Integrand size = 33, antiderivative size = 256 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {(b d-a e)^2 (2 b B d+7 A b e-9 a B e) \sqrt {d+e x}}{b^5}+\frac {(b d-a e) (2 b B d+7 A b e-9 a B e) (d+e x)^{3/2}}{3 b^4}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{5/2}}{5 b^3}+\frac {(2 b B d+7 A b e-9 a B e) (d+e x)^{7/2}}{7 b^2 (b d-a e)}-\frac {(A b-a B) (d+e x)^{9/2}}{b (b d-a e) (a+b x)}-\frac {(b d-a e)^{5/2} (2 b B d+7 A b e-9 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{11/2}} \]
1/3*(-a*e+b*d)*(7*A*b*e-9*B*a*e+2*B*b*d)*(e*x+d)^(3/2)/b^4+1/5*(7*A*b*e-9* B*a*e+2*B*b*d)*(e*x+d)^(5/2)/b^3+1/7*(7*A*b*e-9*B*a*e+2*B*b*d)*(e*x+d)^(7/ 2)/b^2/(-a*e+b*d)-(A*b-B*a)*(e*x+d)^(9/2)/b/(-a*e+b*d)/(b*x+a)-(-a*e+b*d)^ (5/2)*(7*A*b*e-9*B*a*e+2*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^( 1/2))/b^(11/2)+(-a*e+b*d)^2*(7*A*b*e-9*B*a*e+2*B*b*d)*(e*x+d)^(1/2)/b^5
Time = 0.16 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.19 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {\sqrt {d+e x} \left (-7 A b \left (-105 a^3 e^3+35 a^2 b e^2 (7 d-2 e x)+7 a b^2 e \left (-23 d^2+24 d e x+2 e^2 x^2\right )+b^3 \left (15 d^3-116 d^2 e x-32 d e^2 x^2-6 e^3 x^3\right )\right )+B \left (-945 a^4 e^3+105 a^3 b e^2 (23 d-6 e x)+7 a^2 b^2 e \left (-277 d^2+236 d e x+18 e^2 x^2\right )+a b^3 \left (457 d^3-1380 d^2 e x-316 d e^2 x^2-54 e^3 x^3\right )+2 b^4 x \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )\right )}{105 b^5 (a+b x)}-\frac {(-b d+a e)^{5/2} (2 b B d+7 A b e-9 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{11/2}} \]
(Sqrt[d + e*x]*(-7*A*b*(-105*a^3*e^3 + 35*a^2*b*e^2*(7*d - 2*e*x) + 7*a*b^ 2*e*(-23*d^2 + 24*d*e*x + 2*e^2*x^2) + b^3*(15*d^3 - 116*d^2*e*x - 32*d*e^ 2*x^2 - 6*e^3*x^3)) + B*(-945*a^4*e^3 + 105*a^3*b*e^2*(23*d - 6*e*x) + 7*a ^2*b^2*e*(-277*d^2 + 236*d*e*x + 18*e^2*x^2) + a*b^3*(457*d^3 - 1380*d^2*e *x - 316*d*e^2*x^2 - 54*e^3*x^3) + 2*b^4*x*(176*d^3 + 122*d^2*e*x + 66*d*e ^2*x^2 + 15*e^3*x^3))))/(105*b^5*(a + b*x)) - ((-(b*d) + a*e)^(5/2)*(2*b*B *d + 7*A*b*e - 9*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]] )/b^(11/2)
Time = 0.38 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.87, 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, 60, 60, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^2 \int \frac {(A+B x) (d+e x)^{7/2}}{b^2 (a+b x)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^2}dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-9 a B e+7 A b e+2 b B d) \int \frac {(d+e x)^{7/2}}{a+b x}dx}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-9 a B e+7 A b e+2 b B d) \left (\frac {(b d-a e) \int \frac {(d+e x)^{5/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-9 a B e+7 A b e+2 b B d) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {(d+e x)^{3/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-9 a B e+7 A b e+2 b B d) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(-9 a B e+7 A b e+2 b B d) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(-9 a B e+7 A b e+2 b B d) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(-9 a B e+7 A b e+2 b B d) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{2 b (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{b (a+b x) (b d-a e)}\) |
