\(\int \frac {(d+e x)^{11/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\) [1666]

Optimal result

Integrand size = 28, antiderivative size = 197 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}-\frac {693 e^5 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}} \]

[Out]

-231/128*e^4*(e*x+d)^(3/2)/b^5/(b*x+a)-231/320*e^3*(e*x+d)^(5/2)/b^4/(b*x+a)^2-33/80*e^2*(e*x+d)^(7/2)/b^3/(b*
x+a)^3-11/40*e*(e*x+d)^(9/2)/b^2/(b*x+a)^4-1/5*(e*x+d)^(11/2)/b/(b*x+a)^5-693/128*e^5*arctanh(b^(1/2)*(e*x+d)^
(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(13/2)+693/128*e^5*(e*x+d)^(1/2)/b^6

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 197, 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} \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {693 e^5 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {693 e^5 \sqrt {d+e x}}{128 b^6} \]

[In]

Int[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(693*e^5*Sqrt[d + e*x])/(128*b^6) - (231*e^4*(d + e*x)^(3/2))/(128*b^5*(a + b*x)) - (231*e^3*(d + e*x)^(5/2))/
(320*b^4*(a + b*x)^2) - (33*e^2*(d + e*x)^(7/2))/(80*b^3*(a + b*x)^3) - (11*e*(d + e*x)^(9/2))/(40*b^2*(a + b*
x)^4) - (d + e*x)^(11/2)/(5*b*(a + b*x)^5) - (693*e^5*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d
- a*e]])/(128*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)^6} \, dx \\ & = -\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {(11 e) \int \frac {(d+e x)^{9/2}}{(a+b x)^5} \, dx}{10 b} \\ & = -\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (99 e^2\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^4} \, dx}{80 b^2} \\ & = -\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (231 e^3\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{160 b^3} \\ & = -\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (231 e^4\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{128 b^4} \\ & = -\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^5\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^5} \\ & = \frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^5 (b d-a e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^6} \\ & = \frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}+\frac {\left (693 e^4 (b d-a e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^6} \\ & = \frac {693 e^5 \sqrt {d+e x}}{128 b^6}-\frac {231 e^4 (d+e x)^{3/2}}{128 b^5 (a+b x)}-\frac {231 e^3 (d+e x)^{5/2}}{320 b^4 (a+b x)^2}-\frac {33 e^2 (d+e x)^{7/2}}{80 b^3 (a+b x)^3}-\frac {11 e (d+e x)^{9/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{11/2}}{5 b (a+b x)^5}-\frac {693 e^5 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{13/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.56 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.39 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (-3465 a^5 e^5+1155 a^4 b e^4 (d-14 e x)+462 a^3 b^2 e^3 \left (d^2+12 d e x-64 e^2 x^2\right )+66 a^2 b^3 e^2 \left (4 d^3+33 d^2 e x+159 d e^2 x^2-395 e^3 x^3\right )+11 a b^4 e \left (16 d^4+112 d^3 e x+366 d^2 e^2 x^2+880 d e^3 x^3-965 e^4 x^4\right )+b^5 \left (128 d^5+816 d^4 e x+2248 d^3 e^2 x^2+3590 d^2 e^3 x^3+4215 d e^4 x^4-1280 e^5 x^5\right )\right )}{640 b^6 (a+b x)^5}-\frac {693 e^5 \sqrt {-b d+a e} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{13/2}} \]

[In]

Integrate[(d + e*x)^(11/2)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/640*(Sqrt[d + e*x]*(-3465*a^5*e^5 + 1155*a^4*b*e^4*(d - 14*e*x) + 462*a^3*b^2*e^3*(d^2 + 12*d*e*x - 64*e^2*
x^2) + 66*a^2*b^3*e^2*(4*d^3 + 33*d^2*e*x + 159*d*e^2*x^2 - 395*e^3*x^3) + 11*a*b^4*e*(16*d^4 + 112*d^3*e*x +
366*d^2*e^2*x^2 + 880*d*e^3*x^3 - 965*e^4*x^4) + b^5*(128*d^5 + 816*d^4*e*x + 2248*d^3*e^2*x^2 + 3590*d^2*e^3*
x^3 + 4215*d*e^4*x^4 - 1280*e^5*x^5)))/(b^6*(a + b*x)^5) - (693*e^5*Sqrt[-(b*d) + a*e]*ArcTan[(Sqrt[b]*Sqrt[d
+ e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(13/2))

Maple [A] (verified)

