Integrand size = 27, antiderivative size = 195 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right ) (d+e x)^2}+\frac {\left (c \left (d^3-4 d e^2\right )-e \left (3 a d e-b \left (d^2+2 e^2\right )\right )\right ) \sqrt {-1+x^2}}{2 e \left (d^2-e^2\right )^2 (d+e x)}-\frac {\left (3 b d e-a \left (2 d^2+e^2\right )-c \left (d^2+2 e^2\right )\right ) \text {arctanh}\left (\frac {e+d x}{\sqrt {d^2-e^2} \sqrt {-1+x^2}}\right )}{2 \left (d^2-e^2\right )^{5/2}} \]
-1/2*(3*b*d*e-a*(2*d^2+e^2)-c*(d^2+2*e^2))*arctanh((d*x+e)/(d^2-e^2)^(1/2) /(x^2-1)^(1/2))/(d^2-e^2)^(5/2)-1/2*(a*e^2-b*d*e+c*d^2)*(x^2-1)^(1/2)/e/(d ^2-e^2)/(e*x+d)^2+1/2*(c*(d^3-4*d*e^2)-e*(3*a*d*e-b*(d^2+2*e^2)))*(x^2-1)^ (1/2)/e/(d^2-e^2)^2/(e*x+d)
Time = 0.93 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx=\frac {\frac {(d-e) (d+e) \sqrt {-1+x^2} \left (a e \left (-4 d^2+e^2-3 d e x\right )+c d \left (-3 d e+d^2 x-4 e^2 x\right )+b \left (2 d^3+d e^2+d^2 e x+2 e^3 x\right )\right )}{(d+e x)^2}+2 \sqrt {-d^2+e^2} \left (-3 b d e+a \left (2 d^2+e^2\right )+c \left (d^2+2 e^2\right )\right ) \arctan \left (\frac {d+e \left (x-\sqrt {-1+x^2}\right )}{\sqrt {-d^2+e^2}}\right )}{2 (d-e)^3 (d+e)^3} \]
(((d - e)*(d + e)*Sqrt[-1 + x^2]*(a*e*(-4*d^2 + e^2 - 3*d*e*x) + c*d*(-3*d *e + d^2*x - 4*e^2*x) + b*(2*d^3 + d*e^2 + d^2*e*x + 2*e^3*x)))/(d + e*x)^ 2 + 2*Sqrt[-d^2 + e^2]*(-3*b*d*e + a*(2*d^2 + e^2) + c*(d^2 + 2*e^2))*ArcT an[(d + e*(x - Sqrt[-1 + x^2]))/Sqrt[-d^2 + e^2]])/(2*(d - e)^3*(d + e)^3)
Time = 0.40 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2182, 25, 679, 488, 219}
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.
| \(\displaystyle \int \frac {a+b x+c x^2}{\sqrt {x^2-1} (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\int -\frac {2 (a d+c d-b e)+\left (\frac {c d^2}{e}+b d-a e-2 c e\right ) x}{(d+e x)^2 \sqrt {x^2-1}}dx}{2 \left (d^2-e^2\right )}-\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 (a d+c d-b e)+\left (\frac {c d^2}{e}+b d-a e-2 c e\right ) x}{(d+e x)^2 \sqrt {x^2-1}}dx}{2 \left (d^2-e^2\right )}-\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {\frac {\sqrt {x^2-1} \left (-3 a d e^2+b d^2 e+2 b e^3+c d^3-4 c d e^2\right )}{e \left (d^2-e^2\right ) (d+e x)}-\frac {\left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {x^2-1}}dx}{d^2-e^2}}{2 \left (d^2-e^2\right )}-\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\frac {\left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right ) \int \frac {1}{d^2-e^2-\frac {(-e-d x)^2}{x^2-1}}d\frac {-e-d x}{\sqrt {x^2-1}}}{d^2-e^2}+\frac {\sqrt {x^2-1} \left (-3 a d e^2+b d^2 e+2 b e^3+c d^3-4 c d e^2\right )}{e \left (d^2-e^2\right ) (d+e x)}}{2 \left (d^2-e^2\right )}-\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {-d x-e}{\sqrt {x^2-1} \sqrt {d^2-e^2}}\right ) \left (-a \left (2 d^2+e^2\right )+3 b d e-c \left (d^2+2 e^2\right )\right )}{\left (d^2-e^2\right )^{3/2}}+\frac {\sqrt {x^2-1} \left (-3 a d e^2+b d^2 e+2 b e^3+c d^3-4 c d e^2\right )}{e \left (d^2-e^2\right ) (d+e x)}}{2 \left (d^2-e^2\right )}-\frac {\sqrt {x^2-1} \left (a e^2-b d e+c d^2\right )}{2 e \left (d^2-e^2\right ) (d+e x)^2}\) |