-(((A*b - a*B)*(d + e*x)^(9/2))/(b*(b*d - a*e)*(a + b*x))) + ((2*b*B*d + 7 *A*b*e - 9*a*B*e)*((2*(d + e*x)^(7/2))/(7*b) + ((b*d - a*e)*((2*(d + e*x)^ (5/2))/(5*b) + ((b*d - a*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))/b))/b))/(2*b*(b*d - a*e))
3.19.8.3.1 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 + 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 = 0.54 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.09
| method | result | size |
| pseudoelliptic | \(\frac {-7 \left (a e -b d \right )^{3} \left (b x +a \right ) \left (b \left (A e +\frac {2 B d}{7}\right )-\frac {9 B a e}{7}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+7 \sqrt {\left (a e -b d \right ) b}\, \left (\left (\frac {2 x^{3} \left (\frac {5 B x}{7}+A \right ) e^{3}}{35}+\frac {32 x^{2} d \left (\frac {33 B x}{56}+A \right ) e^{2}}{105}+\frac {116 x \left (\frac {61 B x}{203}+A \right ) d^{2} e}{105}-\frac {\left (-\frac {352 B x}{105}+A \right ) d^{3}}{7}\right ) b^{4}+\frac {23 \left (-\frac {2 x^{2} \left (\frac {27 B x}{49}+A \right ) e^{3}}{23}-\frac {24 x \left (\frac {79 B x}{294}+A \right ) d \,e^{2}}{23}+d^{2} \left (-\frac {60 B x}{49}+A \right ) e +\frac {457 B \,d^{3}}{1127}\right ) a \,b^{3}}{15}-\frac {7 \left (-\frac {2 \left (\frac {9 B x}{35}+A \right ) x \,e^{2}}{7}+d \left (-\frac {236 B x}{245}+A \right ) e +\frac {277 B \,d^{2}}{245}\right ) e \,a^{2} b^{2}}{3}+\left (\left (-\frac {6 B x}{7}+A \right ) e +\frac {23 B d}{7}\right ) e^{2} a^{3} b -\frac {9 B \,e^{3} a^{4}}{7}\right ) \sqrt {e x +d}}{b^{5} \left (b x +a \right ) \sqrt {\left (a e -b d \right ) b}}\) | \(278\) |
| risch | \(\frac {2 \left (15 x^{3} B \,b^{3} e^{3}+21 A \,b^{3} e^{3} x^{2}-42 B a \,b^{2} e^{3} x^{2}+66 B \,b^{3} d \,e^{2} x^{2}-70 A x a \,b^{2} e^{3}+112 A \,b^{3} d \,e^{2} x +105 B x \,a^{2} b \,e^{3}-224 B a \,b^{2} d \,e^{2} x +122 B \,b^{3} d^{2} e x +315 A \,a^{2} b \,e^{3}-700 A a \,b^{2} d \,e^{2}+406 A \,b^{3} d^{2} e -420 B \,e^{3} a^{3}+1050 B \,a^{2} b d \,e^{2}-812 B a \,b^{2} d^{2} e +176 B \,b^{3} d^{3}\right ) \sqrt {e x +d}}{105 b^{5}}-\frac {\left (2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) \left (\frac {\left (-\frac {1}{2} A b e +\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (7 A b e -9 B a e +2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\) | \(323\) |
