Integrand size = 33, antiderivative size = 313 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {(A b-a B) \sqrt {d+e x}}{5 b (b d-a e) (a+b x)^5}-\frac {(10 b B d-9 A b e-a B e) \sqrt {d+e x}}{40 b (b d-a e)^2 (a+b x)^4}+\frac {7 e (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{240 b (b d-a e)^3 (a+b x)^3}-\frac {7 e^2 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{192 b (b d-a e)^4 (a+b x)^2}+\frac {7 e^3 (10 b B d-9 A b e-a B e) \sqrt {d+e x}}{128 b (b d-a e)^5 (a+b x)}-\frac {7 e^4 (10 b B d-9 A b e-a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{3/2} (b d-a e)^{11/2}} \]
-7/128*e^4*(-9*A*b*e-B*a*e+10*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b *d)^(1/2))/b^(3/2)/(-a*e+b*d)^(11/2)-1/5*(A*b-B*a)*(e*x+d)^(1/2)/b/(-a*e+b *d)/(b*x+a)^5-1/40*(-9*A*b*e-B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b/(-a*e+b*d)^2/ (b*x+a)^4+7/240*e*(-9*A*b*e-B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b/(-a*e+b*d)^3/( b*x+a)^3-7/192*e^2*(-9*A*b*e-B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b/(-a*e+b*d)^4/ (b*x+a)^2+7/128*e^3*(-9*A*b*e-B*a*e+10*B*b*d)*(e*x+d)^(1/2)/b/(-a*e+b*d)^5 /(b*x+a)
Time = 1.95 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.35 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {d+e x} \left (B \left (-105 a^5 e^4+790 a^4 b e^3 (-2 d+e x)+2 a^3 b^2 e^2 \left (578 d^2-4239 d e x+448 e^2 x^2\right )+10 b^5 d x \left (48 d^3-56 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )+2 a^2 b^3 e \left (-256 d^3+3018 d^2 e x-4641 d e^2 x^2+245 e^3 x^3\right )+a b^4 \left (96 d^4-2608 d^3 e x+3276 d^2 e^2 x^2-4970 d e^3 x^3+105 e^4 x^4\right )\right )+3 A b \left (965 a^4 e^4+10 a^3 b e^3 (-149 d+237 e x)+6 a^2 b^2 e^2 \left (228 d^2-289 d e x+448 e^2 x^2\right )+2 a b^3 e \left (-328 d^3+384 d^2 e x-483 d e^2 x^2+735 e^3 x^3\right )+b^4 \left (128 d^4-144 d^3 e x+168 d^2 e^2 x^2-210 d e^3 x^3+315 e^4 x^4\right )\right )\right )}{1920 b (-b d+a e)^5 (a+b x)^5}+\frac {7 e^4 (-10 b B d+9 A b e+a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{3/2} (-b d+a e)^{11/2}} \]
(Sqrt[d + e*x]*(B*(-105*a^5*e^4 + 790*a^4*b*e^3*(-2*d + e*x) + 2*a^3*b^2*e ^2*(578*d^2 - 4239*d*e*x + 448*e^2*x^2) + 10*b^5*d*x*(48*d^3 - 56*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3) + 2*a^2*b^3*e*(-256*d^3 + 3018*d^2*e*x - 464 1*d*e^2*x^2 + 245*e^3*x^3) + a*b^4*(96*d^4 - 2608*d^3*e*x + 3276*d^2*e^2*x ^2 - 4970*d*e^3*x^3 + 105*e^4*x^4)) + 3*A*b*(965*a^4*e^4 + 10*a^3*b*e^3*(- 149*d + 237*e*x) + 6*a^2*b^2*e^2*(228*d^2 - 289*d*e*x + 448*e^2*x^2) + 2*a *b^3*e*(-328*d^3 + 384*d^2*e*x - 483*d*e^2*x^2 + 735*e^3*x^3) + b^4*(128*d ^4 - 144*d^3*e*x + 168*d^2*e^2*x^2 - 210*d*e^3*x^3 + 315*e^4*x^4))))/(1920 *b*(-(b*d) + a*e)^5*(a + b*x)^5) + (7*e^4*(-10*b*B*d + 9*A*b*e + a*B*e)*Ar cTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(3/2)*(-(b*d) + a *e)^(11/2))
Time = 0.40 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1184, 27, 87, 52, 52, 52, 52, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^3 \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {A+B x}{b^6 (a+b x)^6 \sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {A+B x}{(a+b x)^6 \sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-a B e-9 A b e+10 b B d) \int \frac {1}{(a+b x)^5 \sqrt {d+e x}}dx}{10 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(-a B e-9 A b e+10 b B d) \left (-\frac {7 e \int \frac {1}{(a+b x)^4 \sqrt {d+e x}}dx}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\right )}{10 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(-a B e-9 A b e+10 b B d) \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x)^3 \sqrt {d+e x}}dx}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\right )}{10 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(-a