Integrand size = 33, antiderivative size = 250 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {5 e^2 (6 b B d-7 A b e+a B e)}{8 b (b d-a e)^4 \sqrt {d+e x}}-\frac {A b-a B}{3 b (b d-a e) (a+b x)^3 \sqrt {d+e x}}-\frac {6 b B d-7 A b e+a B e}{12 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x}}+\frac {5 e (6 b B d-7 A b e+a B e)}{24 b (b d-a e)^3 (a+b x) \sqrt {d+e x}}-\frac {5 e^2 (6 b B d-7 A b e+a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 \sqrt {b} (b d-a e)^{9/2}} \]
-5/8*e^2*(-7*A*b*e+B*a*e+6*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d) ^(1/2))/(-a*e+b*d)^(9/2)/b^(1/2)+5/8*e^2*(-7*A*b*e+B*a*e+6*B*b*d)/b/(-a*e+ b*d)^4/(e*x+d)^(1/2)+1/3*(-A*b+B*a)/b/(-a*e+b*d)/(b*x+a)^3/(e*x+d)^(1/2)+1 /12*(7*A*b*e-B*a*e-6*B*b*d)/b/(-a*e+b*d)^2/(b*x+a)^2/(e*x+d)^(1/2)+5/24*e* (-7*A*b*e+B*a*e+6*B*b*d)/b/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(1/2)
Time = 1.36 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {B \left (3 a^3 e^2 (27 d+11 e x)+4 a^2 b e \left (7 d^2+53 d e x+10 e^2 x^2\right )+6 b^3 d x \left (-2 d^2+5 d e x+15 e^2 x^2\right )+a b^2 \left (-4 d^3+82 d^2 e x+245 d e^2 x^2+15 e^3 x^3\right )\right )-A \left (48 a^3 e^3+3 a^2 b e^2 (29 d+77 e x)+2 a b^2 e \left (-19 d^2+49 d e x+140 e^2 x^2\right )+b^3 \left (8 d^3-14 d^2 e x+35 d e^2 x^2+105 e^3 x^3\right )\right )}{24 (b d-a e)^4 (a+b x)^3 \sqrt {d+e x}}+\frac {5 e^2 (6 b B d-7 A b e+a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 \sqrt {b} (-b d+a e)^{9/2}} \]
(B*(3*a^3*e^2*(27*d + 11*e*x) + 4*a^2*b*e*(7*d^2 + 53*d*e*x + 10*e^2*x^2) + 6*b^3*d*x*(-2*d^2 + 5*d*e*x + 15*e^2*x^2) + a*b^2*(-4*d^3 + 82*d^2*e*x + 245*d*e^2*x^2 + 15*e^3*x^3)) - A*(48*a^3*e^3 + 3*a^2*b*e^2*(29*d + 77*e*x ) + 2*a*b^2*e*(-19*d^2 + 49*d*e*x + 140*e^2*x^2) + b^3*(8*d^3 - 14*d^2*e*x + 35*d*e^2*x^2 + 105*e^3*x^3)))/(24*(b*d - a*e)^4*(a + b*x)^3*Sqrt[d + e* x]) + (5*e^2*(6*b*B*d - 7*A*b*e + a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sq rt[-(b*d) + a*e]])/(8*Sqrt[b]*(-(b*d) + a*e)^(9/2))
Time = 0.36 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1184, 27, 87, 52, 52, 61, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^4 \int \frac {A+B x}{b^4 (a+b x)^4 (d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {A+B x}{(a+b x)^4 (d+e x)^{3/2}}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a B e-7 A b e+6 b B d) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}}dx}{6 b (b d-a e)}-\frac {A b-a B}{3 b (a+b x)^3 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a B e-7 A b e+6 b B d) \left (-\frac {5 e \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 b (b d-a e)}-\frac {A b-a B}{3 b (a+b x)^3 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a B e-7 A b e+6 b B d) \left (-\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 b (b d-a e)}-\frac {A b-a B}{3 b (a+b x)^3 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a B e-7 A b e+6 b B d) \left (-\frac {5 e \left (-\frac {3 e \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 b (b d-a e)}-\frac {A b-a B}{3 b (a+b x)^3 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a B e-7 A b e+6 b B d) \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 b (b d-a e)}-\frac {A b-a B}{3 b (a+b x)^3 \sqrt {d+e x} (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a B e-7 A b e+6 b B d) \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 b (b d-a e)}-\frac {A b-a B}{3 b (a+b x)^3 \sqrt {d+e x} (b d-a e)}\) |