| derivativedivides | \(\frac {\frac {2 B \left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 A \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,b^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 A \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {8 B a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A \,a^{2} b \,e^{3} \sqrt {e x +d}-12 A a \,b^{2} d \,e^{2} \sqrt {e x +d}+6 A \,b^{3} d^{2} e \sqrt {e x +d}-8 B \,a^{3} e^{3} \sqrt {e x +d}+18 B \,a^{2} b d \,e^{2} \sqrt {e x +d}-12 B a \,b^{2} d^{2} e \sqrt {e x +d}+2 B \,b^{3} d^{3} \sqrt {e x +d}}{b^{5}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A \,a^{3} b \,e^{4}+\frac {3}{2} A \,a^{2} b^{2} d \,e^{3}-\frac {3}{2} A a \,b^{3} d^{2} e^{2}+\frac {1}{2} A \,b^{4} d^{3} e +\frac {1}{2} B \,a^{4} e^{4}-\frac {3}{2} B \,a^{3} b d \,e^{3}+\frac {3}{2} B \,a^{2} b^{2} d^{2} e^{2}-\frac {1}{2} B a \,b^{3} d^{3} e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (7 A \,a^{3} b \,e^{4}-21 A \,a^{2} b^{2} d \,e^{3}+21 A a \,b^{3} d^{2} e^{2}-7 A \,b^{4} d^{3} e -9 B \,a^{4} e^{4}+29 B \,a^{3} b d \,e^{3}-33 B \,a^{2} b^{2} d^{2} e^{2}+15 B a \,b^{3} d^{3} e -2 b^{4} B \,d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\) | \(525\) |
| default | \(\frac {\frac {2 B \left (e x +d \right )^{\frac {7}{2}} b^{3}}{7}+\frac {2 A \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 B \,b^{3} d \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {4 A \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {8 B a \,b^{2} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B \,b^{3} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 A \,a^{2} b \,e^{3} \sqrt {e x +d}-12 A a \,b^{2} d \,e^{2} \sqrt {e x +d}+6 A \,b^{3} d^{2} e \sqrt {e x +d}-8 B \,a^{3} e^{3} \sqrt {e x +d}+18 B \,a^{2} b d \,e^{2} \sqrt {e x +d}-12 B a \,b^{2} d^{2} e \sqrt {e x +d}+2 B \,b^{3} d^{3} \sqrt {e x +d}}{b^{5}}-\frac {2 \left (\frac {\left (-\frac {1}{2} A \,a^{3} b \,e^{4}+\frac {3}{2} A \,a^{2} b^{2} d \,e^{3}-\frac {3}{2} A a \,b^{3} d^{2} e^{2}+\frac {1}{2} A \,b^{4} d^{3} e +\frac {1}{2} B \,a^{4} e^{4}-\frac {3}{2} B \,a^{3} b d \,e^{3}+\frac {3}{2} B \,a^{2} b^{2} d^{2} e^{2}-\frac {1}{2} B a \,b^{3} d^{3} e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (7 A \,a^{3} b \,e^{4}-21 A \,a^{2} b^{2} d \,e^{3}+21 A a \,b^{3} d^{2} e^{2}-7 A \,b^{4} d^{3} e -9 B \,a^{4} e^{4}+29 B \,a^{3} b d \,e^{3}-33 B \,a^{2} b^{2} d^{2} e^{2}+15 B a \,b^{3} d^{3} e -2 b^{4} B \,d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\) | \(525\) |
7*(-(a*e-b*d)^3*(b*x+a)*(b*(A*e+2/7*B*d)-9/7*B*a*e)*arctan(b*(e*x+d)^(1/2) /((a*e-b*d)*b)^(1/2))+((a*e-b*d)*b)^(1/2)*((2/35*x^3*(5/7*B*x+A)*e^3+32/10 5*x^2*d*(33/56*B*x+A)*e^2+116/105*x*(61/203*B*x+A)*d^2*e-1/7*(-352/105*B*x +A)*d^3)*b^4+23/15*(-2/23*x^2*(27/49*B*x+A)*e^3-24/23*x*(79/294*B*x+A)*d*e ^2+d^2*(-60/49*B*x+A)*e+457/1127*B*d^3)*a*b^3-7/3*(-2/7*(9/35*B*x+A)*x*e^2 +d*(-236/245*B*x+A)*e+277/245*B*d^2)*e*a^2*b^2+((-6/7*B*x+A)*e+23/7*B*d)*e ^2*a^3*b-9/7*B*e^3*a^4)*(e*x+d)^(1/2))/((a*e-b*d)*b)^(1/2)/b^5/(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (229) = 458\).