B e-9 A b e+10 b B d) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x)^2 \sqrt {d+e x}}dx}{4 (b d-a e)}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)}\right )}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\right )}{10 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(-a B e-9 A b e+10 b B d) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (-\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 (b d-a e)}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 (b d-a e)}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)}\right )}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\right )}{10 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(-a B e-9 A b e+10 b B d) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (-\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b d-a e}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 (b d-a e)}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)}\right )}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\right )}{10 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-a B e-9 A b e+10 b B d) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 (b d-a e)}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)}\right )}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\right )}{10 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{5 b (a+b x)^5 (b d-a e)}\) |
-1/5*((A*b - a*B)*Sqrt[d + e*x])/(b*(b*d - a*e)*(a + b*x)^5) + ((10*b*B*d - 9*A*b*e - a*B*e)*(-1/4*Sqrt[d + e*x]/((b*d - a*e)*(a + b*x)^4) - (7*e*(- 1/3*Sqrt[d + e*x]/((b*d - a*e)*(a + b*x)^3) - (5*e*(-1/2*Sqrt[d + e*x]/((b *d - a*e)*(a + b*x)^2) - (3*e*(-(Sqrt[d + e*x]/((b*d - a*e)*(a + b*x))) + (e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^ (3/2))))/(4*(b*d - a*e))))/(6*(b*d - a*e))))/(8*(b*d - a*e))))/(10*b*(b*d - a*e))
3.19.31.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.38 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(\frac {\frac {193 \left (\frac {63 b^{4} \left (\left (A b +\frac {B a}{9}\right ) e -\frac {10 B b d}{9}\right ) e^{3} x^{4}}{193}+\frac {294 b^{3} \left (\left (A b +\frac {B a}{9}\right ) e -\frac {10 B b d}{9}\right ) \left (a e -\frac {b d}{7}\right ) e^{2} x^{3}}{193}+\frac {2688 b^{2} \left (\left (A b +\frac {B a}{9}\right ) e -\frac {10 B b d}{9}\right ) \left (e^{2} a^{2}-\frac {23}{64} a b d e +\frac {1}{16} b^{2} d^{2}\right ) e \,x^{2}}{965}+\frac {474 b \left (a^{3} e^{3}-\frac {289}{395} a^{2} b d \,e^{2}+\frac {128}{395} a \,b^{2} d^{2} e -\frac {24}{395} b^{3} d^{3}\right ) \left (\left (A b +\frac {B a}{9}\right ) e -\frac {10 B b d}{9}\right ) x}{193}+a^{4} \left (A b -\frac {7 B a}{193}\right ) e^{4}-\frac {298 b d \left (A b +\frac {158 B a}{447}\right ) a^{3} e^{3}}{193}+\frac {1368 b^{2} d^{2} \left (A b +\frac {289 B a}{1026}\right ) a^{2} e^{2}}{965}-\frac {656 b^{3} \left (A b +\frac {32 B a}{123}\right ) d^{3} a e}{965}+\frac {128 b^{4} \left (A b +\frac {B a}{4}\right ) d^{4}}{965}\right ) \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}}{128}+\frac {63 \left (\left (A b +\frac {B a}{9}\right ) e -\frac {10 B b d}{9}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \left (b x +a \right )^{5} e^{4}}{128}}{\sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} b \left (a e -b d \right )^{5}}\) | \(361\) |
| derivativedivides | \(2 e^{4} \left (\frac {\frac {7 \left (9 A b e +B a e -10 B b d \right ) b^{3} \left (e x +d \right )^{\frac {9}{2}}}{256 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}+\frac {49 \left (9 A b e +B a e -10 B b d \right ) b^{2} \left (e x +d \right )^{\frac {7}{2}}}{384 \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}+\frac {7 \left (9 A b e +B a e -10 B b d \right ) b \left (e x +d \right )^{\frac {5}{2}}}{30 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {79 \left (9 A b e +B a e -10 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {\left (193 A b e -7 B a e -186 B b d \right ) \sqrt {e x +d}}{256 b \left (a e -b d \right )}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {7 \left (9 A b e +B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(465\) |
