Integrand size = 33, antiderivative size = 284 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {35 e^2 (2 b B d+A b e-3 a B e) \sqrt {d+e x}}{8 b^5}+\frac {35 e^2 (2 b B d+A b e-3 a B e) (d+e x)^{3/2}}{24 b^4 (b d-a e)}-\frac {7 e (2 b B d+A b e-3 a B e) (d+e x)^{5/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(2 b B d+A b e-3 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^3}-\frac {35 e^2 \sqrt {b d-a e} (2 b B d+A b e-3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{11/2}} \]
35/24*e^2*(A*b*e-3*B*a*e+2*B*b*d)*(e*x+d)^(3/2)/b^4/(-a*e+b*d)-7/8*e*(A*b* e-3*B*a*e+2*B*b*d)*(e*x+d)^(5/2)/b^3/(-a*e+b*d)/(b*x+a)-1/4*(A*b*e-3*B*a*e +2*B*b*d)*(e*x+d)^(7/2)/b^2/(-a*e+b*d)/(b*x+a)^2-1/3*(A*b-B*a)*(e*x+d)^(9/ 2)/b/(-a*e+b*d)/(b*x+a)^3-35/8*e^2*(A*b*e-3*B*a*e+2*B*b*d)*arctanh(b^(1/2) *(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(11/2)+35/8*e^2*(A*b*e -3*B*a*e+2*B*b*d)*(e*x+d)^(1/2)/b^5
Time = 1.42 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} \left (A b \left (-105 a^3 e^3+35 a^2 b e^2 (d-8 e x)+7 a b^2 e \left (2 d^2+14 d e x-33 e^2 x^2\right )+b^3 \left (8 d^3+38 d^2 e x+87 d e^2 x^2-48 e^3 x^3\right )\right )+B \left (315 a^4 e^3+105 a^3 b e^2 (-3 d+8 e x)+7 a^2 b^2 e \left (4 d^2-122 d e x+99 e^2 x^2\right )+2 b^4 x \left (6 d^3+39 d^2 e x-80 d e^2 x^2-8 e^3 x^3\right )+a b^3 \left (4 d^3+82 d^2 e x-723 d e^2 x^2+144 e^3 x^3\right )\right )\right )}{24 b^5 (a+b x)^3}-\frac {35 e^2 \sqrt {-b d+a e} (2 b B d+A b e-3 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{11/2}} \]
-1/24*(Sqrt[d + e*x]*(A*b*(-105*a^3*e^3 + 35*a^2*b*e^2*(d - 8*e*x) + 7*a*b ^2*e*(2*d^2 + 14*d*e*x - 33*e^2*x^2) + b^3*(8*d^3 + 38*d^2*e*x + 87*d*e^2* x^2 - 48*e^3*x^3)) + B*(315*a^4*e^3 + 105*a^3*b*e^2*(-3*d + 8*e*x) + 7*a^2 *b^2*e*(4*d^2 - 122*d*e*x + 99*e^2*x^2) + 2*b^4*x*(6*d^3 + 39*d^2*e*x - 80 *d*e^2*x^2 - 8*e^3*x^3) + a*b^3*(4*d^3 + 82*d^2*e*x - 723*d*e^2*x^2 + 144* e^3*x^3))))/(b^5*(a + b*x)^3) - (35*e^2*Sqrt[-(b*d) + a*e]*(2*b*B*d + A*b* e - 3*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(8*b^(11/ 2))
Time = 0.33 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.80, 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, 51, 51, 60, 60, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^4 \int \frac {(A+B x) (d+e x)^{7/2}}{b^4 (a+b x)^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^4}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \int \frac {(d+e x)^{7/2}}{(a+b x)^3}dx}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)}\) |
