Integrand size = 33, antiderivative size = 171 \[ \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {315 e^4 \sqrt {d+e x}}{64 b^5}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x)^2}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^3}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}-\frac {315 e^4 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2}} \]
-105/64*e^3*(e*x+d)^(3/2)/b^4/(b*x+a)-21/32*e^2*(e*x+d)^(5/2)/b^3/(b*x+a)^ 2-3/8*e*(e*x+d)^(7/2)/b^2/(b*x+a)^3-1/4*(e*x+d)^(9/2)/b/(b*x+a)^4-315/64*e ^4*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*(-a*e+b*d)^(1/2)/b^(11/ 2)+315/64*e^4*(e*x+d)^(1/2)/b^5
Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.24 \[ \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )}{64 b^5 (a+b x)^4}-\frac {315 e^4 \sqrt {-b d+a e} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{64 b^{11/2}} \]
-1/64*(Sqrt[d + e*x]*(-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 + 1 85*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^5*(a + b*x)^4) - (315*e^4*Sqrt[-(b*d) + a*e]*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(64*b^(11/2))
Time = 0.30 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1184, 27, 51, 51, 51, 51, 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.
| \(\displaystyle \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {(d+e x)^{9/2}}{b^6 (a+b x)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^{9/2}}{(a+b x)^5}dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \int \frac {(d+e x)^{7/2}}{(a+b x)^4}dx}{8 b}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^3}dx}{6 b}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{5/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \int \frac {\sqrt {d+e x}}{a+b x}dx}{2 b}-\frac {(d+e x)^{3/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{5/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{2 b}-\frac {(d+e x)^{3/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{5/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{2 b}-\frac {(d+e x)^{3/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{5/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{2 b}-\frac {(d+e x)^{3/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{5/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^4}\) |
-1/4*(d + e*x)^(9/2)/(b*(a + b*x)^4) + (9*e*(-1/3*(d + e*x)^(7/2)/(b*(a + b*x)^3) + (7*e*(-1/2*(d + e*x)^(5/2)/(b*(a + b*x)^2) + (5*e*(-((d + e*x)^( 3/2)/(b*(a + b*x))) + (3*e*((2*Sqrt[d + e*x])/b - (2*Sqrt[b*d - a*e]*ArcTa nh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(3/2)))/(2*b)))/(4*b)))/(6* b)))/(8*b)
3.21.84.3.1 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_)^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 = 0.86 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {2 e^{4} \sqrt {e x +d}}{b^{5}}-\frac {\left (2 a e -2 b d \right ) e^{4} \left (\frac {-\frac {325 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}-\frac {765 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} b^{2}}{128}+\left (-\frac {643}{128} a^{2} b \,e^{2}+\frac {643}{64} a \,b^{2} d e -\frac {643}{128} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} a^{3} e^{3}+\frac {561}{128} a^{2} b d \,e^{2}-\frac {561}{128} a \,b^{2} d^{2} e +\frac {187}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{5}}\) | \(201\) |