| default | \(2 e^{4} \left (\frac {\frac {7 \left (9 A b e +B a e -10 B b d \right ) b^{3} \left (e x +d \right )^{\frac {9}{2}}}{256 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}+\frac {49 \left (9 A b e +B a e -10 B b d \right ) b^{2} \left (e x +d \right )^{\frac {7}{2}}}{384 \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}+\frac {7 \left (9 A b e +B a e -10 B b d \right ) b \left (e x +d \right )^{\frac {5}{2}}}{30 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {79 \left (9 A b e +B a e -10 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{384 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {\left (193 A b e -7 B a e -186 B b d \right ) \sqrt {e x +d}}{256 b \left (a e -b d \right )}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {7 \left (9 A b e +B a e -10 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(465\) |
193/128/((a*e-b*d)*b)^(1/2)*((63/193*b^4*((A*b+1/9*B*a)*e-10/9*B*b*d)*e^3* x^4+294/193*b^3*((A*b+1/9*B*a)*e-10/9*B*b*d)*(a*e-1/7*b*d)*e^2*x^3+2688/96 5*b^2*((A*b+1/9*B*a)*e-10/9*B*b*d)*(e^2*a^2-23/64*a*b*d*e+1/16*b^2*d^2)*e* x^2+474/193*b*(a^3*e^3-289/395*a^2*b*d*e^2+128/395*a*b^2*d^2*e-24/395*b^3* d^3)*((A*b+1/9*B*a)*e-10/9*B*b*d)*x+a^4*(A*b-7/193*B*a)*e^4-298/193*b*d*(A *b+158/447*B*a)*a^3*e^3+1368/965*b^2*d^2*(A*b+289/1026*B*a)*a^2*e^2-656/96 5*b^3*(A*b+32/123*B*a)*d^3*a*e+128/965*b^4*(A*b+1/4*B*a)*d^4)*((a*e-b*d)*b )^(1/2)*(e*x+d)^(1/2)+63/193*((A*b+1/9*B*a)*e-10/9*B*b*d)*arctan(b*(e*x+d) ^(1/2)/((a*e-b*d)*b)^(1/2))*(b*x+a)^5*e^4)/(b*x+a)^5/b/(a*e-b*d)^5
Leaf count of result is larger than twice the leaf count of optimal. 1351 vs. \(2 (281) = 562\).
Time = 0.44 (sec) , antiderivative size = 2715, normalized size of antiderivative = 8.67 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
[1/3840*(105*(10*B*a^5*b*d*e^4 - (B*a^6 + 9*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (B*a*b^5 + 9*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 - (B*a^2*b^4 + 9*A*a *b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (B*a^3*b^3 + 9*A*a^2*b^4)*e^5)*x ^3 + 10*(10*B*a^3*b^3*d*e^4 - (B*a^4*b^2 + 9*A*a^3*b^3)*e^5)*x^2 + 5*(10*B *a^4*b^2*d*e^4 - (B*a^5*b + 9*A*a^4*b^2)*e^5)*x)*sqrt(b^2*d - a*b*e)*log(( b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2* (96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(38*B*a^2*b^5 + 147*A*a*b^6)*d^4*e + 12*( 139*B*a^3*b^4 + 506*A*a^2*b^5)*d^3*e^2 - 6*(456*B*a^4*b^3 + 1429*A*a^3*b^4 )*d^2*e^3 + 5*(295*B*a^5*b^2 + 1473*A*a^4*b^3)*d*e^4 + 15*(7*B*a^6*b - 193 *A*a^5*b^2)*e^5 - 105*(10*B*b^7*d^2*e^3 - (11*B*a*b^6 + 9*A*b^7)*d*e^4 + ( B*a^2*b^5 + 9*A*a*b^6)*e^5)*x^4 + 70*(10*B*b^7*d^3*e^2 - 9*(9*B*a*b^6 + A* b^7)*d^2*e^3 + 6*(13*B*a^2*b^5 + 12*A*a*b^6)*d*e^4 - 7*(B*a^3*b^4 + 9*A*a^ 2*b^5)*e^5)*x^3 - 14*(40*B*b^7*d^4*e - 2*(137*B*a*b^6 + 18*A*b^7)*d^3*e^2 + 3*(299*B*a^2*b^5 + 81*A*a*b^6)*d^2*e^3 - (727*B*a^3*b^4 + 783*A*a^2*b^5) *d*e^4 + 64*(B*a^4*b^3 + 9*A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5 - 8*(193 *B*a*b^6 + 27*A*b^7)*d^4*e + 2*(2161*B*a^2*b^5 + 684*A*a*b^6)*d^3*e^2 - 3* (2419*B*a^3*b^4 + 1251*A*a^2*b^5)*d^2*e^3 + 2*(2317*B*a^4*b^3 + 3078*A*a^3 *b^4)*d*e^4 - 395*(B*a^5*b^2 + 9*A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^ 8*d^6 - 6*a^6*b^7*d^5*e + 15*a^7*b^6*d^4*e^2 - 20*a^8*b^5*d^3*e^3 + 15*a^9 *b^4*d^2*e^4 - 6*a^10*b^3*d*e^5 + a^11*b^2*e^6 + (b^13*d^6 - 6*a*b^12*d...