-1/2*((c*d^2 - b*d*e + a*e^2)*Sqrt[-1 + x^2])/(e*(d^2 - e^2)*(d + e*x)^2) + (((c*d^3 + b*d^2*e - 3*a*d*e^2 - 4*c*d*e^2 + 2*b*e^3)*Sqrt[-1 + x^2])/(e *(d^2 - e^2)*(d + e*x)) + ((3*b*d*e - a*(2*d^2 + e^2) - c*(d^2 + 2*e^2))*A rcTanh[(-e - d*x)/(Sqrt[d^2 - e^2]*Sqrt[-1 + x^2])])/(d^2 - e^2)^(3/2))/(2 *(d^2 - e^2))
3.9.32.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(728\) vs. \(2(179)=358\).
Time = 1.39 (sec) , antiderivative size = 729, normalized size of antiderivative = 3.74
| method | result | size |
| default | \(-\frac {c \ln \left (\frac {\frac {2 d^{2}-2 e^{2}}{e^{2}}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}}+\frac {\left (b e -2 c d \right ) \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{\left (d^{2}-e^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e d \ln \left (\frac {\frac {2 d^{2}-2 e^{2}}{e^{2}}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (d^{2}-e^{2}\right ) \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}}\right )}{e^{4}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{2 \left (d^{2}-e^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 e d \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{\left (d^{2}-e^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {e d \ln \left (\frac {\frac {2 d^{2}-2 e^{2}}{e^{2}}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (d^{2}-e^{2}\right ) \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}}\right )}{2 \left (d^{2}-e^{2}\right )}+\frac {e^{2} \ln \left (\frac {\frac {2 d^{2}-2 e^{2}}{e^{2}}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2}-\frac {2 d \left (x +\frac {d}{e}\right )}{e}+\frac {d^{2}-e^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (d^{2}-e^{2}\right ) \sqrt {\frac {d^{2}-e^{2}}{e^{2}}}}\right )}{e^{5}}\) | \(729\) |
-c/e^3/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2/e*d*(x+1/e*d)+2*((d^2-e ^2)/e^2)^(1/2)*((x+1/e*d)^2-2/e*d*(x+1/e*d)+(d^2-e^2)/e^2)^(1/2))/(x+1/e*d ))+(b*e-2*c*d)/e^4*(-1/(d^2-e^2)*e^2/(x+1/e*d)*((x+1/e*d)^2-2/e*d*(x+1/e*d )+(d^2-e^2)/e^2)^(1/2)-e*d/(d^2-e^2)/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2) /e^2-2/e*d*(x+1/e*d)+2*((d^2-e^2)/e^2)^(1/2)*((x+1/e*d)^2-2/e*d*(x+1/e*d)+ (d^2-e^2)/e^2)^(1/2))/(x+1/e*d)))+(a*e^2-b*d*e+c*d^2)/e^5*(-1/2/(d^2-e^2)* e^2/(x+1/e*d)^2*((x+1/e*d)^2-2/e*d*(x+1/e*d)+(d^2-e^2)/e^2)^(1/2)+3/2*e*d/ (d^2-e^2)*(-1/(d^2-e^2)*e^2/(x+1/e*d)*((x+1/e*d)^2-2/e*d*(x+1/e*d)+(d^2-e^ 2)/e^2)^(1/2)-e*d/(d^2-e^2)/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^2-e^2)/e^2-2/e* d*(x+1/e*d)+2*((d^2-e^2)/e^2)^(1/2)*((x+1/e*d)^2-2/e*d*(x+1/e*d)+(d^2-e^2) /e^2)^(1/2))/(x+1/e*d)))+1/2/(d^2-e^2)*e^2/((d^2-e^2)/e^2)^(1/2)*ln((2*(d^ 2-e^2)/e^2-2/e*d*(x+1/e*d)+2*((d^2-e^2)/e^2)^(1/2)*((x+1/e*d)^2-2/e*d*(x+1 /e*d)+(d^2-e^2)/e^2)^(1/2))/(x+1/e*d)))
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (179) = 358\).
