Integrand size = 33, antiderivative size = 198 \[ \int \frac {(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1155 e^4 (b d-a e) \sqrt {d+e x}}{64 b^6}+\frac {385 e^4 (d+e x)^{3/2}}{64 b^5}-\frac {231 e^3 (d+e x)^{5/2}}{64 b^4 (a+b x)}-\frac {33 e^2 (d+e x)^{7/2}}{32 b^3 (a+b x)^2}-\frac {11 e (d+e x)^{9/2}}{24 b^2 (a+b x)^3}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}-\frac {1155 e^4 (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2}} \]
385/64*e^4*(e*x+d)^(3/2)/b^5-231/64*e^3*(e*x+d)^(5/2)/b^4/(b*x+a)-33/32*e^ 2*(e*x+d)^(7/2)/b^3/(b*x+a)^2-11/24*e*(e*x+d)^(9/2)/b^2/(b*x+a)^3-1/4*(e*x +d)^(11/2)/b/(b*x+a)^4-1155/64*e^4*(-a*e+b*d)^(3/2)*arctanh(b^(1/2)*(e*x+d )^(1/2)/(-a*e+b*d)^(1/2))/b^(13/2)+1155/64*e^4*(-a*e+b*d)*(e*x+d)^(1/2)/b^ 6
Time = 0.15 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (3465 a^5 e^5+1155 a^4 b e^4 (-4 d+11 e x)+231 a^3 b^2 e^3 \left (3 d^2-74 d e x+73 e^2 x^2\right )+99 a^2 b^3 e^2 \left (2 d^3+27 d^2 e x-232 d e^2 x^2+93 e^3 x^3\right )+11 a b^4 e \left (8 d^4+68 d^3 e x+345 d^2 e^2 x^2-1162 d e^3 x^3+128 e^4 x^4\right )+b^5 \left (48 d^5+328 d^4 e x+1030 d^3 e^2 x^2+2295 d^2 e^3 x^3-2048 d e^4 x^4-128 e^5 x^5\right )\right )}{192 b^6 (a+b x)^4}+\frac {1155 e^4 (-b d+a e)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{64 b^{13/2}} \]
-1/192*(Sqrt[d + e*x]*(3465*a^5*e^5 + 1155*a^4*b*e^4*(-4*d + 11*e*x) + 231 *a^3*b^2*e^3*(3*d^2 - 74*d*e*x + 73*e^2*x^2) + 99*a^2*b^3*e^2*(2*d^3 + 27* d^2*e*x - 232*d*e^2*x^2 + 93*e^3*x^3) + 11*a*b^4*e*(8*d^4 + 68*d^3*e*x + 3 45*d^2*e^2*x^2 - 1162*d*e^3*x^3 + 128*e^4*x^4) + b^5*(48*d^5 + 328*d^4*e*x + 1030*d^3*e^2*x^2 + 2295*d^2*e^3*x^3 - 2048*d*e^4*x^4 - 128*e^5*x^5)))/( b^6*(a + b*x)^4) + (1155*e^4*(-(b*d) + a*e)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(64*b^(13/2))
Time = 0.33 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {1184, 27, 51, 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.
| \(\displaystyle \int \frac {(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {(d+e x)^{11/2}}{b^6 (a+b x)^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^{11/2}}{(a+b x)^5}dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {11 e \int \frac {(d+e x)^{9/2}}{(a+b x)^4}dx}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {11 e \left (\frac {3 e \int \frac {(d+e x)^{7/2}}{(a+b x)^3}dx}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {11 e \left (\frac {3 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {11 e \left (\frac {3 e \left (\frac {7 e \left (\frac {5 e \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{2 b}-\frac {(d+e x)^{5/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{7/2}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{9/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{11/2}}{4 b (a+b x)^4}\) |
-1/4*(d + e*x)^(11/2)/(b*(a + b*x)^4) + (11*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])/Sqr t[b*d - a*e]])/b^(3/2)))/b))/(2*b)))/(4*b)))/(2*b)))/(8*b)
3.21.83.3.1 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 = 0.87 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.15
| method | result | size |
| risch | \(-\frac {2 e^{4} \left (-b e x +15 a e -16 b d \right ) \sqrt {e x +d}}{3 b^{6}}+\frac {\left (2 e^{2} a^{2}-4 a b d e +2 b^{2} d^{2}\right ) e^{4} \left (\frac {-\frac {765 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}-\frac {5855 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}} b^{2}}{384}+\left (-\frac {5153}{384} a^{2} b \,e^{2}+\frac {5153}{192} a \,b^{2} d e -\frac {5153}{384} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {515}{128} a^{3} e^{3}+\frac {1545}{128} a^{2} b d \,e^{2}-\frac {1545}{128} a \,b^{2} d^{2} e +\frac {515}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {1155 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{6}}\) | \(228\) |