-1/3*((A*b - a*B)*(d + e*x)^(9/2))/(b*(b*d - a*e)*(a + b*x)^3) + ((2*b*B*d + A*b*e - 3*a*B*e)*(-1/2*(d + e*x)^(7/2)/(b*(a + b*x)^2) + (7*e*(-((d + e *x)^(5/2)/(b*(a + b*x))) + (5*e*((2*(d + e*x)^(3/2))/(3*b) + ((b*d - a*e)* ((2*Sqrt[d + e*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/ Sqrt[b*d - a*e]])/b^(3/2)))/b))/(2*b)))/(4*b)))/(2*b*(b*d - a*e))
3.19.17.3.1 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/(b*(m + 1))), x] - Simp[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 ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[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 = 0.55 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {35 \left (a e -b d \right ) \left (b \left (A e +2 B d \right )-3 B a e \right ) \left (b x +a \right )^{3} e^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8}+\frac {35 \left (\left (\frac {16 \left (\frac {B x}{3}+A \right ) x^{3} e^{3}}{35}-\frac {29 \left (-\frac {160 B x}{87}+A \right ) x^{2} d \,e^{2}}{35}-\frac {38 x \,d^{2} \left (\frac {39 B x}{19}+A \right ) e}{105}-\frac {8 d^{3} \left (\frac {3 B x}{2}+A \right )}{105}\right ) b^{4}-\frac {2 \left (\left (\frac {72}{7} B \,x^{3}-\frac {33}{2} A \,x^{2}\right ) e^{3}+7 \left (-\frac {723 B x}{98}+A \right ) x d \,e^{2}+d^{2} \left (\frac {41 B x}{7}+A \right ) e +\frac {2 B \,d^{3}}{7}\right ) a \,b^{3}}{15}-\frac {e \left (\left (\frac {99}{5} B \,x^{2}-8 A x \right ) e^{2}+d \left (-\frac {122 B x}{5}+A \right ) e +\frac {4 B \,d^{2}}{5}\right ) a^{2} b^{2}}{3}+\left (\left (-8 B x +A \right ) e +3 B d \right ) e^{2} a^{3} b -3 B \,e^{3} a^{4}\right ) \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}}{8}}{\left (b x +a \right )^{3} b^{5} \sqrt {\left (a e -b d \right ) b}}\) | \(287\) |
| risch | \(\frac {2 e^{2} \left (B b e x +3 A b e -12 B a e +10 B b d \right ) \sqrt {e x +d}}{3 b^{5}}-\frac {\left (2 a e -2 b d \right ) e^{2} \left (\frac {\left (-\frac {29}{16} A \,b^{3} e +\frac {55}{16} B e \,b^{2} a -\frac {13}{8} B \,b^{3} d \right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {b \left (17 A a b \,e^{2}-17 A \,b^{2} d e -35 a^{2} B \,e^{2}+53 B a b d e -18 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} A \,a^{2} b \,e^{3}+\frac {19}{8} A a \,b^{2} d \,e^{2}-\frac {19}{16} A \,b^{3} d^{2} e +\frac {41}{16} B \,e^{3} a^{3}-\frac {13}{2} B \,a^{2} b d \,e^{2}+\frac {85}{16} B a \,b^{2} d^{2} e -\frac {11}{8} B \,b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {35 \left (A b e -3 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\) | \(289\) |