Time = 3.87 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.35

method	result	size
risch	\(\frac {2 e^{5} \sqrt {e x +d}}{b^{6}}-\frac {\left (2 a e -2 b d \right ) e^{5} \left (\frac {-\frac {843 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}-\frac {1327 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {131}{10} a^{2} b^{2} e^{2}+\frac {131}{5} a \,b^{3} d e -\frac {131}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {977}{128} a^{3} b \,e^{3}+\frac {2931}{128} a^{2} b^{2} d \,e^{2}-\frac {2931}{128} a \,b^{3} d^{2} e +\frac {977}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {437}{256} e^{4} a^{4}+\frac {437}{64} b \,e^{3} d \,a^{3}-\frac {1311}{128} b^{2} e^{2} d^{2} a^{2}+\frac {437}{64} a \,b^{3} d^{3} e -\frac {437}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {693 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{6}}\)	\(265\)
pseudoelliptic	\(-\frac {693 \left (e^{5} \left (b x +a \right )^{5} \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )-\left (\left (\frac {256}{693} e^{5} x^{5}-\frac {281}{231} x^{4} d \,e^{4}-\frac {718}{693} d^{2} e^{3} x^{3}-\frac {2248}{3465} d^{3} e^{2} x^{2}-\frac {272}{1155} d^{4} e x -\frac {128}{3465} d^{5}\right ) b^{5}-\frac {16 e a \left (-\frac {965}{16} e^{4} x^{4}+55 d \,e^{3} x^{3}+\frac {183}{8} d^{2} e^{2} x^{2}+7 d^{3} e x +d^{4}\right ) b^{4}}{315}-\frac {8 \left (-\frac {395}{4} e^{3} x^{3}+\frac {159}{4} d \,e^{2} x^{2}+\frac {33}{4} d^{2} e x +d^{3}\right ) e^{2} a^{2} b^{3}}{105}-\frac {2 a^{3} e^{3} \left (16 e x +d \right ) \left (-4 e x +d \right ) b^{2}}{15}-\frac {a^{4} e^{4} \left (-14 e x +d \right ) b}{3}+a^{5} e^{5}\right ) \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\right )}{128 \sqrt {\left (a e -b d \right ) b}\, b^{6} \left (b x +a \right )^{5}}\)	\(275\)
derivativedivides	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{b^{6}}-\frac {\frac {\left (-\frac {843}{256} e a \,b^{4}+\frac {843}{256} b^{5} d \right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {1327 b^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {131}{10} a^{3} b^{2} e^{3}+\frac {393}{10} a^{2} b^{3} d \,e^{2}-\frac {393}{10} a \,b^{4} d^{2} e +\frac {131}{10} b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {977}{128} e^{4} a^{4} b +\frac {977}{32} d \,e^{3} a^{3} b^{2}-\frac {2931}{64} d^{2} e^{2} a^{2} b^{3}+\frac {977}{32} a \,b^{4} d^{3} e -\frac {977}{128} d^{4} b^{5}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {437}{256} a^{5} e^{5}+\frac {2185}{256} a^{4} b d \,e^{4}-\frac {2185}{128} a^{3} b^{2} d^{2} e^{3}+\frac {2185}{128} a^{2} b^{3} d^{3} e^{2}-\frac {2185}{256} a \,b^{4} d^{4} e +\frac {437}{256} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {693 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}}{b^{6}}\right )\)	\(327\)
default	\(2 e^{5} \left (\frac {\sqrt {e x +d}}{b^{6}}-\frac {\frac {\left (-\frac {843}{256} e a \,b^{4}+\frac {843}{256} b^{5} d \right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {1327 b^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {131}{10} a^{3} b^{2} e^{3}+\frac {393}{10} a^{2} b^{3} d \,e^{2}-\frac {393}{10} a \,b^{4} d^{2} e +\frac {131}{10} b^{5} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {977}{128} e^{4} a^{4} b +\frac {977}{32} d \,e^{3} a^{3} b^{2}-\frac {2931}{64} d^{2} e^{2} a^{2} b^{3}+\frac {977}{32} a \,b^{4} d^{3} e -\frac {977}{128} d^{4} b^{5}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {437}{256} a^{5} e^{5}+\frac {2185}{256} a^{4} b d \,e^{4}-\frac {2185}{128} a^{3} b^{2} d^{2} e^{3}+\frac {2185}{128} a^{2} b^{3} d^{3} e^{2}-\frac {2185}{256} a \,b^{4} d^{4} e +\frac {437}{256} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {693 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}}{b^{6}}\right )\)	\(327\)