| pseudoelliptic | \(\frac {\frac {1155 e^{4} \left (b x +a \right )^{4} \left (a e -b d \right )^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{64}-\frac {1155 \left (\left (-\frac {128}{3465} e^{5} x^{5}-\frac {2048}{3465} d \,e^{4} x^{4}+\frac {51}{77} d^{2} e^{3} x^{3}+\frac {206}{693} d^{3} e^{2} x^{2}+\frac {328}{3465} d^{4} e x +\frac {16}{1155} d^{5}\right ) b^{5}+\frac {8 e \left (16 e^{4} x^{4}-\frac {581}{4} d \,e^{3} x^{3}+\frac {345}{8} d^{2} e^{2} x^{2}+\frac {17}{2} d^{3} e x +d^{4}\right ) a \,b^{4}}{315}+\frac {2 e^{2} a^{2} \left (\frac {93}{2} e^{3} x^{3}-116 d \,e^{2} x^{2}+\frac {27}{2} d^{2} e x +d^{3}\right ) b^{3}}{35}+\frac {e^{3} a^{3} \left (\frac {73}{3} e^{2} x^{2}-\frac {74}{3} d e x +d^{2}\right ) b^{2}}{5}-\frac {4 e^{4} \left (-\frac {11 e x}{4}+d \right ) a^{4} b}{3}+e^{5} a^{5}\right ) \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}}{64}}{b^{6} \left (b x +a \right )^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(282\) |
| derivativedivides | \(2 e^{4} \left (-\frac {-\frac {b \left (e x +d \right )^{\frac {3}{2}}}{3}+5 a e \sqrt {e x +d}-5 b d \sqrt {e x +d}}{b^{6}}+\frac {\frac {\left (-\frac {765}{128} e^{2} a^{2} b^{3}+\frac {765}{64} d e \,b^{4} a -\frac {765}{128} d^{2} b^{5}\right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {5855 b^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {5153}{384} e^{4} a^{4} b +\frac {5153}{96} e^{3} d \,a^{3} b^{2}-\frac {5153}{64} d^{2} e^{2} a^{2} b^{3}+\frac {5153}{96} d^{3} e \,b^{4} a -\frac {5153}{384} d^{4} b^{5}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {515}{128} e^{5} a^{5}+\frac {2575}{128} b d \,e^{4} a^{4}-\frac {2575}{64} b^{2} d^{2} e^{3} a^{3}+\frac {2575}{64} b^{3} d^{3} e^{2} a^{2}-\frac {2575}{128} b^{4} d^{4} e a +\frac {515}{128} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {1155 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}}{b^{6}}\right )\) | \(345\) |
| default | \(2 e^{4} \left (-\frac {-\frac {b \left (e x +d \right )^{\frac {3}{2}}}{3}+5 a e \sqrt {e x +d}-5 b d \sqrt {e x +d}}{b^{6}}+\frac {\frac {\left (-\frac {765}{128} e^{2} a^{2} b^{3}+\frac {765}{64} d e \,b^{4} a -\frac {765}{128} d^{2} b^{5}\right ) \left (e x +d \right )^{\frac {7}{2}}-\frac {5855 b^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {5}{2}}}{384}+\left (-\frac {5153}{384} e^{4} a^{4} b +\frac {5153}{96} e^{3} d \,a^{3} b^{2}-\frac {5153}{64} d^{2} e^{2} a^{2} b^{3}+\frac {5153}{96} d^{3} e \,b^{4} a -\frac {5153}{384} d^{4} b^{5}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {515}{128} e^{5} a^{5}+\frac {2575}{128} b d \,e^{4} a^{4}-\frac {2575}{64} b^{2} d^{2} e^{3} a^{3}+\frac {2575}{64} b^{3} d^{3} e^{2} a^{2}-\frac {2575}{128} b^{4} d^{4} e a +\frac {515}{128} b^{5} d^{5}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {1155 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}}{b^{6}}\right )\) | \(345\) |
-2/3*e^4*(-b*e*x+15*a*e-16*b*d)*(e*x+d)^(1/2)/b^6+1/b^6*(2*a^2*e^2-4*a*b*d *e+2*b^2*d^2)*e^4*((-765/128*(e*x+d)^(7/2)*b^3-5855/384*(a*e-b*d)*(e*x+d)^ (5/2)*b^2+(-5153/384*a^2*b*e^2+5153/192*a*b^2*d*e-5153/384*b^3*d^2)*(e*x+d )^(3/2)+(-515/128*a^3*e^3+1545/128*a^2*b*d*e^2-1545/128*a*b^2*d^2*e+515/12 8*b^3*d^3)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^4+1155/128/((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. 479 vs. \(2 (162) = 324\).