Time = 0.30 (sec) , antiderivative size = 1174, normalized size of antiderivative = 6.02 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx=\text {Too large to display} \]
[1/2*(c*d^7 + b*d^6*e - (3*a + 5*c)*d^5*e^2 + b*d^4*e^3 + (3*a + 4*c)*d^3* e^4 - 2*b*d^2*e^5 + (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e ^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x^2 + ((2*a + c)*d^4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4 + ((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^2 + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5)*x)*sqrt(d^2 - e^2)* log((d^2*x + d*e + sqrt(d^2 - e^2)*(d*x + e) + (d^2 - e^2 + sqrt(d^2 - e^2 )*d)*sqrt(x^2 - 1))/(e*x + d)) + 2*(c*d^6*e + b*d^5*e^2 - (3*a + 5*c)*d^4* e^3 + b*d^3*e^4 + (3*a + 4*c)*d^2*e^5 - 2*b*d*e^6)*x + (2*b*d^5*e^2 - (4*a + 3*c)*d^4*e^3 - b*d^3*e^4 + (5*a + 3*c)*d^2*e^5 - b*d*e^6 - a*e^7 + (c*d ^5*e^2 + b*d^4*e^3 - (3*a + 5*c)*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x)*sqrt(x^2 - 1))/(d^8*e^2 - 3*d^6*e^4 + 3*d^4*e^6 - d^2*e^8 + ( d^6*e^4 - 3*d^4*e^6 + 3*d^2*e^8 - e^10)*x^2 + 2*(d^7*e^3 - 3*d^5*e^5 + 3*d ^3*e^7 - d*e^9)*x), 1/2*(c*d^7 + b*d^6*e - (3*a + 5*c)*d^5*e^2 + b*d^4*e^3 + (3*a + 4*c)*d^3*e^4 - 2*b*d^2*e^5 + (c*d^5*e^2 + b*d^4*e^3 - (3*a + 5*c )*d^3*e^4 + b*d^2*e^5 + (3*a + 4*c)*d*e^6 - 2*b*e^7)*x^2 - 2*((2*a + c)*d^ 4*e^2 - 3*b*d^3*e^3 + (a + 2*c)*d^2*e^4 + ((2*a + c)*d^2*e^4 - 3*b*d*e^5 + (a + 2*c)*e^6)*x^2 + 2*((2*a + c)*d^3*e^3 - 3*b*d^2*e^4 + (a + 2*c)*d*e^5 )*x)*sqrt(-d^2 + e^2)*arctan(-(sqrt(-d^2 + e^2)*sqrt(x^2 - 1)*e - sqrt(-d^ 2 + e^2)*(e*x + d))/(d^2 - e^2)) + 2*(c*d^6*e + b*d^5*e^2 - (3*a + 5*c)*d^ 4*e^3 + b*d^3*e^4 + (3*a + 4*c)*d^2*e^5 - 2*b*d*e^6)*x + (2*b*d^5*e^2 -...
\[ \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx=\int \frac {a + b x + c x^{2}}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((e-d)*(e+d)>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (179) = 358\).