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 885 vs. \(2 (281) = 562\).
Time = 0.28 (sec) , antiderivative size = 885, normalized size of antiderivative = 2.83 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {7 \, {\left (10 \, B b d e^{4} - B a e^{5} - 9 \, A b e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} \sqrt {-b^{2} d + a b e}} + \frac {1050 \, {\left (e x + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} - 4900 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} + 8960 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} - 7900 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} + 2790 \, \sqrt {e x + d} B b^{5} d^{5} e^{4} - 105 \, {\left (e x + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 945 \, {\left (e x + d\right )}^{\frac {9}{2}} A b^{5} e^{5} + 5390 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 4410 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} - 18816 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} - 8064 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} + 24490 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} + 7110 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} - 11055 \, \sqrt {e x + d} B a b^{4} d^{4} e^{5} - 2895 \, \sqrt {e x + d} A b^{5} d^{4} e^{5} - 490 \, {\left (e x + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 4410 \, {\left (e x + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} + 10752 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} + 16128 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} - 26070 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} - 21330 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} + 16320 \, \sqrt {e x + d} B a^{2} b^{3} d^{3} e^{6} + 11580 \, \sqrt {e x + d} A a b^{4} d^{3} e^{6} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} - 8064 \, {\left (e x + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} + 10270 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} + 21330 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} - 10530 \, \sqrt {e x + d} B a^{3} b^{2} d^{2} e^{7} - 17370 \, \sqrt {e x + d} A a^{2} b^{3} d^{2} e^{7} - 790 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} - 7110 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} + 2370 \, \sqrt {e x + d} B a^{4} b d e^{8} + 11580 \, \sqrt {e x + d} A a^{3} b^{2} d e^{8} + 105 \, \sqrt {e x + d} B a^{5} e^{9} - 2895 \, \sqrt {e x + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]
7/128*(10*B*b*d*e^4 - B*a*e^5 - 9*A*b*e^5)*arctan(sqrt(e*x + d)*b/sqrt(-b^ 2*d + a*b*e))/((b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3* d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*sqrt(-b^2*d + a*b*e)) + 1/1920*(105 0*(e*x + d)^(9/2)*B*b^5*d*e^4 - 4900*(e*x + d)^(7/2)*B*b^5*d^2*e^4 + 8960* (e*x + d)^(5/2)*B*b^5*d^3*e^4 - 7900*(e*x + d)^(3/2)*B*b^5*d^4*e^4 + 2790* sqrt(e*x + d)*B*b^5*d^5*e^4 - 105*(e*x + d)^(9/2)*B*a*b^4*e^5 - 945*(e*x + d)^(9/2)*A*b^5*e^5 + 5390*(e*x + d)^(7/2)*B*a*b^4*d*e^5 + 4410*(e*x + d)^ (7/2)*A*b^5*d*e^5 - 18816*(e*x + d)^(5/2)*B*a*b^4*d^2*e^5 - 8064*(e*x + d) ^(5/2)*A*b^5*d^2*e^5 + 24490*(e*x + d)^(3/2)*B*a*b^4*d^3*e^5 + 7110*(e*x + d)^(3/2)*A*b^5*d^3*e^5 - 11055*sqrt(e*x + d)*B*a*b^4*d^4*e^5 - 2895*sqrt( e*x + d)*A*b^5*d^4*e^5 - 490*(e*x + d)^(7/2)*B*a^2*b^3*e^6 - 4410*(e*x + d )^(7/2)*A*a*b^4*e^6 + 10752*(e*x + d)^(5/2)*B*a^2*b^3*d*e^6 + 16128*(e*x + d)^(5/2)*A*a*b^4*d*e^6 - 26070*(e*x + d)^(3/2)*B*a^2*b^3*d^2*e^6 - 21330* (e*x + d)^(3/2)*A*a*b^4*d^2*e^6 + 16320*sqrt(e*x + d)*B*a^2*b^3*d^3*e^6 + 11580*sqrt(e*x + d)*A*a*b^4*d^3*e^6 - 896*(e*x + d)^(5/2)*B*a^3*b^2*e^7 - 8064*(e*x + d)^(5/2)*A*a^2*b^3*e^7 + 10270*(e*x + d)^(3/2)*B*a^3*b^2*d*e^7 + 21330*(e*x + d)^(3/2)*A*a^2*b^3*d*e^7 - 10530*sqrt(e*x + d)*B*a^3*b^2*d ^2*e^7 - 17370*sqrt(e*x + d)*A*a^2*b^3*d^2*e^7 - 790*(e*x + d)^(3/2)*B*a^4 *b*e^8 - 7110*(e*x + d)^(3/2)*A*a^3*b^2*e^8 + 2370*sqrt(e*x + d)*B*a^4*b*d *e^8 + 11580*sqrt(e*x + d)*A*a^3*b^2*d*e^8 + 105*sqrt(e*x + d)*B*a^5*e^...
Time = 11.20 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.81 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {49\,{\left (d+e\,x\right )}^{7/2}\,\left (9\,A\,b^3\,e^5-10\,B\,d\,b^3\,e^4+B\,a\,b^2\,e^5\right )}{192\,{\left (a\,e-b\,d\right )}^4}+\frac {79\,{\left (d+e\,x\right )}^{3/2}\,\left (9\,A\,b\,e^5+B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,{\left (a\,e-b\,d\right )}^2}+\frac {7\,b\,{\left (d+e\,x\right )}^{5/2}\,\left (9\,A\,b\,e^5+B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,{\left (a\,e-b\,d\right )}^3}+\frac {7\,b^3\,{\left (d+e\,x\right )}^{9/2}\,\left (9\,A\,b\,e^5+B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^5}-\frac {\sqrt {d+e\,x}\,\left (7\,B\,a\,e^5-193\,A\,b\,e^5+186\,B\,b\,d\,e^4\right )}{128\,b\,\left (a\,e-b\,d\right )}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4}+\frac {7\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (9\,A\,b\,e+B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (9\,A\,b\,e^5+B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (9\,A\,b\,e+B\,a\,e-10\,B\,b\,d\right )}{128\,b^{3/2}\,{\left (a\,e-b\,d\right )}^{11/2}} \]
((49*(d + e*x)^(7/2)*(9*A*b^3*e^5 + B*a*b^2*e^5 - 10*B*b^3*d*e^4))/(192*(a *e - b*d)^4) + (79*(d + e*x)^(3/2)*(9*A*b*e^5 + B*a*e^5 - 10*B*b*d*e^4))/( 192*(a*e - b*d)^2) + (7*b*(d + e*x)^(5/2)*(9*A*b*e^5 + B*a*e^5 - 10*B*b*d* e^4))/(15*(a*e - b*d)^3) + (7*b^3*(d + e*x)^(9/2)*(9*A*b*e^5 + B*a*e^5 - 1 0*B*b*d*e^4))/(128*(a*e - b*d)^5) - ((d + e*x)^(1/2)*(7*B*a*e^5 - 193*A*b* e^5 + 186*B*b*d*e^4))/(128*b*(a*e - b*d)))/((d + e*x)*(5*b^5*d^4 + 5*a^4*b *e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 - 20*a*b^4*d^3*e) - (d + e*x) ^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + b^5 *(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) + (7*e^4*atan((b^(1 /2)*e^4*(d + e*x)^(1/2)*(9*A*b*e + B*a*e - 10*B*b*d))/((a*e - b*d)^(1/2)*( 9*A*b*e^5 + B*a*e^5 - 10*B*b*d*e^4)))*(9*A*b*e + B*a*e - 10*B*b*d))/(128*b ^(3/2)*(a*e - b*d)^(11/2))