[In]

int((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*(e*x+d)^(1/2)/b^6-1/b^6*(2*a*e-2*b*d)*e^5*((-843/256*b^4*(e*x+d)^(9/2)-1327/128*(a*e-b*d)*b^3*(e*x+d)^(7
/2)+(-131/10*a^2*b^2*e^2+131/5*a*b^3*d*e-131/10*b^4*d^2)*(e*x+d)^(5/2)+(-977/128*a^3*b*e^3+2931/128*a^2*b^2*d*
e^2-2931/128*a*b^3*d^2*e+977/128*b^4*d^3)*(e*x+d)^(3/2)+(-437/256*e^4*a^4+437/64*b*e^3*d*a^3-1311/128*b^2*e^2*
d^2*a^2+437/64*a*b^3*d^3*e-437/256*b^4*d^4)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^5+693/256/((a*e-b*d)*b)^(1/2)*a
rctan(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. 440 vs. \(2 (161) = 322\).

Time = 0.33 (sec) , antiderivative size = 890, normalized size of antiderivative = 4.52 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [\frac {3465 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\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 (1280 \, b^{5} e^{5} x^{5} - 128 \, b^{5} d^{5} - 176 \, a b^{4} d^{4} e - 264 \, a^{2} b^{3} d^{3} e^{2} - 462 \, a^{3} b^{2} d^{2} e^{3} - 1155 \, a^{4} b d e^{4} + 3465 \, a^{5} e^{5} - 5 \, {\left (843 \, b^{5} d e^{4} - 2123 \, a b^{4} e^{5}\right )} x^{4} - 10 \, {\left (359 \, b^{5} d^{2} e^{3} + 968 \, a b^{4} d e^{4} - 2607 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (1124 \, b^{5} d^{3} e^{2} + 2013 \, a b^{4} d^{2} e^{3} + 5247 \, a^{2} b^{3} d e^{4} - 14784 \, a^{3} b^{2} e^{5}\right )} x^{2} - 2 \, {\left (408 \, b^{5} d^{4} e + 616 \, a b^{4} d^{3} e^{2} + 1089 \, a^{2} b^{3} d^{2} e^{3} + 2772 \, a^{3} b^{2} d e^{4} - 8085 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{1280 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}, -\frac {3465 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\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 (1280 \, b^{5} e^{5} x^{5} - 128 \, b^{5} d^{5} - 176 \, a b^{4} d^{4} e - 264 \, a^{2} b^{3} d^{3} e^{2} - 462 \, a^{3} b^{2} d^{2} e^{3} - 1155 \, a^{4} b d e^{4} + 3465 \, a^{5} e^{5} - 5 \, {\left (843 \, b^{5} d e^{4} - 2123 \, a b^{4} e^{5}\right )} x^{4} - 10 \, {\left (359 \, b^{5} d^{2} e^{3} + 968 \, a b^{4} d e^{4} - 2607 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \, {\left (1124 \, b^{5} d^{3} e^{2} + 2013 \, a b^{4} d^{2} e^{3} + 5247 \, a^{2} b^{3} d e^{4} - 14784 \, a^{3} b^{2} e^{5}\right )} x^{2} - 2 \, {\left (408 \, b^{5} d^{4} e + 616 \, a b^{4} d^{3} e^{2} + 1089 \, a^{2} b^{3} d^{2} e^{3} + 2772 \, a^{3} b^{2} d e^{4} - 8085 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{640 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}}\right ] \]

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[1/1280*(3465*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e
^5)*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*(1280
*b^5*e^5*x^5 - 128*b^5*d^5 - 176*a*b^4*d^4*e - 264*a^2*b^3*d^3*e^2 - 462*a^3*b^2*d^2*e^3 - 1155*a^4*b*d*e^4 +
3465*a^5*e^5 - 5*(843*b^5*d*e^4 - 2123*a*b^4*e^5)*x^4 - 10*(359*b^5*d^2*e^3 + 968*a*b^4*d*e^4 - 2607*a^2*b^3*e
^5)*x^3 - 2*(1124*b^5*d^3*e^2 + 2013*a*b^4*d^2*e^3 + 5247*a^2*b^3*d*e^4 - 14784*a^3*b^2*e^5)*x^2 - 2*(408*b^5*
d^4*e + 616*a*b^4*d^3*e^2 + 1089*a^2*b^3*d^2*e^3 + 2772*a^3*b^2*d*e^4 - 8085*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^1
1*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^7*x + a^5*b^6), -1/640*(3465*(b^5*e^5*x^5 + 5
*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*sqrt(-(b*d - a*e)/b)*arcta
n(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (1280*b^5*e^5*x^5 - 128*b^5*d^5 - 176*a*b^4*d^4*e - 264
*a^2*b^3*d^3*e^2 - 462*a^3*b^2*d^2*e^3 - 1155*a^4*b*d*e^4 + 3465*a^5*e^5 - 5*(843*b^5*d*e^4 - 2123*a*b^4*e^5)*
x^4 - 10*(359*b^5*d^2*e^3 + 968*a*b^4*d*e^4 - 2607*a^2*b^3*e^5)*x^3 - 2*(1124*b^5*d^3*e^2 + 2013*a*b^4*d^2*e^3
+ 5247*a^2*b^3*d*e^4 - 14784*a^3*b^2*e^5)*x^2 - 2*(408*b^5*d^4*e + 616*a*b^4*d^3*e^2 + 1089*a^2*b^3*d^2*e^3 +
2772*a^3*b^2*d*e^4 - 8085*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8
*x^2 + 5*a^4*b^7*x + a^5*b^6)]