Time = 0.38 (sec) , antiderivative size = 968, normalized size of antiderivative = 4.89 \[ \int \frac {(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {3465 \, {\left (a^{4} b d e^{4} - a^{5} e^{5} + {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \, {\left (a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{2} d e^{4} - a^{4} b e^{5}\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 (128 \, b^{5} e^{5} x^{5} - 48 \, b^{5} d^{5} - 88 \, a b^{4} d^{4} e - 198 \, a^{2} b^{3} d^{3} e^{2} - 693 \, a^{3} b^{2} d^{2} e^{3} + 4620 \, a^{4} b d e^{4} - 3465 \, a^{5} e^{5} + 128 \, {\left (16 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} - {\left (2295 \, b^{5} d^{2} e^{3} - 12782 \, a b^{4} d e^{4} + 9207 \, a^{2} b^{3} e^{5}\right )} x^{3} - {\left (1030 \, b^{5} d^{3} e^{2} + 3795 \, a b^{4} d^{2} e^{3} - 22968 \, a^{2} b^{3} d e^{4} + 16863 \, a^{3} b^{2} e^{5}\right )} x^{2} - {\left (328 \, b^{5} d^{4} e + 748 \, a b^{4} d^{3} e^{2} + 2673 \, a^{2} b^{3} d^{2} e^{3} - 17094 \, a^{3} b^{2} d e^{4} + 12705 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{384 \, {\left (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}\right )}}, -\frac {3465 \, {\left (a^{4} b d e^{4} - a^{5} e^{5} + {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \, {\left (a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{2} d e^{4} - a^{4} b e^{5}\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 (128 \, b^{5} e^{5} x^{5} - 48 \, b^{5} d^{5} - 88 \, a b^{4} d^{4} e - 198 \, a^{2} b^{3} d^{3} e^{2} - 693 \, a^{3} b^{2} d^{2} e^{3} + 4620 \, a^{4} b d e^{4} - 3465 \, a^{5} e^{5} + 128 \, {\left (16 \, b^{5} d e^{4} - 11 \, a b^{4} e^{5}\right )} x^{4} - {\left (2295 \, b^{5} d^{2} e^{3} - 12782 \, a b^{4} d e^{4} + 9207 \, a^{2} b^{3} e^{5}\right )} x^{3} - {\left (1030 \, b^{5} d^{3} e^{2} + 3795 \, a b^{4} d^{2} e^{3} - 22968 \, a^{2} b^{3} d e^{4} + 16863 \, a^{3} b^{2} e^{5}\right )} x^{2} - {\left (328 \, b^{5} d^{4} e + 748 \, a b^{4} d^{3} e^{2} + 2673 \, a^{2} b^{3} d^{2} e^{3} - 17094 \, a^{3} b^{2} d e^{4} + 12705 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{192 \, {\left (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}\right )}}\right ] \]
[-1/384*(3465*(a^4*b*d*e^4 - a^5*e^5 + (b^5*d*e^4 - a*b^4*e^5)*x^4 + 4*(a* b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 6*(a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 4*(a^ 3*b^2*d*e^4 - a^4*b*e^5)*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^5*e^5*x^5 - 48*b^5*d^5 - 88*a*b^4*d^4*e - 198*a^2*b^3*d^3*e^2 - 693*a^3*b^2*d^2*e^3 + 4620*a^4*b*d*e^4 - 3465*a^5*e^5 + 128*(16*b^5*d*e^4 - 11*a*b^4*e^5)*x^4 - (2295*b^5*d^2*e^3 - 12782*a*b^4*d*e^4 + 9207*a^2*b^3*e^5)*x^3 - (1030*b^5* d^3*e^2 + 3795*a*b^4*d^2*e^3 - 22968*a^2*b^3*d*e^4 + 16863*a^3*b^2*e^5)*x^ 2 - (328*b^5*d^4*e + 748*a*b^4*d^3*e^2 + 2673*a^2*b^3*d^2*e^3 - 17094*a^3* b^2*d*e^4 + 12705*a^4*b*e^5)*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*(3465*(a^4*b*d*e^4 - a^5*e^5 + (b^5*d*e^4 - a*b^4*e^5)*x^4 + 4*(a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 6*(a^ 2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 4*(a^3*b^2*d*e^4 - a^4*b*e^5)*x)*sqrt(-(b *d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - ( 128*b^5*e^5*x^5 - 48*b^5*d^5 - 88*a*b^4*d^4*e - 198*a^2*b^3*d^3*e^2 - 693* a^3*b^2*d^2*e^3 + 4620*a^4*b*d*e^4 - 3465*a^5*e^5 + 128*(16*b^5*d*e^4 - 11 *a*b^4*e^5)*x^4 - (2295*b^5*d^2*e^3 - 12782*a*b^4*d*e^4 + 9207*a^2*b^3*e^5 )*x^3 - (1030*b^5*d^3*e^2 + 3795*a*b^4*d^2*e^3 - 22968*a^2*b^3*d*e^4 + 168 63*a^3*b^2*e^5)*x^2 - (328*b^5*d^4*e + 748*a*b^4*d^3*e^2 + 2673*a^2*b^3*d^ 2*e^3 - 17094*a^3*b^2*d*e^4 + 12705*a^4*b*e^5)*x)*sqrt(e*x + d))/(b^10*...