-1/3*(A*b - a*B)/(b*(b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x]) + ((6*b*B*d - 7 *A*b*e + a*B*e)*(-1/2*1/((b*d - a*e)*(a + b*x)^2*Sqrt[d + e*x]) - (5*e*(-( 1/((b*d - a*e)*(a + b*x)*Sqrt[d + e*x])) - (3*e*(2/((b*d - a*e)*Sqrt[d + e *x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^(3/2)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/(6*b*(b*d - a*e))
3.19.22.3.1 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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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 = 267, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\frac {35 \left (\left (A b -\frac {B a}{7}\right ) e -\frac {6 B b d}{7}\right ) \left (b x +a \right )^{3} e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {e x +d}}{16}+\left (\frac {35 b^{2} \left (\left (A b -\frac {B a}{7}\right ) e -\frac {6 B b d}{7}\right ) e^{2} x^{3}}{16}+\frac {35 b \left (a e +\frac {b d}{8}\right ) \left (\left (A b -\frac {B a}{7}\right ) e -\frac {6 B b d}{7}\right ) e \,x^{2}}{6}+\frac {77 \left (e^{2} a^{2}+\frac {14}{33} a b d e -\frac {2}{33} b^{2} d^{2}\right ) \left (\left (A b -\frac {B a}{7}\right ) e -\frac {6 B b d}{7}\right ) x}{16}+A \,a^{3} e^{3}+\frac {29 d \left (A b -\frac {27 B a}{29}\right ) a^{2} e^{2}}{16}-\frac {19 b \,d^{2} \left (A b +\frac {14 B a}{19}\right ) a e}{24}+\frac {b^{2} d^{3} \left (A b +\frac {B a}{2}\right )}{6}\right ) \sqrt {\left (a e -b d \right ) b}\right )}{\sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, \left (b x +a \right )^{3} \left (a e -b d \right )^{4}}\) | \(267\) |
| derivativedivides | \(2 e^{2} \left (-\frac {A e -B d}{\left (a e -b d \right )^{4} \sqrt {e x +d}}-\frac {\frac {\left (\frac {19}{16} A \,b^{3} e -\frac {7}{8} B \,b^{3} d -\frac {5}{16} B e \,b^{2} a \right ) \left (e x +d \right )^{\frac {5}{2}}+\frac {b \left (17 A a b \,e^{2}-17 A \,b^{2} d e -5 a^{2} B \,e^{2}-7 B a b d e +12 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (\frac {29}{16} A \,a^{2} b \,e^{3}-\frac {29}{8} A a \,b^{2} d \,e^{2}+\frac {29}{16} A \,b^{3} d^{2} e -\frac {11}{16} B \,e^{3} a^{3}+\frac {1}{4} B \,a^{2} b d \,e^{2}+\frac {25}{16} B a \,b^{2} d^{2} e -\frac {9}{8} B \,b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {5 \left (7 A b e -B a e -6 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}}{\left (a e -b d \right )^{4}}\right )\) | \(281\) |
| default | \(2 e^{2} \left (-\frac {A e -B d}{\left (a e -b d \right )^{4} \sqrt {e x +d}}-\frac {\frac {\left (\frac {19}{16} A \,b^{3} e -\frac {7}{8} B \,b^{3} d -\frac {5}{16} B e \,b^{2} a \right ) \left (e x +d \right )^{\frac {5}{2}}+\frac {b \left (17 A a b \,e^{2}-17 A \,b^{2} d e -5 a^{2} B \,e^{2}-7 B a b d e +12 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (\frac {29}{16} A \,a^{2} b \,e^{3}-\frac {29}{8} A a \,b^{2} d \,e^{2}+\frac {29}{16} A \,b^{3} d^{2} e -\frac {11}{16} B \,e^{3} a^{3}+\frac {1}{4} B \,a^{2} b d \,e^{2}+\frac {25}{16} B a \,b^{2} d^{2} e -\frac {9}{8} B \,b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {5 \left (7 A b e -B a e -6 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}}{\left (a e -b d \right )^{4}}\right )\) | \(281\) |
-2*(35/16*((A*b-1/7*B*a)*e-6/7*B*b*d)*(b*x+a)^3*e^2*arctan(b*(e*x+d)^(1/2) /((a*e-b*d)*b)^(1/2))*(e*x+d)^(1/2)+(35/16*b^2*((A*b-1/7*B*a)*e-6/7*B*b*d) *e^2*x^3+35/6*b*(a*e+1/8*b*d)*((A*b-1/7*B*a)*e-6/7*B*b*d)*e*x^2+77/16*(e^2 *a^2+14/33*a*b*d*e-2/33*b^2*d^2)*((A*b-1/7*B*a)*e-6/7*B*b*d)*x+A*a^3*e^3+2 9/16*d*(A*b-27/29*B*a)*a^2*e^2-19/24*b*d^2*(A*b+14/19*B*a)*a*e+1/6*b^2*d^3 *(A*b+1/2*B*a))*((a*e-b*d)*b)^(1/2))/((a*e-b*d)*b)^(1/2)/(e*x+d)^(1/2)/(b* x+a)^3/(a*e-b*d)^4
Leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (222) = 444\).