| pseudoelliptic | \(-\frac {315 \left (-\sqrt {\left (a e -b d \right ) b}\, \left (\left (\frac {128}{315} x^{4} b^{4}+\frac {93}{35} a \,b^{3} x^{3}+\frac {73}{15} x^{2} b^{2} a^{2}+\frac {11}{3} b \,a^{3} x +a^{4}\right ) e^{4}-\frac {b \left (\frac {65}{21} x^{3} b^{3}+\frac {37}{7} a \,b^{2} x^{2}+\frac {19}{5} b \,a^{2} x +a^{3}\right ) d \,e^{3}}{3}-\frac {2 b^{2} \left (5 b^{2} x^{2}+\frac {26}{7} a b x +a^{2}\right ) d^{2} e^{2}}{15}-\frac {8 \left (\frac {11 b x}{3}+a \right ) b^{3} d^{3} e}{105}-\frac {16 b^{4} d^{4}}{315}\right ) \sqrt {e x +d}+e^{4} \left (b x +a \right )^{4} \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )\right )}{64 \sqrt {\left (a e -b d \right ) b}\, b^{5} \left (b x +a \right )^{4}}\) | \(218\) |
| derivativedivides | \(2 e^{4} \left (\frac {\sqrt {e x +d}}{b^{5}}-\frac {\frac {\left (-\frac {325}{128} a \,b^{3} e +\frac {325}{128} b^{4} d \right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {765 b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{128}+\left (-\frac {643}{128} e^{3} b \,a^{3}+\frac {1929}{128} a^{2} b^{2} d \,e^{2}-\frac {1929}{128} d^{2} e \,b^{3} a +\frac {643}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} e^{4} a^{4}+\frac {187}{32} b d \,e^{3} a^{3}-\frac {561}{64} b^{2} d^{2} e^{2} a^{2}+\frac {187}{32} b^{3} d^{3} e a -\frac {187}{128} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}}{b^{5}}\right )\) | \(249\) |
| default | \(2 e^{4} \left (\frac {\sqrt {e x +d}}{b^{5}}-\frac {\frac {\left (-\frac {325}{128} a \,b^{3} e +\frac {325}{128} b^{4} d \right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {765 b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{128}+\left (-\frac {643}{128} e^{3} b \,a^{3}+\frac {1929}{128} a^{2} b^{2} d \,e^{2}-\frac {1929}{128} d^{2} e \,b^{3} a +\frac {643}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} e^{4} a^{4}+\frac {187}{32} b d \,e^{3} a^{3}-\frac {561}{64} b^{2} d^{2} e^{2} a^{2}+\frac {187}{32} b^{3} d^{3} e a -\frac {187}{128} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}}{b^{5}}\right )\) | \(249\) |
2*e^4*(e*x+d)^(1/2)/b^5-1/b^5*(2*a*e-2*b*d)*e^4*((-325/128*(e*x+d)^(7/2)*b ^3-765/128*(a*e-b*d)*(e*x+d)^(5/2)*b^2+(-643/128*a^2*b*e^2+643/64*a*b^2*d* e-643/128*b^3*d^2)*(e*x+d)^(3/2)+(-187/128*a^3*e^3+561/128*a^2*b*d*e^2-561 /128*a*b^2*d^2*e+187/128*b^3*d^3)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^4+315 /128/((a*e-b*d)*b)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (139) = 278\).
Time = 0.45 (sec) , antiderivative size = 680, normalized size of antiderivative = 3.98 \[ \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [\frac {315 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{128 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac {315 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}\right ] \]
[1/128*(315*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e ^4*x + a^4*e^4)*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^4*e^4*x^4 - 16*b^4*d^4 - 24*a*b^3*d^3*e - 42*a^2*b^2*d^2*e^2 - 105*a^3*b*d*e^3 + 315*a^4*e^4 - (32 5*b^4*d*e^3 - 837*a*b^3*e^4)*x^3 - 3*(70*b^4*d^2*e^2 + 185*a*b^3*d*e^3 - 5 11*a^2*b^2*e^4)*x^2 - (88*b^4*d^3*e + 156*a*b^3*d^2*e^2 + 399*a^2*b^2*d*e^ 3 - 1155*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x ^2 + 4*a^3*b^6*x + a^4*b^5), -1/64*(315*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6 *a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt(-(b*d - a*e)/b)*arctan(-s qrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (128*b^4*e^4*x^4 - 16*b ^4*d^4 - 24*a*b^3*d^3*e - 42*a^2*b^2*d^2*e^2 - 105*a^3*b*d*e^3 + 315*a^4*e ^4 - (325*b^4*d*e^3 - 837*a*b^3*e^4)*x^3 - 3*(70*b^4*d^2*e^2 + 185*a*b^3*d *e^3 - 511*a^2*b^2*e^4)*x^2 - (88*b^4*d^3*e + 156*a*b^3*d^2*e^2 + 399*a^2* b^2*d*e^3 - 1155*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^9*x^4 + 4*a*b^8*x^3 + 6*a ^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5)]
Timed out. \[ \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 340 vs. \(2 (139) = 278\).