Timed out. \[ \int \frac {(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 474 vs. \(2 (162) = 324\).
Time = 0.30 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.39 \[ \int \frac {(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1155 \, {\left (b^{2} d^{2} e^{4} - 2 \, a b d e^{5} + a^{2} e^{6}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{6}} - \frac {2295 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} d^{2} e^{4} - 5855 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d^{3} e^{4} + 5153 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{4} e^{4} - 1545 \, \sqrt {e x + d} b^{5} d^{5} e^{4} - 4590 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{4} d e^{5} + 17565 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} d^{2} e^{5} - 20612 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d^{3} e^{5} + 7725 \, \sqrt {e x + d} a b^{4} d^{4} e^{5} + 2295 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{3} e^{6} - 17565 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{3} d e^{6} + 30918 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} d^{2} e^{6} - 15450 \, \sqrt {e x + d} a^{2} b^{3} d^{3} e^{6} + 5855 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{2} e^{7} - 20612 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{2} d e^{7} + 15450 \, \sqrt {e x + d} a^{3} b^{2} d^{2} e^{7} + 5153 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b e^{8} - 7725 \, \sqrt {e x + d} a^{4} b d e^{8} + 1545 \, \sqrt {e x + d} a^{5} e^{9}}{192 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{6}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{10} e^{4} + 15 \, \sqrt {e x + d} b^{10} d e^{4} - 15 \, \sqrt {e x + d} a b^{9} e^{5}\right )}}{3 \, b^{15}} \]
1155/64*(b^2*d^2*e^4 - 2*a*b*d*e^5 + a^2*e^6)*arctan(sqrt(e*x + d)*b/sqrt( -b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6) - 1/192*(2295*(e*x + d)^(7/2)* b^5*d^2*e^4 - 5855*(e*x + d)^(5/2)*b^5*d^3*e^4 + 5153*(e*x + d)^(3/2)*b^5* d^4*e^4 - 1545*sqrt(e*x + d)*b^5*d^5*e^4 - 4590*(e*x + d)^(7/2)*a*b^4*d*e^ 5 + 17565*(e*x + d)^(5/2)*a*b^4*d^2*e^5 - 20612*(e*x + d)^(3/2)*a*b^4*d^3* e^5 + 7725*sqrt(e*x + d)*a*b^4*d^4*e^5 + 2295*(e*x + d)^(7/2)*a^2*b^3*e^6 - 17565*(e*x + d)^(5/2)*a^2*b^3*d*e^6 + 30918*(e*x + d)^(3/2)*a^2*b^3*d^2* e^6 - 15450*sqrt(e*x + d)*a^2*b^3*d^3*e^6 + 5855*(e*x + d)^(5/2)*a^3*b^2*e ^7 - 20612*(e*x + d)^(3/2)*a^3*b^2*d*e^7 + 15450*sqrt(e*x + d)*a^3*b^2*d^2 *e^7 + 5153*(e*x + d)^(3/2)*a^4*b*e^8 - 7725*sqrt(e*x + d)*a^4*b*d*e^8 + 1 545*sqrt(e*x + d)*a^5*e^9)/(((e*x + d)*b - b*d + a*e)^4*b^6) + 2/3*((e*x + d)^(3/2)*b^10*e^4 + 15*sqrt(e*x + d)*b^10*d*e^4 - 15*sqrt(e*x + d)*a*b^9* e^5)/b^15