| derivativedivides | \(2 e^{2} \left (\frac {\frac {B \left (e x +d \right )^{\frac {3}{2}} b}{3}+A b e \sqrt {e x +d}-4 B a e \sqrt {e x +d}+3 B b d \sqrt {e x +d}}{b^{5}}-\frac {\frac {\left (-\frac {29}{16} A a \,b^{3} e^{2}+\frac {29}{16} A \,b^{4} d e +\frac {55}{16} B \,e^{2} b^{2} a^{2}-\frac {81}{16} B a \,b^{3} d e +\frac {13}{8} B \,b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {b \left (17 A \,a^{2} b \,e^{3}-34 A a \,b^{2} d \,e^{2}+17 A \,b^{3} d^{2} e -35 B \,e^{3} a^{3}+88 B \,a^{2} b d \,e^{2}-71 B a \,b^{2} d^{2} e +18 B \,b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} A \,a^{3} b \,e^{4}+\frac {57}{16} A \,a^{2} b^{2} d \,e^{3}-\frac {57}{16} A a \,b^{3} d^{2} e^{2}+\frac {19}{16} A \,b^{4} d^{3} e +\frac {41}{16} B \,a^{4} e^{4}-\frac {145}{16} B \,a^{3} b d \,e^{3}+\frac {189}{16} B \,a^{2} b^{2} d^{2} e^{2}-\frac {107}{16} B a \,b^{3} d^{3} e +\frac {11}{8} b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {35 \left (A a b \,e^{2}-A \,b^{2} d e -3 a^{2} B \,e^{2}+5 B a b d e -2 B \,b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}}{b^{5}}\right )\) | \(409\) |
| default | \(2 e^{2} \left (\frac {\frac {B \left (e x +d \right )^{\frac {3}{2}} b}{3}+A b e \sqrt {e x +d}-4 B a e \sqrt {e x +d}+3 B b d \sqrt {e x +d}}{b^{5}}-\frac {\frac {\left (-\frac {29}{16} A a \,b^{3} e^{2}+\frac {29}{16} A \,b^{4} d e +\frac {55}{16} B \,e^{2} b^{2} a^{2}-\frac {81}{16} B a \,b^{3} d e +\frac {13}{8} B \,b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {b \left (17 A \,a^{2} b \,e^{3}-34 A a \,b^{2} d \,e^{2}+17 A \,b^{3} d^{2} e -35 B \,e^{3} a^{3}+88 B \,a^{2} b d \,e^{2}-71 B a \,b^{2} d^{2} e +18 B \,b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} A \,a^{3} b \,e^{4}+\frac {57}{16} A \,a^{2} b^{2} d \,e^{3}-\frac {57}{16} A a \,b^{3} d^{2} e^{2}+\frac {19}{16} A \,b^{4} d^{3} e +\frac {41}{16} B \,a^{4} e^{4}-\frac {145}{16} B \,a^{3} b d \,e^{3}+\frac {189}{16} B \,a^{2} b^{2} d^{2} e^{2}-\frac {107}{16} B a \,b^{3} d^{3} e +\frac {11}{8} b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {35 \left (A a b \,e^{2}-A \,b^{2} d e -3 a^{2} B \,e^{2}+5 B a b d e -2 B \,b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}}{b^{5}}\right )\) | \(409\) |
35/8/((a*e-b*d)*b)^(1/2)*(-(a*e-b*d)*(b*(A*e+2*B*d)-3*B*a*e)*(b*x+a)^3*e^2 *arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))+((16/35*(1/3*B*x+A)*x^3*e^3-2 9/35*(-160/87*B*x+A)*x^2*d*e^2-38/105*x*d^2*(39/19*B*x+A)*e-8/105*d^3*(3/2 *B*x+A))*b^4-2/15*((72/7*B*x^3-33/2*A*x^2)*e^3+7*(-723/98*B*x+A)*x*d*e^2+d ^2*(41/7*B*x+A)*e+2/7*B*d^3)*a*b^3-1/3*e*((99/5*B*x^2-8*A*x)*e^2+d*(-122/5 *B*x+A)*e+4/5*B*d^2)*a^2*b^2+((-8*B*x+A)*e+3*B*d)*e^2*a^3*b-3*B*e^3*a^4)*( (a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2))/(b*x+a)^3/b^5
Leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (252) = 504\).