Time = 0.48 (sec) , antiderivative size = 1006, normalized size of antiderivative = 3.93 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left [\frac {105 \, {\left (2 \, B a b^{3} d^{3} - {\left (13 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d^{2} e + 2 \, {\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d e^{2} - {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d^{3} - {\left (13 \, B a b^{3} - 7 \, A b^{4}\right )} d^{2} e + 2 \, {\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} - {\left (9 \, B a^{3} b - 7 \, 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 (30 \, B b^{4} e^{3} x^{4} + {\left (457 \, B a b^{3} - 105 \, A b^{4}\right )} d^{3} - 7 \, {\left (277 \, B a^{2} b^{2} - 161 \, A a b^{3}\right )} d^{2} e + 35 \, {\left (69 \, B a^{3} b - 49 \, A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + 6 \, {\left (22 \, B b^{4} d e^{2} - {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} e^{3}\right )} x^{3} + 2 \, {\left (122 \, B b^{4} d^{2} e - 2 \, {\left (79 \, B a b^{3} - 56 \, A b^{4}\right )} d e^{2} + 7 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (176 \, B b^{4} d^{3} - 2 \, {\left (345 \, B a b^{3} - 203 \, A b^{4}\right )} d^{2} e + 14 \, {\left (59 \, B a^{2} b^{2} - 42 \, A a b^{3}\right )} d e^{2} - 35 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{210 \, {\left (b^{6} x + a b^{5}\right )}}, -\frac {105 \, {\left (2 \, B a b^{3} d^{3} - {\left (13 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d^{2} e + 2 \, {\left (10 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} d e^{2} - {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d^{3} - {\left (13 \, B a b^{3} - 7 \, A b^{4}\right )} d^{2} e + 2 \, {\left (10 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} - {\left (9 \, B a^{3} b - 7 \, 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 (30 \, B b^{4} e^{3} x^{4} + {\left (457 \, B a b^{3} - 105 \, A b^{4}\right )} d^{3} - 7 \, {\left (277 \, B a^{2} b^{2} - 161 \, A a b^{3}\right )} d^{2} e + 35 \, {\left (69 \, B a^{3} b - 49 \, A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} e^{3} + 6 \, {\left (22 \, B b^{4} d e^{2} - {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} e^{3}\right )} x^{3} + 2 \, {\left (122 \, B b^{4} d^{2} e - 2 \, {\left (79 \, B a b^{3} - 56 \, A b^{4}\right )} d e^{2} + 7 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} e^{3}\right )} x^{2} + 2 \, {\left (176 \, B b^{4} d^{3} - 2 \, {\left (345 \, B a b^{3} - 203 \, A b^{4}\right )} d^{2} e + 14 \, {\left (59 \, B a^{2} b^{2} - 42 \, A a b^{3}\right )} d e^{2} - 35 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \]
[1/210*(105*(2*B*a*b^3*d^3 - (13*B*a^2*b^2 - 7*A*a*b^3)*d^2*e + 2*(10*B*a^ 3*b - 7*A*a^2*b^2)*d*e^2 - (9*B*a^4 - 7*A*a^3*b)*e^3 + (2*B*b^4*d^3 - (13* B*a*b^3 - 7*A*b^4)*d^2*e + 2*(10*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 - (9*B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*s qrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(30*B*b^4*e^3*x^4 + (45 7*B*a*b^3 - 105*A*b^4)*d^3 - 7*(277*B*a^2*b^2 - 161*A*a*b^3)*d^2*e + 35*(6 9*B*a^3*b - 49*A*a^2*b^2)*d*e^2 - 105*(9*B*a^4 - 7*A*a^3*b)*e^3 + 6*(22*B* b^4*d*e^2 - (9*B*a*b^3 - 7*A*b^4)*e^3)*x^3 + 2*(122*B*b^4*d^2*e - 2*(79*B* a*b^3 - 56*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 7*A*a*b^3)*e^3)*x^2 + 2*(176*B* b^4*d^3 - 2*(345*B*a*b^3 - 203*A*b^4)*d^2*e + 14*(59*B*a^2*b^2 - 42*A*a*b^ 3)*d*e^2 - 35*(9*B*a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^6*x + a* b^5), -1/105*(105*(2*B*a*b^3*d^3 - (13*B*a^2*b^2 - 7*A*a*b^3)*d^2*e + 2*(1 0*B*a^3*b - 7*A*a^2*b^2)*d*e^2 - (9*B*a^4 - 7*A*a^3*b)*e^3 + (2*B*b^4*d^3 - (13*B*a*b^3 - 7*A*b^4)*d^2*e + 2*(10*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 - (9*B *a^3*b - 7*A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b *sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (30*B*b^4*e^3*x^4 + (457*B*a*b^3 - 10 5*A*b^4)*d^3 - 7*(277*B*a^2*b^2 - 161*A*a*b^3)*d^2*e + 35*(69*B*a^3*b - 49 *A*a^2*b^2)*d*e^2 - 105*(9*B*a^4 - 7*A*a^3*b)*e^3 + 6*(22*B*b^4*d*e^2 - (9 *B*a*b^3 - 7*A*b^4)*e^3)*x^3 + 2*(122*B*b^4*d^2*e - 2*(79*B*a*b^3 - 56*A*b ^4)*d*e^2 + 7*(9*B*a^2*b^2 - 7*A*a*b^3)*e^3)*x^2 + 2*(176*B*b^4*d^3 - 2...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (229) = 458\).
