Integrand size = 35, antiderivative size = 489 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 \sqrt {b d-a e} (8 b B d+3 A b e-11 a B e) (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
35/64*e^3*(3*A*b*e-11*B*a*e+8*B*b*d)*(b*x+a)*(e*x+d)^(3/2)/b^5/(-a*e+b*d)/ ((b*x+a)^2)^(1/2)-21/64*e^2*(3*A*b*e-11*B*a*e+8*B*b*d)*(e*x+d)^(5/2)/b^4/( -a*e+b*d)/((b*x+a)^2)^(1/2)-3/32*e*(3*A*b*e-11*B*a*e+8*B*b*d)*(e*x+d)^(7/2 )/b^3/(-a*e+b*d)/(b*x+a)/((b*x+a)^2)^(1/2)-1/24*(3*A*b*e-11*B*a*e+8*B*b*d) *(e*x+d)^(9/2)/b^2/(-a*e+b*d)/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/4*(A*b-B*a)*(e *x+d)^(11/2)/b/(-a*e+b*d)/(b*x+a)^3/((b*x+a)^2)^(1/2)-105/64*e^3*(3*A*b*e- 11*B*a*e+8*B*b*d)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))* (-a*e+b*d)^(1/2)/b^(13/2)/((b*x+a)^2)^(1/2)+105/64*e^3*(3*A*b*e-11*B*a*e+8 *B*b*d)*(b*x+a)*(e*x+d)^(1/2)/b^6/((b*x+a)^2)^(1/2)
Time = 2.28 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3 (a+b x) \left (-\frac {\sqrt {b} \sqrt {d+e x} \left (3 A b \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )+B \left (3465 a^5 e^4+105 a^4 b e^3 (-35 d+121 e x)+21 a^3 b^2 e^2 \left (18 d^2-649 d e x+803 e^2 x^2\right )+9 a^2 b^3 e \left (8 d^3+164 d^2 e x-2041 d e^2 x^2+1023 e^3 x^3\right )+8 b^5 x \left (8 d^4+50 d^3 e x+165 d^2 e^2 x^2-208 d e^3 x^3-16 e^4 x^4\right )+a b^4 \left (16 d^4+280 d^3 e x+2130 d^2 e^2 x^2-10271 d e^3 x^3+1408 e^4 x^4\right )\right )\right )}{e^3 (a+b x)^4}-315 \sqrt {-b d+a e} (8 b B d+3 A b e-11 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{192 b^{13/2} \sqrt {(a+b x)^2}} \]
(e^3*(a + b*x)*(-((Sqrt[b]*Sqrt[d + e*x]*(3*A*b*(-315*a^4*e^4 + 105*a^3*b* e^3*(d - 11*e*x) + 21*a^2*b^2*e^2*(2*d^2 + 19*d*e*x - 73*e^2*x^2) + 3*a*b^ 3*e*(8*d^3 + 52*d^2*e*x + 185*d*e^2*x^2 - 279*e^3*x^3) + b^4*(16*d^4 + 88* d^3*e*x + 210*d^2*e^2*x^2 + 325*d*e^3*x^3 - 128*e^4*x^4)) + B*(3465*a^5*e^ 4 + 105*a^4*b*e^3*(-35*d + 121*e*x) + 21*a^3*b^2*e^2*(18*d^2 - 649*d*e*x + 803*e^2*x^2) + 9*a^2*b^3*e*(8*d^3 + 164*d^2*e*x - 2041*d*e^2*x^2 + 1023*e ^3*x^3) + 8*b^5*x*(8*d^4 + 50*d^3*e*x + 165*d^2*e^2*x^2 - 208*d*e^3*x^3 - 16*e^4*x^4) + a*b^4*(16*d^4 + 280*d^3*e*x + 2130*d^2*e^2*x^2 - 10271*d*e^3 *x^3 + 1408*e^4*x^4))))/(e^3*(a + b*x)^4)) - 315*Sqrt[-(b*d) + a*e]*(8*b*B *d + 3*A*b*e - 11*a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e] ]))/(192*b^(13/2)*Sqrt[(a + b*x)^2])
Time = 0.40 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.58, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1187, 27, 87, 51, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {(A+B x) (d+e x)^{9/2}}{b^5 (a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(A+B x) (d+e x)^{9/2}}{(a+b x)^5}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \int \frac {(d+e x)^{9/2}}{(a+b x)^4}dx}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \int \frac {(d+e x)^{7/2}}{(a+b