Time = 0.40 (sec) , antiderivative size = 1026, normalized size of antiderivative = 3.61 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [-\frac {105 \, {\left (2 \, B a^{3} b d e^{2} - {\left (3 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d e^{2} - {\left (3 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (2 \, B a b^{3} d e^{2} - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (2 \, B a^{2} b^{2} d e^{2} - {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\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 (16 \, B b^{4} e^{3} x^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} - 14 \, {\left (2 \, B a^{2} b^{2} + A a b^{3}\right )} d^{2} e + 35 \, {\left (9 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (3 \, B a^{4} - A a^{3} b\right )} e^{3} + 16 \, {\left (10 \, B b^{4} d e^{2} - 3 \, {\left (3 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} - 3 \, {\left (26 \, B b^{4} d^{2} e - {\left (241 \, B a b^{3} - 29 \, A b^{4}\right )} d e^{2} + 77 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \, {\left (6 \, B b^{4} d^{3} + {\left (41 \, B a b^{3} + 19 \, A b^{4}\right )} d^{2} e - 7 \, {\left (61 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} + 140 \, {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}, -\frac {105 \, {\left (2 \, B a^{3} b d e^{2} - {\left (3 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d e^{2} - {\left (3 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (2 \, B a b^{3} d e^{2} - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (2 \, B a^{2} b^{2} d e^{2} - {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\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 (16 \, B b^{4} e^{3} x^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} - 14 \, {\left (2 \, B a^{2} b^{2} + A a b^{3}\right )} d^{2} e + 35 \, {\left (9 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (3 \, B a^{4} - A a^{3} b\right )} e^{3} + 16 \, {\left (10 \, B b^{4} d e^{2} - 3 \, {\left (3 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} - 3 \, {\left (26 \, B b^{4} d^{2} e - {\left (241 \, B a b^{3} - 29 \, A b^{4}\right )} d e^{2} + 77 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \, {\left (6 \, B b^{4} d^{3} + {\left (41 \, B a b^{3} + 19 \, A b^{4}\right )} d^{2} e - 7 \, {\left (61 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} + 140 \, {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}\right ] \]
[-1/48*(105*(2*B*a^3*b*d*e^2 - (3*B*a^4 - A*a^3*b)*e^3 + (2*B*b^4*d*e^2 - (3*B*a*b^3 - A*b^4)*e^3)*x^3 + 3*(2*B*a*b^3*d*e^2 - (3*B*a^2*b^2 - A*a*b^3 )*e^3)*x^2 + 3*(2*B*a^2*b^2*d*e^2 - (3*B*a^3*b - A*a^2*b^2)*e^3)*x)*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*(16*B*b^4*e^3*x^4 - 4*(B*a*b^3 + 2*A*b^4)*d^3 - 14*(2 *B*a^2*b^2 + A*a*b^3)*d^2*e + 35*(9*B*a^3*b - A*a^2*b^2)*d*e^2 - 105*(3*B* a^4 - A*a^3*b)*e^3 + 16*(10*B*b^4*d*e^2 - 3*(3*B*a*b^3 - A*b^4)*e^3)*x^3 - 3*(26*B*b^4*d^2*e - (241*B*a*b^3 - 29*A*b^4)*d*e^2 + 77*(3*B*a^2*b^2 - A* a*b^3)*e^3)*x^2 - 2*(6*B*b^4*d^3 + (41*B*a*b^3 + 19*A*b^4)*d^2*e - 7*(61*B *a^2*b^2 - 7*A*a*b^3)*d*e^2 + 140*(3*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x + a^3*b^5), -1/24*(105*(2*B*a^ 3*b*d*e^2 - (3*B*a^4 - A*a^3*b)*e^3 + (2*B*b^4*d*e^2 - (3*B*a*b^3 - A*b^4) *e^3)*x^3 + 3*(2*B*a*b^3*d*e^2 - (3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 3*(2*B *a^2*b^2*d*e^2 - (3*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arct an(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (16*B*b^4*e^3*x^4 - 4*(B*a*b^3 + 2*A*b^4)*d^3 - 14*(2*B*a^2*b^2 + A*a*b^3)*d^2*e + 35*(9*B*a ^3*b - A*a^2*b^2)*d*e^2 - 105*(3*B*a^4 - A*a^3*b)*e^3 + 16*(10*B*b^4*d*e^2 - 3*(3*B*a*b^3 - A*b^4)*e^3)*x^3 - 3*(26*B*b^4*d^2*e - (241*B*a*b^3 - 29* A*b^4)*d*e^2 + 77*(3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 - 2*(6*B*b^4*d^3 + (41* B*a*b^3 + 19*A*b^4)*d^2*e - 7*(61*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 + 140*(3...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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. 577 vs. \(2 (252) = 504\).