Time = 0.34 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.86 \[ \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx=\frac {{\left (2 \, a d^{2} + c d^{2} - 3 \, b d e + a e^{2} + 2 \, c e^{2}\right )} \arctan \left (-\frac {e {\left (x - \sqrt {x^{2} - 1}\right )} + d}{\sqrt {-d^{2} + e^{2}}}\right )}{{\left (d^{4} - 2 \, d^{2} e^{2} + e^{4}\right )} \sqrt {-d^{2} + e^{2}}} + \frac {2 \, c d^{4} e {\left (x - \sqrt {x^{2} - 1}\right )}^{3} - 2 \, a d^{2} e^{3} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} - 5 \, c d^{2} e^{3} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} + 3 \, b d e^{4} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} - a e^{5} {\left (x - \sqrt {x^{2} - 1}\right )}^{3} + 2 \, c d^{5} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 2 \, b d^{4} e {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 6 \, a d^{3} e^{2} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 7 \, c d^{3} e^{2} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 5 \, b d^{2} e^{3} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 3 \, a d e^{4} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} - 4 \, c d e^{4} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 2 \, b e^{5} {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 2 \, c d^{4} e {\left (x - \sqrt {x^{2} - 1}\right )} + 4 \, b d^{3} e^{2} {\left (x - \sqrt {x^{2} - 1}\right )} - 10 \, a d^{2} e^{3} {\left (x - \sqrt {x^{2} - 1}\right )} - 11 \, c d^{2} e^{3} {\left (x - \sqrt {x^{2} - 1}\right )} + 5 \, b d e^{4} {\left (x - \sqrt {x^{2} - 1}\right )} + a e^{5} {\left (x - \sqrt {x^{2} - 1}\right )} + c d^{3} e^{2} + b d^{2} e^{3} - 3 \, a d e^{4} - 4 \, c d e^{4} + 2 \, b e^{5}}{{\left (d^{4} e^{2} - 2 \, d^{2} e^{4} + e^{6}\right )} {\left (e {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 2 \, d {\left (x - \sqrt {x^{2} - 1}\right )} + e\right )}^{2}} \]
(2*a*d^2 + c*d^2 - 3*b*d*e + a*e^2 + 2*c*e^2)*arctan(-(e*(x - sqrt(x^2 - 1 )) + d)/sqrt(-d^2 + e^2))/((d^4 - 2*d^2*e^2 + e^4)*sqrt(-d^2 + e^2)) + (2* c*d^4*e*(x - sqrt(x^2 - 1))^3 - 2*a*d^2*e^3*(x - sqrt(x^2 - 1))^3 - 5*c*d^ 2*e^3*(x - sqrt(x^2 - 1))^3 + 3*b*d*e^4*(x - sqrt(x^2 - 1))^3 - a*e^5*(x - sqrt(x^2 - 1))^3 + 2*c*d^5*(x - sqrt(x^2 - 1))^2 + 2*b*d^4*e*(x - sqrt(x^ 2 - 1))^2 - 6*a*d^3*e^2*(x - sqrt(x^2 - 1))^2 - 7*c*d^3*e^2*(x - sqrt(x^2 - 1))^2 + 5*b*d^2*e^3*(x - sqrt(x^2 - 1))^2 - 3*a*d*e^4*(x - sqrt(x^2 - 1) )^2 - 4*c*d*e^4*(x - sqrt(x^2 - 1))^2 + 2*b*e^5*(x - sqrt(x^2 - 1))^2 + 2* c*d^4*e*(x - sqrt(x^2 - 1)) + 4*b*d^3*e^2*(x - sqrt(x^2 - 1)) - 10*a*d^2*e ^3*(x - sqrt(x^2 - 1)) - 11*c*d^2*e^3*(x - sqrt(x^2 - 1)) + 5*b*d*e^4*(x - sqrt(x^2 - 1)) + a*e^5*(x - sqrt(x^2 - 1)) + c*d^3*e^2 + b*d^2*e^3 - 3*a* d*e^4 - 4*c*d*e^4 + 2*b*e^5)/((d^4*e^2 - 2*d^2*e^4 + e^6)*(e*(x - sqrt(x^2 - 1))^2 + 2*d*(x - sqrt(x^2 - 1)) + e)^2)
Timed out. \[ \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {-1+x^2}} \, dx=\int \frac {c\,x^2+b\,x+a}{\sqrt {x^2-1}\,{\left (d+e\,x\right )}^3} \,d x \]