x)^3}dx}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-11 a B e+3 A b e+8 b B d) \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b (b d-a e)}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^4 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/4*((A*b - a*B)*(d + e*x)^(11/2))/(b*(b*d - a*e)*(a + b*x)^4 ) + ((8*b*B*d + 3*A*b*e - 11*a*B*e)*(-1/3*(d + e*x)^(9/2)/(b*(a + b*x)^3) + (3*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)))/(8*b*(b*d - a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.19.75.3.1 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[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.33 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(\frac {2 e^{3} \left (B b e x +3 A b e -15 B a e +13 B b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{6} \left (b x +a \right )}-\frac {\left (2 a e -2 b d \right ) e^{3} \left (\frac {\left (-\frac {325}{128} A \,b^{4} e +\frac {765}{128} B e \,b^{3} a -\frac {55}{16} B \,b^{4} d \right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {5 b^{2} \left (459 A a b \,e^{2}-459 A \,b^{2} d e -1171 a^{2} B \,e^{2}+1883 B a b d e -712 B \,b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {643}{128} A \,a^{2} b^{2} e^{3}+\frac {643}{64} A a \,b^{3} d \,e^{2}-\frac {643}{128} A \,b^{4} d^{2} e +\frac {5153}{384} B \,e^{3} b \,a^{3}-\frac {2255}{64} B \,a^{2} b^{2} d \,e^{2}+\frac {3867}{128} B a \,b^{3} d^{2} e -\frac {403}{48} B \,b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} A \,a^{3} b \,e^{4}+\frac {561}{128} A \,a^{2} b^{2} d \,e^{3}-\frac {561}{128} A a \,b^{3} d^{2} e^{2}+\frac {187}{128} A \,b^{4} d^{3} e +\frac {515}{128} B \,a^{4} e^{4}-\frac {1873}{128} B \,a^{3} b d \,e^{3}+\frac {2529}{128} B \,a^{2} b^{2} d^{2} e^{2}-\frac {1499}{128} B a \,b^{3} d^{3} e +\frac {41}{16} b^{4} B \,d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {105 \left (3 A b e -11 B a e +8 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{6} \left (b x +a \right )}\) | \(439\) |
| default | \(\text {Expression too large to display}\) | \(2430\) |
2/3*e^3*(B*b*e*x+3*A*b*e-15*B*a*e+13*B*b*d)*(e*x+d)^(1/2)/b^6*((b*x+a)^2)^ (1/2)/(b*x+a)-1/b^6*(2*a*e-2*b*d)*e^3*(((-325/128*A*b^4*e+765/128*B*e*b^3* a-55/16*B*b^4*d)*(e*x+d)^(7/2)-5/384*b^2*(459*A*a*b*e^2-459*A*b^2*d*e-1171 *B*a^2*e^2+1883*B*a*b*d*e-712*B*b^2*d^2)*(e*x+d)^(5/2)+(-643/128*A*a^2*b^2 *e^3+643/64*A*a*b^3*d*e^2-643/128*A*b^4*d^2*e+5153/384*B*e^3*b*a^3-2255/64 *B*a^2*b^2*d*e^2+3867/128*B*a*b^3*d^2*e-403/48*B*b^4*d^3)*(e*x+d)^(3/2)+(- 187/128*A*a^3*b*e^4+561/128*A*a^2*b^2*d*e^3-561/128*A*a*b^3*d^2*e^2+187/12 8*A*b^4*d^3*e+515/128*B*a^4*e^4-1873/128*B*a^3*b*d*e^3+2529/128*B*a^2*b^2* d^2*e^2-1499/128*B*a*b^3*d^3*e+41/16*b^4*B*d^4)*(e*x+d)^(1/2))/(b*(e*x+d)+ a*e-b*d)^4+105/128*(3*A*b*e-11*B*a*e+8*B*b*d)/((a*e-b*d)*b)^(1/2)*arctan(b *(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))*((b*x+a)^2)^(1/2)/(b*x+a)