Time = 0.30 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.03 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {35 \, {\left (2 \, B b^{2} d^{2} e^{2} - 5 \, B a b d e^{3} + A b^{2} d e^{3} + 3 \, B a^{2} e^{4} - A a b e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {78 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{2} - 144 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{2} + 66 \, \sqrt {e x + d} B b^{4} d^{4} e^{2} - 243 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{3} d e^{3} + 87 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{4} d e^{3} + 568 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{3} - 136 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{3} - 321 \, \sqrt {e x + d} B a b^{3} d^{3} e^{3} + 57 \, \sqrt {e x + d} A b^{4} d^{3} e^{3} + 165 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{4} - 87 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{3} e^{4} - 704 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{4} + 272 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{3} d e^{4} + 567 \, \sqrt {e x + d} B a^{2} b^{2} d^{2} e^{4} - 171 \, \sqrt {e x + d} A a b^{3} d^{2} e^{4} + 280 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b e^{5} - 136 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{5} - 435 \, \sqrt {e x + d} B a^{3} b d e^{5} + 171 \, \sqrt {e x + d} A a^{2} b^{2} d e^{5} + 123 \, \sqrt {e x + d} B a^{4} e^{6} - 57 \, \sqrt {e x + d} A a^{3} b e^{6}}{24 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3} b^{5}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{8} e^{2} + 9 \, \sqrt {e x + d} B b^{8} d e^{2} - 12 \, \sqrt {e x + d} B a b^{7} e^{3} + 3 \, \sqrt {e x + d} A b^{8} e^{3}\right )}}{3 \, b^{12}} \]
35/8*(2*B*b^2*d^2*e^2 - 5*B*a*b*d*e^3 + A*b^2*d*e^3 + 3*B*a^2*e^4 - A*a*b* e^4)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^ 5) - 1/24*(78*(e*x + d)^(5/2)*B*b^4*d^2*e^2 - 144*(e*x + d)^(3/2)*B*b^4*d^ 3*e^2 + 66*sqrt(e*x + d)*B*b^4*d^4*e^2 - 243*(e*x + d)^(5/2)*B*a*b^3*d*e^3 + 87*(e*x + d)^(5/2)*A*b^4*d*e^3 + 568*(e*x + d)^(3/2)*B*a*b^3*d^2*e^3 - 136*(e*x + d)^(3/2)*A*b^4*d^2*e^3 - 321*sqrt(e*x + d)*B*a*b^3*d^3*e^3 + 57 *sqrt(e*x + d)*A*b^4*d^3*e^3 + 165*(e*x + d)^(5/2)*B*a^2*b^2*e^4 - 87*(e*x + d)^(5/2)*A*a*b^3*e^4 - 704*(e*x + d)^(3/2)*B*a^2*b^2*d*e^4 + 272*(e*x + d)^(3/2)*A*a*b^3*d*e^4 + 567*sqrt(e*x + d)*B*a^2*b^2*d^2*e^4 - 171*sqrt(e *x + d)*A*a*b^3*d^2*e^4 + 280*(e*x + d)^(3/2)*B*a^3*b*e^5 - 136*(e*x + d)^ (3/2)*A*a^2*b^2*e^5 - 435*sqrt(e*x + d)*B*a^3*b*d*e^5 + 171*sqrt(e*x + d)* A*a^2*b^2*d*e^5 + 123*sqrt(e*x + d)*B*a^4*e^6 - 57*sqrt(e*x + d)*A*a^3*b*e ^6)/(((e*x + d)*b - b*d + a*e)^3*b^5) + 2/3*((e*x + d)^(3/2)*B*b^8*e^2 + 9 *sqrt(e*x + d)*B*b^8*d*e^2 - 12*sqrt(e*x + d)*B*a*b^7*e^3 + 3*sqrt(e*x + d )*A*b^8*e^3)/b^12