Time = 0.46 (sec) , antiderivative size = 2114, normalized size of antiderivative = 8.46 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Too large to display} \]
[-1/48*(15*(6*B*a^3*b*d^2*e^2 + (B*a^4 - 7*A*a^3*b)*d*e^3 + (6*B*b^4*d*e^3 + (B*a*b^3 - 7*A*b^4)*e^4)*x^4 + (6*B*b^4*d^2*e^2 + (19*B*a*b^3 - 7*A*b^4 )*d*e^3 + 3*(B*a^2*b^2 - 7*A*a*b^3)*e^4)*x^3 + 3*(6*B*a*b^3*d^2*e^2 + 7*(B *a^2*b^2 - A*a*b^3)*d*e^3 + (B*a^3*b - 7*A*a^2*b^2)*e^4)*x^2 + (18*B*a^2*b ^2*d^2*e^2 + 3*(3*B*a^3*b - 7*A*a^2*b^2)*d*e^3 + (B*a^4 - 7*A*a^3*b)*e^4)* 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)*sq rt(e*x + d))/(b*x + a)) - 2*(48*A*a^4*b*e^4 - 4*(B*a*b^4 + 2*A*b^5)*d^4 + 2*(16*B*a^2*b^3 + 23*A*a*b^4)*d^3*e + (53*B*a^3*b^2 - 125*A*a^2*b^3)*d^2*e ^2 - 3*(27*B*a^4*b - 13*A*a^3*b^2)*d*e^3 + 15*(6*B*b^5*d^2*e^2 - (5*B*a*b^ 4 + 7*A*b^5)*d*e^3 - (B*a^2*b^3 - 7*A*a*b^4)*e^4)*x^3 + 5*(6*B*b^5*d^3*e + (43*B*a*b^4 - 7*A*b^5)*d^2*e^2 - (41*B*a^2*b^3 + 49*A*a*b^4)*d*e^3 - 8*(B *a^3*b^2 - 7*A*a^2*b^3)*e^4)*x^2 - (12*B*b^5*d^4 - 2*(47*B*a*b^4 + 7*A*b^5 )*d^3*e - 2*(65*B*a^2*b^3 - 56*A*a*b^4)*d^2*e^2 + (179*B*a^3*b^2 + 133*A*a ^2*b^3)*d*e^3 + 33*(B*a^4*b - 7*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(a^3*b^6 *d^6 - 5*a^4*b^5*d^5*e + 10*a^5*b^4*d^4*e^2 - 10*a^6*b^3*d^3*e^3 + 5*a^7*b ^2*d^2*e^4 - a^8*b*d*e^5 + (b^9*d^5*e - 5*a*b^8*d^4*e^2 + 10*a^2*b^7*d^3*e ^3 - 10*a^3*b^6*d^2*e^4 + 5*a^4*b^5*d*e^5 - a^5*b^4*e^6)*x^4 + (b^9*d^6 - 2*a*b^8*d^5*e - 5*a^2*b^7*d^4*e^2 + 20*a^3*b^6*d^3*e^3 - 25*a^4*b^5*d^2*e^ 4 + 14*a^5*b^4*d*e^5 - 3*a^6*b^3*e^6)*x^3 + 3*(a*b^8*d^6 - 4*a^2*b^7*d^5*e + 5*a^3*b^6*d^4*e^2 - 5*a^5*b^4*d^2*e^4 + 4*a^6*b^3*d*e^5 - a^7*b^2*e^...
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (222) = 444\).