Time = 0.50 (sec) , antiderivative size = 1414, normalized size of antiderivative = 2.89 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[-1/384*(315*(8*B*a^4*b*d*e^3 - (11*B*a^5 - 3*A*a^4*b)*e^4 + (8*B*b^5*d*e^ 3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (11*B*a^3*b^2 - 3*A*a^2*b^3) *e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (11*B*a^4*b - 3*A*a^3*b^2)*e^4)*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*(128*B*b^5*e^4*x^5 - 16*(B*a*b^4 + 3*A*b^5)*d^4 - 7 2*(B*a^2*b^3 + A*a*b^4)*d^3*e - 126*(3*B*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 10 5*(35*B*a^4*b - 3*A*a^3*b^2)*d*e^3 - 315*(11*B*a^5 - 3*A*a^4*b)*e^4 + 128* (13*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 - (1320*B*b^5*d^2*e^2 - (10271*B*a*b^4 - 975*A*b^5)*d*e^3 + 837*(11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^ 3 - (400*B*b^5*d^3*e + 30*(71*B*a*b^4 + 21*A*b^5)*d^2*e^2 - 9*(2041*B*a^2* b^3 - 185*A*a*b^4)*d*e^3 + 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 - (6 4*B*b^5*d^4 + 8*(35*B*a*b^4 + 33*A*b^5)*d^3*e + 36*(41*B*a^2*b^3 + 13*A*a* b^4)*d^2*e^2 - 21*(649*B*a^3*b^2 - 57*A*a^2*b^3)*d*e^3 + 1155*(11*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8* x^2 + 4*a^3*b^7*x + a^4*b^6), -1/192*(315*(8*B*a^4*b*d*e^3 - (11*B*a^5 - 3 *A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 + 4*(8*B* a*b^4*d*e^3 - (11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (11*B*a^4* b - 3*A*a^3*b^2)*e^4)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*s...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 824 vs. \(2 (376) = 752\).
Time = 0.32 (sec) , antiderivative size = 824, normalized size of antiderivative = 1.69 \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {105 \, {\left (8 \, B b^{2} d^{2} e^{3} - 19 \, B a b d e^{4} + 3 \, A b^{2} d e^{4} + 11 \, B a^{2} e^{5} - 3 \, A a b e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{6} \mathrm {sgn}\left (b x + a\right )} - \frac {1320 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{3} - 3560 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{3} + 3224 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{3} - 984 \, \sqrt {e x + d} B b^{5} d^{5} e^{3} - 3615 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{4} d e^{4} + 975 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{5} d e^{4} + 12975 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{4} - 2295 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{4} - 14825 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{4} + 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{4} + 5481 \, \sqrt {e x + d} B a b^{4} d^{4} e^{4} - 561 \, \sqrt {e x + d} A b^{5} d^{4} e^{4} + 2295 \, {\left (e x + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{5} - 975 \, {\left (e x + d\right )}^{\frac {7}{2}} A a b^{4} e^{5} - 15270 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{5} + 