Time = 0.29 (sec) , antiderivative size = 583, normalized size of antiderivative = 2.28 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {{\left (2 \, B b^{4} d^{4} - 15 \, B a b^{3} d^{3} e + 7 \, A b^{4} d^{3} e + 33 \, B a^{2} b^{2} d^{2} e^{2} - 21 \, A a b^{3} d^{2} e^{2} - 29 \, B a^{3} b d e^{3} + 21 \, A a^{2} b^{2} d e^{3} + 9 \, B a^{4} e^{4} - 7 \, A a^{3} b e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{5}} + \frac {\sqrt {e x + d} B a b^{3} d^{3} e - \sqrt {e x + d} A b^{4} d^{3} e - 3 \, \sqrt {e x + d} B a^{2} b^{2} d^{2} e^{2} + 3 \, \sqrt {e x + d} A a b^{3} d^{2} e^{2} + 3 \, \sqrt {e x + d} B a^{3} b d e^{3} - 3 \, \sqrt {e x + d} A a^{2} b^{2} d e^{3} - \sqrt {e x + d} B a^{4} e^{4} + \sqrt {e x + d} A a^{3} b e^{4}}{{\left ({\left (e x + d\right )} b - b d + a e\right )} b^{5}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{12} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{12} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{12} d^{2} + 105 \, \sqrt {e x + d} B b^{12} d^{3} - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{11} e + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{12} e - 140 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{11} d e + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{12} d e - 630 \, \sqrt {e x + d} B a b^{11} d^{2} e + 315 \, \sqrt {e x + d} A b^{12} d^{2} e + 105 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{10} e^{2} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{11} e^{2} + 945 \, \sqrt {e x + d} B a^{2} b^{10} d e^{2} - 630 \, \sqrt {e x + d} A a b^{11} d e^{2} - 420 \, \sqrt {e x + d} B a^{3} b^{9} e^{3} + 315 \, \sqrt {e x + d} A a^{2} b^{10} e^{3}\right )}}{105 \, b^{14}} \]
(2*B*b^4*d^4 - 15*B*a*b^3*d^3*e + 7*A*b^4*d^3*e + 33*B*a^2*b^2*d^2*e^2 - 2 1*A*a*b^3*d^2*e^2 - 29*B*a^3*b*d*e^3 + 21*A*a^2*b^2*d*e^3 + 9*B*a^4*e^4 - 7*A*a^3*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) + (sqrt(e*x + d)*B*a*b^3*d^3*e - sqrt(e*x + d)*A*b^4*d^3*e - 3*sqrt(e*x + d)*B*a^2*b^2*d^2*e^2 + 3*sqrt(e*x + d)*A*a*b^3*d^2*e^2 + 3*sq rt(e*x + d)*B*a^3*b*d*e^3 - 3*sqrt(e*x + d)*A*a^2*b^2*d*e^3 - sqrt(e*x + d )*B*a^4*e^4 + sqrt(e*x + d)*A*a^3*b*e^4)/(((e*x + d)*b - b*d + a*e)*b^5) + 2/105*(15*(e*x + d)^(7/2)*B*b^12 + 21*(e*x + d)^(5/2)*B*b^12*d + 35*(e*x + d)^(3/2)*B*b^12*d^2 + 105*sqrt(e*x + d)*B*b^12*d^3 - 42*(e*x + d)^(5/2)* B*a*b^11*e + 21*(e*x + d)^(5/2)*A*b^12*e - 140*(e*x + d)^(3/2)*B*a*b^11*d* e + 70*(e*x + d)^(3/2)*A*b^12*d*e - 630*sqrt(e*x + d)*B*a*b^11*d^2*e + 315 *sqrt(e*x + d)*A*b^12*d^2*e + 105*(e*x + d)^(3/2)*B*a^2*b^10*e^2 - 70*(e*x + d)^(3/2)*A*a*b^11*e^2 + 945*sqrt(e*x + d)*B*a^2*b^10*d*e^2 - 630*sqrt(e *x + d)*A*a*b^11*d*e^2 - 420*sqrt(e*x + d)*B*a^3*b^9*e^3 + 315*sqrt(e*x + d)*A*a^2*b^10*e^3)/b^14