Time = 0.30 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.99 \[ \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2 \, \sqrt {e x + d} e^{4}}{b^{5}} + \frac {315 \, {\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {325 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{4} - 765 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{4} + 643 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt {e x + d} b^{4} d^{4} e^{4} - 325 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{5} + 1530 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{5} - 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt {e x + d} a b^{3} d^{3} e^{5} - 765 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{6} + 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{6} - 643 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{7} + 748 \, \sqrt {e x + d} a^{3} b d e^{7} - 187 \, \sqrt {e x + d} a^{4} e^{8}}{64 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{5}} \]
2*sqrt(e*x + d)*e^4/b^5 + 315/64*(b*d*e^4 - a*e^5)*arctan(sqrt(e*x + d)*b/ sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - 1/64*(325*(e*x + d)^(7/ 2)*b^4*d*e^4 - 765*(e*x + d)^(5/2)*b^4*d^2*e^4 + 643*(e*x + d)^(3/2)*b^4*d ^3*e^4 - 187*sqrt(e*x + d)*b^4*d^4*e^4 - 325*(e*x + d)^(7/2)*a*b^3*e^5 + 1 530*(e*x + d)^(5/2)*a*b^3*d*e^5 - 1929*(e*x + d)^(3/2)*a*b^3*d^2*e^5 + 748 *sqrt(e*x + d)*a*b^3*d^3*e^5 - 765*(e*x + d)^(5/2)*a^2*b^2*e^6 + 1929*(e*x + d)^(3/2)*a^2*b^2*d*e^6 - 1122*sqrt(e*x + d)*a^2*b^2*d^2*e^6 - 643*(e*x + d)^(3/2)*a^3*b*e^7 + 748*sqrt(e*x + d)*a^3*b*d*e^7 - 187*sqrt(e*x + d)*a ^4*e^8)/(((e*x + d)*b - b*d + a*e)^4*b^5)
Time = 11.07 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.55 \[ \int \frac {(a+b x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {{\left (d+e\,x\right )}^{5/2}\,\left (\frac {765\,a^2\,b^2\,e^6}{64}-\frac {765\,a\,b^3\,d\,e^5}{32}+\frac {765\,b^4\,d^2\,e^4}{64}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {643\,a^3\,b\,e^7}{64}-\frac {1929\,a^2\,b^2\,d\,e^6}{64}+\frac {1929\,a\,b^3\,d^2\,e^5}{64}-\frac {643\,b^4\,d^3\,e^4}{64}\right )+\left (\frac {325\,a\,b^3\,e^5}{64}-\frac {325\,b^4\,d\,e^4}{64}\right )\,{\left (d+e\,x\right )}^{7/2}+\sqrt {d+e\,x}\,\left (\frac {187\,a^4\,e^8}{64}-\frac {187\,a^3\,b\,d\,e^7}{16}+\frac {561\,a^2\,b^2\,d^2\,e^6}{32}-\frac {187\,a\,b^3\,d^3\,e^5}{16}+\frac {187\,b^4\,d^4\,e^4}{64}\right )}{b^9\,{\left (d+e\,x\right )}^4-\left (4\,b^9\,d-4\,a\,b^8\,e\right )\,{\left (d+e\,x\right )}^3+b^9\,d^4+{\left (d+e\,x\right )}^2\,\left (6\,a^2\,b^7\,e^2-12\,a\,b^8\,d\,e+6\,b^9\,d^2\right )-\left (d+e\,x\right )\,\left (-4\,a^3\,b^6\,e^3+12\,a^2\,b^7\,d\,e^2-12\,a\,b^8\,d^2\,e+4\,b^9\,d^3\right )+a^4\,b^5\,e^4-4\,a^3\,b^6\,d\,e^3+6\,a^2\,b^7\,d^2\,e^2-4\,a\,b^8\,d^3\,e}+\frac {2\,e^4\,\sqrt {d+e\,x}}{b^5}-\frac {315\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^5-b\,d\,e^4}\right )\,\sqrt {a\,e-b\,d}}{64\,b^{11/2}} \]
((d + e*x)^(5/2)*((765*a^2*b^2*e^6)/64 + (765*b^4*d^2*e^4)/64 - (765*a*b^3 *d*e^5)/32) + (d + e*x)^(3/2)*((643*a^3*b*e^7)/64 - (643*b^4*d^3*e^4)/64 + (1929*a*b^3*d^2*e^5)/64 - (1929*a^2*b^2*d*e^6)/64) + ((325*a*b^3*e^5)/64 - (325*b^4*d*e^4)/64)*(d + e*x)^(7/2) + (d + e*x)^(1/2)*((187*a^4*e^8)/64 + (187*b^4*d^4*e^4)/64 - (187*a*b^3*d^3*e^5)/16 + (561*a^2*b^2*d^2*e^6)/32 - (187*a^3*b*d*e^7)/16))/(b^9*(d + e*x)^4 - (4*b^9*d - 4*a*b^8*e)*(d + e* x)^3 + b^9*d^4 + (d + e*x)^2*(6*b^9*d^2 + 6*a^2*b^7*e^2 - 12*a*b^8*d*e) - (d + e*x)*(4*b^9*d^3 - 4*a^3*b^6*e^3 + 12*a^2*b^7*d*e^2 - 12*a*b^8*d^2*e) + a^4*b^5*e^4 - 4*a^3*b^6*d*e^3 + 6*a^2*b^7*d^2*e^2 - 4*a*b^8*d^3*e) + (2* e^4*(d + e*x)^(1/2))/b^5 - (315*e^4*atan((b^(1/2)*e^4*(a*e - b*d)^(1/2)*(d + e*x)^(1/2))/(a*e^5 - b*d*e^4))*(a*e - b*d)^(1/2))/(64*b^(11/2))