Time = 11.08 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.70 \[ \int \frac {(a+b x) (d+e x)^{11/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {2\,e^4\,{\left (d+e\,x\right )}^{3/2}}{3\,b^5}-\frac {{\left (d+e\,x\right )}^{7/2}\,\left (\frac {765\,a^2\,b^3\,e^6}{64}-\frac {765\,a\,b^4\,d\,e^5}{32}+\frac {765\,b^5\,d^2\,e^4}{64}\right )+\sqrt {d+e\,x}\,\left (\frac {515\,a^5\,e^9}{64}-\frac {2575\,a^4\,b\,d\,e^8}{64}+\frac {2575\,a^3\,b^2\,d^2\,e^7}{32}-\frac {2575\,a^2\,b^3\,d^3\,e^6}{32}+\frac {2575\,a\,b^4\,d^4\,e^5}{64}-\frac {515\,b^5\,d^5\,e^4}{64}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {5855\,a^3\,b^2\,e^7}{192}-\frac {5855\,a^2\,b^3\,d\,e^6}{64}+\frac {5855\,a\,b^4\,d^2\,e^5}{64}-\frac {5855\,b^5\,d^3\,e^4}{192}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {5153\,a^4\,b\,e^8}{192}-\frac {5153\,a^3\,b^2\,d\,e^7}{48}+\frac {5153\,a^2\,b^3\,d^2\,e^6}{32}-\frac {5153\,a\,b^4\,d^3\,e^5}{48}+\frac {5153\,b^5\,d^4\,e^4}{192}\right )}{b^{10}\,{\left (d+e\,x\right )}^4-\left (4\,b^{10}\,d-4\,a\,b^9\,e\right )\,{\left (d+e\,x\right )}^3+b^{10}\,d^4+{\left (d+e\,x\right )}^2\,\left (6\,a^2\,b^8\,e^2-12\,a\,b^9\,d\,e+6\,b^{10}\,d^2\right )-\left (d+e\,x\right )\,\left (-4\,a^3\,b^7\,e^3+12\,a^2\,b^8\,d\,e^2-12\,a\,b^9\,d^2\,e+4\,b^{10}\,d^3\right )+a^4\,b^6\,e^4-4\,a^3\,b^7\,d\,e^3+6\,a^2\,b^8\,d^2\,e^2-4\,a\,b^9\,d^3\,e}+\frac {2\,e^4\,\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,\sqrt {d+e\,x}}{b^{10}}+\frac {1155\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,{\left (a\,e-b\,d\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^6-2\,a\,b\,d\,e^5+b^2\,d^2\,e^4}\right )\,{\left (a\,e-b\,d\right )}^{3/2}}{64\,b^{13/2}} \]
(2*e^4*(d + e*x)^(3/2))/(3*b^5) - ((d + e*x)^(7/2)*((765*a^2*b^3*e^6)/64 + (765*b^5*d^2*e^4)/64 - (765*a*b^4*d*e^5)/32) + (d + e*x)^(1/2)*((515*a^5* e^9)/64 - (515*b^5*d^5*e^4)/64 + (2575*a*b^4*d^4*e^5)/64 - (2575*a^2*b^3*d ^3*e^6)/32 + (2575*a^3*b^2*d^2*e^7)/32 - (2575*a^4*b*d*e^8)/64) + (d + e*x )^(5/2)*((5855*a^3*b^2*e^7)/192 - (5855*b^5*d^3*e^4)/192 + (5855*a*b^4*d^2 *e^5)/64 - (5855*a^2*b^3*d*e^6)/64) + (d + e*x)^(3/2)*((5153*a^4*b*e^8)/19 2 + (5153*b^5*d^4*e^4)/192 - (5153*a*b^4*d^3*e^5)/48 - (5153*a^3*b^2*d*e^7 )/48 + (5153*a^2*b^3*d^2*e^6)/32))/(b^10*(d + e*x)^4 - (4*b^10*d - 4*a*b^9 *e)*(d + e*x)^3 + b^10*d^4 + (d + e*x)^2*(6*b^10*d^2 + 6*a^2*b^8*e^2 - 12* a*b^9*d*e) - (d + e*x)*(4*b^10*d^3 - 4*a^3*b^7*e^3 + 12*a^2*b^8*d*e^2 - 12 *a*b^9*d^2*e) + a^4*b^6*e^4 - 4*a^3*b^7*d*e^3 + 6*a^2*b^8*d^2*e^2 - 4*a*b^ 9*d^3*e) + (2*e^4*(5*b^5*d - 5*a*b^4*e)*(d + e*x)^(1/2))/b^10 + (1155*e^4* atan((b^(1/2)*e^4*(a*e - b*d)^(3/2)*(d + e*x)^(1/2))/(a^2*e^6 + b^2*d^2*e^ 4 - 2*a*b*d*e^5))*(a*e - b*d)^(3/2))/(64*b^(13/2))