Time = 10.74 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.81 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left (\frac {2\,A\,e^3-2\,B\,d\,e^2}{b^4}+\frac {2\,B\,e^2\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{b^8}\right )\,\sqrt {d+e\,x}-\frac {\sqrt {d+e\,x}\,\left (\frac {41\,B\,a^4\,e^6}{8}-\frac {145\,B\,a^3\,b\,d\,e^5}{8}-\frac {19\,A\,a^3\,b\,e^6}{8}+\frac {189\,B\,a^2\,b^2\,d^2\,e^4}{8}+\frac {57\,A\,a^2\,b^2\,d\,e^5}{8}-\frac {107\,B\,a\,b^3\,d^3\,e^3}{8}-\frac {57\,A\,a\,b^3\,d^2\,e^4}{8}+\frac {11\,B\,b^4\,d^4\,e^2}{4}+\frac {19\,A\,b^4\,d^3\,e^3}{8}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (-\frac {35\,B\,a^3\,b\,e^5}{3}+\frac {88\,B\,a^2\,b^2\,d\,e^4}{3}+\frac {17\,A\,a^2\,b^2\,e^5}{3}-\frac {71\,B\,a\,b^3\,d^2\,e^3}{3}-\frac {34\,A\,a\,b^3\,d\,e^4}{3}+6\,B\,b^4\,d^3\,e^2+\frac {17\,A\,b^4\,d^2\,e^3}{3}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {55\,B\,a^2\,b^2\,e^4}{8}-\frac {81\,B\,a\,b^3\,d\,e^3}{8}-\frac {29\,A\,a\,b^3\,e^4}{8}+\frac {13\,B\,b^4\,d^2\,e^2}{4}+\frac {29\,A\,b^4\,d\,e^3}{8}\right )}{b^8\,{\left (d+e\,x\right )}^3-\left (3\,b^8\,d-3\,a\,b^7\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^6\,e^2-6\,a\,b^7\,d\,e+3\,b^8\,d^2\right )-b^8\,d^3+a^3\,b^5\,e^3-3\,a^2\,b^6\,d\,e^2+3\,a\,b^7\,d^2\,e}+\frac {2\,B\,e^2\,{\left (d+e\,x\right )}^{3/2}}{3\,b^4}+\frac {e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,1{}\mathrm {i}}{\sqrt {b\,d-a\,e}}\right )\,\sqrt {b\,d-a\,e}\,\left (A\,b\,e-3\,B\,a\,e+2\,B\,b\,d\right )\,35{}\mathrm {i}}{8\,b^{11/2}} \]
((2*A*e^3 - 2*B*d*e^2)/b^4 + (2*B*e^2*(4*b^4*d - 4*a*b^3*e))/b^8)*(d + e*x )^(1/2) - ((d + e*x)^(1/2)*((41*B*a^4*e^6)/8 - (19*A*a^3*b*e^6)/8 + (19*A* b^4*d^3*e^3)/8 + (11*B*b^4*d^4*e^2)/4 - (57*A*a*b^3*d^2*e^4)/8 + (57*A*a^2 *b^2*d*e^5)/8 - (107*B*a*b^3*d^3*e^3)/8 + (189*B*a^2*b^2*d^2*e^4)/8 - (145 *B*a^3*b*d*e^5)/8) - (d + e*x)^(3/2)*((17*A*a^2*b^2*e^5)/3 - (35*B*a^3*b*e ^5)/3 + (17*A*b^4*d^2*e^3)/3 + 6*B*b^4*d^3*e^2 - (71*B*a*b^3*d^2*e^3)/3 + (88*B*a^2*b^2*d*e^4)/3 - (34*A*a*b^3*d*e^4)/3) + (d + e*x)^(5/2)*((29*A*b^ 4*d*e^3)/8 - (29*A*a*b^3*e^4)/8 + (55*B*a^2*b^2*e^4)/8 + (13*B*b^4*d^2*e^2 )/4 - (81*B*a*b^3*d*e^3)/8))/(b^8*(d + e*x)^3 - (3*b^8*d - 3*a*b^7*e)*(d + e*x)^2 + (d + e*x)*(3*b^8*d^2 + 3*a^2*b^6*e^2 - 6*a*b^7*d*e) - b^8*d^3 + a^3*b^5*e^3 - 3*a^2*b^6*d*e^2 + 3*a*b^7*d^2*e) + (2*B*e^2*(d + e*x)^(3/2)) /(3*b^4) + (e^2*atan((b^(1/2)*(d + e*x)^(1/2)*1i)/(b*d - a*e)^(1/2))*(b*d - a*e)^(1/2)*(A*b*e - 3*B*a*e + 2*B*b*d)*35i)/(8*b^(11/2))