Time = 0.16 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.20 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left (\frac {2\,A\,e-2\,B\,d}{5\,b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{5\,b^4}\right )\,{\left (d+e\,x\right )}^{5/2}+\left (\frac {\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^2}-\frac {{\left (a\,e-b\,d\right )}^2\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{b^2}\right )\,\sqrt {d+e\,x}+\left (\frac {\left (2\,b^2\,d-2\,a\,b\,e\right )\,\left (\frac {2\,A\,e-2\,B\,d}{b^2}+\frac {2\,B\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{b^4}\right )}{3\,b^2}-\frac {2\,B\,{\left (a\,e-b\,d\right )}^2}{3\,b^4}\right )\,{\left (d+e\,x\right )}^{3/2}-\frac {\sqrt {d+e\,x}\,\left (B\,a^4\,e^4-3\,B\,a^3\,b\,d\,e^3-A\,a^3\,b\,e^4+3\,B\,a^2\,b^2\,d^2\,e^2+3\,A\,a^2\,b^2\,d\,e^3-B\,a\,b^3\,d^3\,e-3\,A\,a\,b^3\,d^2\,e^2+A\,b^4\,d^3\,e\right )}{b^6\,\left (d+e\,x\right )-b^6\,d+a\,b^5\,e}+\frac {2\,B\,{\left (d+e\,x\right )}^{7/2}}{7\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}\,\left (7\,A\,b\,e-9\,B\,a\,e+2\,B\,b\,d\right )}{9\,B\,a^4\,e^4-29\,B\,a^3\,b\,d\,e^3-7\,A\,a^3\,b\,e^4+33\,B\,a^2\,b^2\,d^2\,e^2+21\,A\,a^2\,b^2\,d\,e^3-15\,B\,a\,b^3\,d^3\,e-21\,A\,a\,b^3\,d^2\,e^2+2\,B\,b^4\,d^4+7\,A\,b^4\,d^3\,e}\right )\,{\left (a\,e-b\,d\right )}^{5/2}\,\left (7\,A\,b\,e-9\,B\,a\,e+2\,B\,b\,d\right )}{b^{11/2}} \]
((2*A*e - 2*B*d)/(5*b^2) + (2*B*(2*b^2*d - 2*a*b*e))/(5*b^4))*(d + e*x)^(5 /2) + (((((2*b^2*d - 2*a*b*e)*((2*A*e - 2*B*d)/b^2 + (2*B*(2*b^2*d - 2*a*b *e))/b^4))/b^2 - (2*B*(a*e - b*d)^2)/b^4)*(2*b^2*d - 2*a*b*e))/b^2 - ((a*e - b*d)^2*((2*A*e - 2*B*d)/b^2 + (2*B*(2*b^2*d - 2*a*b*e))/b^4))/b^2)*(d + e*x)^(1/2) + (((2*b^2*d - 2*a*b*e)*((2*A*e - 2*B*d)/b^2 + (2*B*(2*b^2*d - 2*a*b*e))/b^4))/(3*b^2) - (2*B*(a*e - b*d)^2)/(3*b^4))*(d + e*x)^(3/2) - ((d + e*x)^(1/2)*(B*a^4*e^4 - A*a^3*b*e^4 + A*b^4*d^3*e - 3*A*a*b^3*d^2*e^ 2 + 3*A*a^2*b^2*d*e^3 + 3*B*a^2*b^2*d^2*e^2 - B*a*b^3*d^3*e - 3*B*a^3*b*d* e^3))/(b^6*(d + e*x) - b^6*d + a*b^5*e) + (2*B*(d + e*x)^(7/2))/(7*b^2) + (atan((b^(1/2)*(a*e - b*d)^(5/2)*(d + e*x)^(1/2)*(7*A*b*e - 9*B*a*e + 2*B* b*d))/(9*B*a^4*e^4 + 2*B*b^4*d^4 - 7*A*a^3*b*e^4 + 7*A*b^4*d^3*e - 21*A*a* b^3*d^2*e^2 + 21*A*a^2*b^2*d*e^3 + 33*B*a^2*b^2*d^2*e^2 - 15*B*a*b^3*d^3*e - 29*B*a^3*b*d*e^3))*(a*e - b*d)^(5/2)*(7*A*b*e - 9*B*a*e + 2*B*b*d))/b^( 11/2)