4590 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{4} d e^{5} + 25131 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{5} - 5787 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{5} - 12084 \, \sqrt {e x + d} B a^{2} b^{3} d^{3} e^{5} + 2244 \, \sqrt {e x + d} A a b^{4} d^{3} e^{5} + 5855 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{6} - 2295 \, {\left (e x + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{6} - 18683 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{6} + 5787 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{6} + 13206 \, \sqrt {e x + d} B a^{3} b^{2} d^{2} e^{6} - 3366 \, \sqrt {e x + d} A a^{2} b^{3} d^{2} e^{6} + 5153 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{4} b e^{7} - 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{7} - 7164 \, \sqrt {e x + d} B a^{4} b d e^{7} + 2244 \, \sqrt {e x + d} A a^{3} b^{2} d e^{7} + 1545 \, \sqrt {e x + d} B a^{5} e^{8} - 561 \, \sqrt {e x + d} A a^{4} b e^{8}}{192 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{6} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{10} e^{3} + 12 \, \sqrt {e x + d} B b^{10} d e^{3} - 15 \, \sqrt {e x + d} B a b^{9} e^{4} + 3 \, \sqrt {e x + d} A b^{10} e^{4}\right )}}{3 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \]
105/64*(8*B*b^2*d^2*e^3 - 19*B*a*b*d*e^4 + 3*A*b^2*d*e^4 + 11*B*a^2*e^5 - 3*A*a*b*e^5)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a *b*e)*b^6*sgn(b*x + a)) - 1/192*(1320*(e*x + d)^(7/2)*B*b^5*d^2*e^3 - 3560 *(e*x + d)^(5/2)*B*b^5*d^3*e^3 + 3224*(e*x + d)^(3/2)*B*b^5*d^4*e^3 - 984* sqrt(e*x + d)*B*b^5*d^5*e^3 - 3615*(e*x + d)^(7/2)*B*a*b^4*d*e^4 + 975*(e* x + d)^(7/2)*A*b^5*d*e^4 + 12975*(e*x + d)^(5/2)*B*a*b^4*d^2*e^4 - 2295*(e *x + d)^(5/2)*A*b^5*d^2*e^4 - 14825*(e*x + d)^(3/2)*B*a*b^4*d^3*e^4 + 1929 *(e*x + d)^(3/2)*A*b^5*d^3*e^4 + 5481*sqrt(e*x + d)*B*a*b^4*d^4*e^4 - 561* sqrt(e*x + d)*A*b^5*d^4*e^4 + 2295*(e*x + d)^(7/2)*B*a^2*b^3*e^5 - 975*(e* x + d)^(7/2)*A*a*b^4*e^5 - 15270*(e*x + d)^(5/2)*B*a^2*b^3*d*e^5 + 4590*(e *x + d)^(5/2)*A*a*b^4*d*e^5 + 25131*(e*x + d)^(3/2)*B*a^2*b^3*d^2*e^5 - 57 87*(e*x + d)^(3/2)*A*a*b^4*d^2*e^5 - 12084*sqrt(e*x + d)*B*a^2*b^3*d^3*e^5 + 2244*sqrt(e*x + d)*A*a*b^4*d^3*e^5 + 5855*(e*x + d)^(5/2)*B*a^3*b^2*e^6 - 2295*(e*x + d)^(5/2)*A*a^2*b^3*e^6 - 18683*(e*x + d)^(3/2)*B*a^3*b^2*d* e^6 + 5787*(e*x + d)^(3/2)*A*a^2*b^3*d*e^6 + 13206*sqrt(e*x + d)*B*a^3*b^2 *d^2*e^6 - 3366*sqrt(e*x + d)*A*a^2*b^3*d^2*e^6 + 5153*(e*x + d)^(3/2)*B*a ^4*b*e^7 - 1929*(e*x + d)^(3/2)*A*a^3*b^2*e^7 - 7164*sqrt(e*x + d)*B*a^4*b *d*e^7 + 2244*sqrt(e*x + d)*A*a^3*b^2*d*e^7 + 1545*sqrt(e*x + d)*B*a^5*e^8 - 561*sqrt(e*x + d)*A*a^4*b*e^8)/(((e*x + d)*b - b*d + a*e)^4*b^6*sgn(b*x + a)) + 2/3*((e*x + d)^(3/2)*B*b^10*e^3 + 12*sqrt(e*x + d)*B*b^10*d*e^...
Timed out. \[ \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]