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(11/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,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. 456 vs. \(2 (161) = 322\).

Time = 0.30 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.31 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2 \, \sqrt {e x + d} e^{5}}{b^{6}} + \frac {693 \, {\left (b d e^{5} - a e^{6}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {4215 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{5} d e^{5} - 13270 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} d^{2} e^{5} + 16768 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{5} - 9770 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{5} + 2185 \, \sqrt {e x + d} b^{5} d^{5} e^{5} - 4215 \, {\left (e x + d\right )}^{\frac {9}{2}} a b^{4} e^{6} + 26540 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{4} d e^{6} - 50304 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{6} + 39080 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{6} - 10925 \, \sqrt {e x + d} a b^{4} d^{4} e^{6} - 13270 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{3} e^{7} + 50304 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{7} - 58620 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{7} + 21850 \, \sqrt {e x + d} a^{2} b^{3} d^{3} e^{7} - 16768 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{8} + 39080 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{8} - 21850 \, \sqrt {e x + d} a^{3} b^{2} d^{2} e^{8} - 9770 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b e^{9} + 10925 \, \sqrt {e x + d} a^{4} b d e^{9} - 2185 \, \sqrt {e x + d} a^{5} e^{10}}{640 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5} b^{6}} \]

[In]

integrate((e*x+d)^(11/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

2*sqrt(e*x + d)*e^5/b^6 + 693/128*(b*d*e^5 - a*e^6)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d
+ a*b*e)*b^6) - 1/640*(4215*(e*x + d)^(9/2)*b^5*d*e^5 - 13270*(e*x + d)^(7/2)*b^5*d^2*e^5 + 16768*(e*x + d)^(5
/2)*b^5*d^3*e^5 - 9770*(e*x + d)^(3/2)*b^5*d^4*e^5 + 2185*sqrt(e*x + d)*b^5*d^5*e^5 - 4215*(e*x + d)^(9/2)*a*b
^4*e^6 + 26540*(e*x + d)^(7/2)*a*b^4*d*e^6 - 50304*(e*x + d)^(5/2)*a*b^4*d^2*e^6 + 39080*(e*x + d)^(3/2)*a*b^4
*d^3*e^6 - 10925*sqrt(e*x + d)*a*b^4*d^4*e^6 - 13270*(e*x + d)^(7/2)*a^2*b^3*e^7 + 50304*(e*x + d)^(5/2)*a^2*b
^3*d*e^7 - 58620*(e*x + d)^(3/2)*a^2*b^3*d^2*e^7 + 21850*sqrt(e*x + d)*a^2*b^3*d^3*e^7 - 16768*(e*x + d)^(5/2)
*a^3*b^2*e^8 + 39080*(e*x + d)^(3/2)*a^3*b^2*d*e^8 - 21850*sqrt(e*x + d)*a^3*b^2*d^2*e^8 - 9770*(e*x + d)^(3/2
)*a^4*b*e^9 + 10925*sqrt(e*x + d)*a^4*b*d*e^9 - 2185*sqrt(e*x + d)*a^5*e^10)/(((e*x + d)*b - b*d + a*e)^5*b^6)