Time = 0.30 (sec) , antiderivative size = 521, normalized size of antiderivative = 2.08 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {5 \, {\left (6 \, B b d e^{2} + B a e^{3} - 7 \, A b e^{3}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (B d e^{2} - A e^{3}\right )}}{{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {e x + d}} + \frac {42 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{3} d e^{2} - 96 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e^{2} + 54 \, \sqrt {e x + d} B b^{3} d^{3} e^{2} + 15 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{2} e^{3} - 57 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{3} e^{3} + 56 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{2} d e^{3} + 136 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{3} d e^{3} - 75 \, \sqrt {e x + d} B a b^{2} d^{2} e^{3} - 87 \, \sqrt {e x + d} A b^{3} d^{2} e^{3} + 40 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b e^{4} - 136 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{2} e^{4} - 12 \, \sqrt {e x + d} B a^{2} b d e^{4} + 174 \, \sqrt {e x + d} A a b^{2} d e^{4} + 33 \, \sqrt {e x + d} B a^{3} e^{5} - 87 \, \sqrt {e x + d} A a^{2} b e^{5}}{24 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3}} \]
5/8*(6*B*b*d*e^2 + B*a*e^3 - 7*A*b*e^3)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(-b^2*d + a*b*e)) + 2*(B*d*e^2 - A*e^3)/((b^4*d^4 - 4*a*b^3* d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(e*x + d)) + 1/24 *(42*(e*x + d)^(5/2)*B*b^3*d*e^2 - 96*(e*x + d)^(3/2)*B*b^3*d^2*e^2 + 54*s qrt(e*x + d)*B*b^3*d^3*e^2 + 15*(e*x + d)^(5/2)*B*a*b^2*e^3 - 57*(e*x + d) ^(5/2)*A*b^3*e^3 + 56*(e*x + d)^(3/2)*B*a*b^2*d*e^3 + 136*(e*x + d)^(3/2)* A*b^3*d*e^3 - 75*sqrt(e*x + d)*B*a*b^2*d^2*e^3 - 87*sqrt(e*x + d)*A*b^3*d^ 2*e^3 + 40*(e*x + d)^(3/2)*B*a^2*b*e^4 - 136*(e*x + d)^(3/2)*A*a*b^2*e^4 - 12*sqrt(e*x + d)*B*a^2*b*d*e^4 + 174*sqrt(e*x + d)*A*a*b^2*d*e^4 + 33*sqr t(e*x + d)*B*a^3*e^5 - 87*sqrt(e*x + d)*A*a^2*b*e^5)/((b^4*d^4 - 4*a*b^3*d ^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*((e*x + d)*b - b*d + a *e)^3)
Time = 10.90 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.68 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {5\,{\left (d+e\,x\right )}^2\,\left (-7\,A\,b^2\,e^3+6\,B\,d\,b^2\,e^2+B\,a\,b\,e^3\right )}{3\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,\left (A\,e^3-B\,d\,e^2\right )}{a\,e-b\,d}+\frac {11\,\left (d+e\,x\right )\,\left (B\,a\,e^3-7\,A\,b\,e^3+6\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^2}+\frac {5\,b^2\,{\left (d+e\,x\right )}^3\,\left (B\,a\,e^3-7\,A\,b\,e^3+6\,B\,b\,d\,e^2\right )}{8\,{\left (a\,e-b\,d\right )}^4}}{\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )+b^3\,{\left (d+e\,x\right )}^{7/2}-\left (3\,b^3\,d-3\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (3\,a^2\,b\,e^2-6\,a\,b^2\,d\,e+3\,b^3\,d^2\right )}+\frac {5\,e^2\,\mathrm {atan}\left (\frac {5\,\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (B\,a\,e-7\,A\,b\,e+6\,B\,b\,d\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^{9/2}\,\left (5\,B\,a\,e^3-35\,A\,b\,e^3+30\,B\,b\,d\,e^2\right )}\right )\,\left (B\,a\,e-7\,A\,b\,e+6\,B\,b\,d\right )}{8\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{9/2}} \]
((5*(d + e*x)^2*(B*a*b*e^3 - 7*A*b^2*e^3 + 6*B*b^2*d*e^2))/(3*(a*e - b*d)^ 3) - (2*(A*e^3 - B*d*e^2))/(a*e - b*d) + (11*(d + e*x)*(B*a*e^3 - 7*A*b*e^ 3 + 6*B*b*d*e^2))/(8*(a*e - b*d)^2) + (5*b^2*(d + e*x)^3*(B*a*e^3 - 7*A*b* e^3 + 6*B*b*d*e^2))/(8*(a*e - b*d)^4))/((d + e*x)^(1/2)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2) + b^3*(d + e*x)^(7/2) - (3*b^3*d - 3*a*b ^2*e)*(d + e*x)^(5/2) + (d + e*x)^(3/2)*(3*b^3*d^2 + 3*a^2*b*e^2 - 6*a*b^2 *d*e)) + (5*e^2*atan((5*b^(1/2)*e^2*(d + e*x)^(1/2)*(B*a*e - 7*A*b*e + 6*B *b*d)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d*e ^3))/((a*e - b*d)^(9/2)*(5*B*a*e^3 - 35*A*b*e^3 + 30*B*b*d*e^2)))*(B*a*e - 7*A*b*e + 6*B*b*d))/(8*b^(1/2)*(a*e - b*d)^(9/2))