x^{2} + 3 \, a^{2} b^{3} d^{2} x + a^{3} b^{2} d^{2}\right )} e^{3}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) - 2 \, {\left (40 \, b^{5} d^{5} - {\left (48 \, b^{5} x^{5} - 176 \, a b^{4} x^{4} + 1584 \, a^{2} b^{3} x^{3} + 7623 \, a^{3} b^{2} x^{2} + 9240 \, a^{4} b x + 3465 \, a^{5}\right )} e^{5} - {\left (416 \, b^{5} d x^{4} - 3872 \, a b^{4} d x^{3} - 18117 \, a^{2} b^{3} d x^{2} - 21714 \, a^{3} b^{2} d x - 8085 \, a^{4} b d\right )} e^{4} - {\left (2768 \, b^{5} d^{2} x^{3} + 12309 \, a b^{4} d^{2} x^{2} + 14454 \, a^{2} b^{3} d^{2} x + 5313 \, a^{3} b^{2} d^{2}\right )} e^{3} + 5 \, {\left (267 \, b^{5} d^{3} x^{2} + 286 \, a b^{4} d^{3} x + 99 \, a^{2} b^{3} d^{3}\right )} e^{2} + 10 \, {\left (31 \, b^{5} d^{4} x + 11 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + 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 ({\left (a^{2} b^{3} x^{3} + 3 \, a^{3} b^{2} x^{2} + 3 \, a^{4} b x + a^{5}\right )} e^{5} - 2 \, {\left (a b^{4} d x^{3} + 3 \, a^{2} b^{3} d x^{2} + 3 \, a^{3} b^{2} d x + a^{4} b d\right )} e^{4} + {\left (b^{5} d^{2} x^{3} + 3 \, a b^{4} d^{2} x^{2} + 3 \, a^{2} b^{3} d^{2} x + a^{3} b^{2} d^{2}\right )} e^{3}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) + {\left (40 \, b^{5} d^{5} - {\left (48 \, b^{5} x^{5} - 176 \, a b^{4} x^{4} + 1584 \, a^{2} b^{3} x^{3} + 7623 \, a^{3} b^{2} x^{2} + 9240 \, a^{4} b x + 3465 \, a^{5}\right )} e^{5} - {\left (416 \, b^{5} d x^{4} - 3872 \, a b^{4} d x^{3} - 18117 \, a^{2} b^{3} d x^{2} - 21714 \, a^{3} b^{2} d x - 8085 \, a^{4} b d\right )} e^{4} - {\left (2768 \, b^{5} d^{2} x^{3} + 12309 \, a b^{4} d^{2} x^{2} + 14454 \, a^{2} b^{3} d^{2} x + 5313 \, a^{3} b^{2} d^{2}\right )} e^{3} + 5 \, {\left (267 \, b^{5} d^{3} x^{2} + 286 \, a b^{4} d^{3} x + 99 \, a^{2} b^{3} d^{3}\right )} e^{2} + 10 \, {\left (31 \, b^{5} d^{4} x + 11 \, a b^{4} d^{4}\right )} e\right )} \sqrt {x e + 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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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^2*b^3*x^3 + 3*a^3*b^2*x^2 + 3*a^4*b*x + a^5)*e^5 - 2*(a*b^4*d*x^3 + 3*a^2*b^3*d*x^2 + 3*a^3*b
^2*d*x + a^4*b*d)*e^4 + (b^5*d^2*x^3 + 3*a*b^4*d^2*x^2 + 3*a^2*b^3*d^2*x + a^3*b^2*d^2)*e^3)*sqrt((b*d - a*e)/
b)*log((2*b*d - 2*sqrt(x*e + d)*b*sqrt((b*d - a*e)/b) + (b*x - a)*e)/(b*x + a)) - 2*(40*b^5*d^5 - (48*b^5*x^5
- 176*a*b^4*x^4 + 1584*a^2*b^3*x^3 + 7623*a^3*b^2*x^2 + 9240*a^4*b*x + 3465*a^5)*e^5 - (416*b^5*d*x^4 - 3872*a
*b^4*d*x^3 - 18117*a^2*b^3*d*x^2 - 21714*a^3*b^2*d*x - 8085*a^4*b*d)*e^4 - (2768*b^5*d^2*x^3 + 12309*a*b^4*d^2
*x^2 + 14454*a^2*b^3*d^2*x + 5313*a^3*b^2*d^2)*e^3 + 5*(267*b^5*d^3*x^2 + 286*a*b^4*d^3*x + 99*a^2*b^3*d^3)*e^
2 + 10*(31*b^5*d^4*x + 11*a*b^4*d^4)*e)*sqrt(x*e + 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^2*b^3*x^3 + 3*a^3*b^2*x^2 + 3*a^4*b*x + a^5)*e^5 - 2*(a*b^4*d*x^3 + 3*a^2*b^3*d*x^2 + 3*a^3*b^2*d*x
+ a^4*b*d)*e^4 + (b^5*d^2*x^3 + 3*a*b^4*d^2*x^2 + 3*a^2*b^3*d^2*x + a^3*b^2*d^2)*e^3)*sqrt(-(b*d - a*e)/b)*ar
ctan(-sqrt(x*e + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) + (40*b^5*d^5 - (48*b^5*x^5 - 176*a*b^4*x^4 + 1584*a^2
*b^3*x^3 + 7623*a^3*b^2*x^2 + 9240*a^4*b*x + 3465*a^5)*e^5 - (416*b^5*d*x^4 - 3872*a*b^4*d*x^3 - 18117*a^2*b^3
*d*x^2 - 21714*a^3*b^2*d*x - 8085*a^4*b*d)*e^4 - (2768*b^5*d^2*x^3 + 12309*a*b^4*d^2*x^2 + 14454*a^2*b^3*d^2*x
+ 5313*a^3*b^2*d^2)*e^3 + 5*(267*b^5*d^3*x^2 + 286*a*b^4*d^3*x + 99*a^2*b^3*d^3)*e^2 + 10*(31*b^5*d^4*x + 11*
a*b^4*d^4)*e)*sqrt(x*e + 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
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (172) = 344\).
time = 0.92, size = 491, normalized size = 2.44 \begin {gather*} \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 {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {267 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{3} - 472 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{3} + 213 \, \sqrt {x e + d} b^{5} d^{5} e^{3} - 801 \, {\left (x e + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{4} + 1888 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{4} - 1065 \, \sqrt {x e + d} a b^{4} d^{4} e^{4} + 801 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{5} - 2832 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{5} + 2130 \, \sqrt {x e + d} a^{2} b^{3} d^{3} e^{5} - 267 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{6} + 1888 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{6} - 2130 \, \sqrt {x e + d} a^{3} b^{2} d^{2} e^{6} - 472 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{4} b e^{7} + 1065 \, \sqrt {x e + d} a^{4} b d e^{7} - 213 \, \sqrt {x e + d} a^{5} e^{8}}{24 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{6}} + \frac {2 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{16} e^{3} + 20 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{16} d e^{3} + 150 \, \sqrt {x e + d} b^{16} d^{2} e^{3} - 20 \, {\left (x e + d\right )}^{\frac {3}{2}} a b^{15} e^{4} - 300 \, \sqrt {x e + d} a b^{15} d e^{4} + 150 \, \sqrt {x e + d} a^{2} b^{14} e^{5}\right )}}{15 \, b^{20}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

________________________________________________________________________________________

Mupad [B]
time = 0.68, size = 495, normalized size = 2.46 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________