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 598, normalized size of antiderivative = 3.04 \[ \int \frac {(d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {{\left (d+e\,x\right )}^{7/2}\,\left (\frac {1327\,a^2\,b^3\,e^7}{64}-\frac {1327\,a\,b^4\,d\,e^6}{32}+\frac {1327\,b^5\,d^2\,e^5}{64}\right )+\sqrt {d+e\,x}\,\left (\frac {437\,a^5\,e^{10}}{128}-\frac {2185\,a^4\,b\,d\,e^9}{128}+\frac {2185\,a^3\,b^2\,d^2\,e^8}{64}-\frac {2185\,a^2\,b^3\,d^3\,e^7}{64}+\frac {2185\,a\,b^4\,d^4\,e^6}{128}-\frac {437\,b^5\,d^5\,e^5}{128}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {131\,a^3\,b^2\,e^8}{5}-\frac {393\,a^2\,b^3\,d\,e^7}{5}+\frac {393\,a\,b^4\,d^2\,e^6}{5}-\frac {131\,b^5\,d^3\,e^5}{5}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {977\,a^4\,b\,e^9}{64}-\frac {977\,a^3\,b^2\,d\,e^8}{16}+\frac {2931\,a^2\,b^3\,d^2\,e^7}{32}-\frac {977\,a\,b^4\,d^3\,e^6}{16}+\frac {977\,b^5\,d^4\,e^5}{64}\right )+\left (\frac {843\,a\,b^4\,e^6}{128}-\frac {843\,b^5\,d\,e^5}{128}\right )\,{\left (d+e\,x\right )}^{9/2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b^7\,e^4-20\,a^3\,b^8\,d\,e^3+30\,a^2\,b^9\,d^2\,e^2-20\,a\,b^{10}\,d^3\,e+5\,b^{11}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^8\,e^3+30\,a^2\,b^9\,d\,e^2-30\,a\,b^{10}\,d^2\,e+10\,b^{11}\,d^3\right )+b^{11}\,{\left (d+e\,x\right )}^5-\left (5\,b^{11}\,d-5\,a\,b^{10}\,e\right )\,{\left (d+e\,x\right )}^4-b^{11}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^9\,e^2-20\,a\,b^{10}\,d\,e+10\,b^{11}\,d^2\right )+a^5\,b^6\,e^5-5\,a^4\,b^7\,d\,e^4-10\,a^2\,b^9\,d^3\,e^2+10\,a^3\,b^8\,d^2\,e^3+5\,a\,b^{10}\,d^4\,e}+\frac {2\,e^5\,\sqrt {d+e\,x}}{b^6}-\frac {693\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^5\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^6-b\,d\,e^5}\right )\,\sqrt {a\,e-b\,d}}{128\,b^{13/2}} \]

[In]

int((d + e*x)^(11/2)/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

((d + e*x)^(7/2)*((1327*a^2*b^3*e^7)/64 + (1327*b^5*d^2*e^5)/64 - (1327*a*b^4*d*e^6)/32) + (d + e*x)^(1/2)*((4
37*a^5*e^10)/128 - (437*b^5*d^5*e^5)/128 + (2185*a*b^4*d^4*e^6)/128 - (2185*a^2*b^3*d^3*e^7)/64 + (2185*a^3*b^
2*d^2*e^8)/64 - (2185*a^4*b*d*e^9)/128) + (d + e*x)^(5/2)*((131*a^3*b^2*e^8)/5 - (131*b^5*d^3*e^5)/5 + (393*a*
b^4*d^2*e^6)/5 - (393*a^2*b^3*d*e^7)/5) + (d + e*x)^(3/2)*((977*a^4*b*e^9)/64 + (977*b^5*d^4*e^5)/64 - (977*a*
b^4*d^3*e^6)/16 - (977*a^3*b^2*d*e^8)/16 + (2931*a^2*b^3*d^2*e^7)/32) + ((843*a*b^4*e^6)/128 - (843*b^5*d*e^5)
/128)*(d + e*x)^(9/2))/((d + e*x)*(5*b^11*d^4 + 5*a^4*b^7*e^4 - 20*a^3*b^8*d*e^3 + 30*a^2*b^9*d^2*e^2 - 20*a*b
^10*d^3*e) - (d + e*x)^2*(10*b^11*d^3 - 10*a^3*b^8*e^3 + 30*a^2*b^9*d*e^2 - 30*a*b^10*d^2*e) + b^11*(d + e*x)^
5 - (5*b^11*d - 5*a*b^10*e)*(d + e*x)^4 - b^11*d^5 + (d + e*x)^3*(10*b^11*d^2 + 10*a^2*b^9*e^2 - 20*a*b^10*d*e
) + a^5*b^6*e^5 - 5*a^4*b^7*d*e^4 - 10*a^2*b^9*d^3*e^2 + 10*a^3*b^8*d^2*e^3 + 5*a*b^10*d^4*e) + (2*e^5*(d + e*
x)^(1/2))/b^6 - (693*e^5*atan((b^(1/2)*e^5*(a*e - b*d)^(1/2)*(d + e*x)^(1/2))/(a*e^6 - b*d*e^5))*(a*e - b*d)^(
1/2))/(128*b^(13/2))