Integrand size = 28, antiderivative size = 188 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {7 e^3 \sqrt {d+e x}}{64 b^4 (a+b x)^2}-\frac {7 e^4 \sqrt {d+e x}}{128 b^4 (b d-a e) (a+b x)}-\frac {7 e^2 (d+e x)^{3/2}}{48 b^3 (a+b x)^3}-\frac {7 e (d+e x)^{5/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{7/2}}{5 b (a+b x)^5}+\frac {7 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{3/2}} \]
-7/48*e^2*(e*x+d)^(3/2)/b^3/(b*x+a)^3-7/40*e*(e*x+d)^(5/2)/b^2/(b*x+a)^4-1 /5*(e*x+d)^(7/2)/b/(b*x+a)^5+7/128*e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e +b*d)^(1/2))/b^(9/2)/(-a*e+b*d)^(3/2)-7/64*e^3*(e*x+d)^(1/2)/b^4/(b*x+a)^2 -7/128*e^4*(e*x+d)^(1/2)/b^4/(-a*e+b*d)/(b*x+a)
Time = 1.27 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\sqrt {d+e x} \left (105 a^4 e^4+70 a^3 b e^3 (d+7 e x)+14 a^2 b^2 e^2 \left (4 d^2+23 d e x+64 e^2 x^2\right )+2 a b^3 e \left (24 d^3+128 d^2 e x+289 d e^2 x^2+395 e^3 x^3\right )-b^4 \left (384 d^4+1488 d^3 e x+2104 d^2 e^2 x^2+1210 d e^3 x^3+105 e^4 x^4\right )\right )}{1920 b^4 (b d-a e) (a+b x)^5}+\frac {7 e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{9/2} (-b d+a e)^{3/2}} \]
(Sqrt[d + e*x]*(105*a^4*e^4 + 70*a^3*b*e^3*(d + 7*e*x) + 14*a^2*b^2*e^2*(4 *d^2 + 23*d*e*x + 64*e^2*x^2) + 2*a*b^3*e*(24*d^3 + 128*d^2*e*x + 289*d*e^ 2*x^2 + 395*e^3*x^3) - b^4*(384*d^4 + 1488*d^3*e*x + 2104*d^2*e^2*x^2 + 12 10*d*e^3*x^3 + 105*e^4*x^4)))/(1920*b^4*(b*d - a*e)*(a + b*x)^5) + (7*e^5* ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(9/2)*(-(b*d) + a*e)^(3/2))
Time = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1098, 27, 51, 51, 51, 51, 52, 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.
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle b^6 \int \frac {(d+e x)^{7/2}}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^5}dx}{10 b}-\frac {(d+e x)^{7/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{(a+b x)^4}dx}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{7/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {e \int \frac {\sqrt {d+e x}}{(a+b x)^3}dx}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{7/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {e \left (\frac {e \int \frac {1}{(a+b x)^2 \sqrt {d+e x}}dx}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{7/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {e \left (\frac {e \left (-\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 (b d-a e)}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{7/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {e \left (\frac {e \left (-\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b d-a e}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{7/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {7 e \left (\frac {5 e \left (\frac {e \left (\frac {e \left (\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b}-\frac {\sqrt {d+e x}}{2 b (a+b x)^2}\right )}{2 b}-\frac {(d+e x)^{3/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{7/2}}{5 b (a+b x)^5}\) |
-1/5*(d + e*x)^(7/2)/(b*(a + b*x)^5) + (7*e*(-1/4*(d + e*x)^(5/2)/(b*(a + b*x)^4) + (5*e*(-1/3*(d + e*x)^(3/2)/(b*(a + b*x)^3) + (e*(-1/2*Sqrt[d + e *x]/(b*(a + b*x)^2) + (e*(-(Sqrt[d + e*x]/((b*d - a*e)*(a + b*x))) + (e*Ar cTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^(3/2) )))/(4*b)))/(2*b)))/(8*b)))/(10*b)
3.17.68.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 3.59 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(2 e^{5} \left (\frac {\frac {7 \left (e x +d \right )^{\frac {9}{2}}}{256 \left (a e -b d \right )}-\frac {79 \left (e x +d \right )^{\frac {7}{2}}}{384 b}-\frac {7 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 b^{2}}-\frac {49 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b^{3}}-\frac {7 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {e x +d}}{256 b^{4}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \left (a e -b d \right ) b^{4} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(207\) |
| default | \(2 e^{5} \left (\frac {\frac {7 \left (e x +d \right )^{\frac {9}{2}}}{256 \left (a e -b d \right )}-\frac {79 \left (e x +d \right )^{\frac {7}{2}}}{384 b}-\frac {7 \left (a e -b d \right ) \left (e x +d \right )^{\frac {5}{2}}}{30 b^{2}}-\frac {49 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{384 b^{3}}-\frac {7 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {e x +d}}{256 b^{4}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \left (a e -b d \right ) b^{4} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(207\) |
| pseudoelliptic | \(\frac {\frac {7 e^{5} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}-\frac {7 \sqrt {e x +d}\, \left (\left (-e^{4} x^{4}-\frac {242}{21} d \,e^{3} x^{3}-\frac {2104}{105} d^{2} e^{2} x^{2}-\frac {496}{35} d^{3} e x -\frac {128}{35} d^{4}\right ) b^{4}+\frac {16 \left (\frac {395}{24} e^{3} x^{3}+\frac {289}{24} d \,e^{2} x^{2}+\frac {16}{3} d^{2} e x +d^{3}\right ) e a \,b^{3}}{35}+\frac {8 e^{2} \left (16 x^{2} e^{2}+\frac {23}{4} d e x +d^{2}\right ) a^{2} b^{2}}{15}+\frac {2 a^{3} e^{3} \left (7 e x +d \right ) b}{3}+e^{4} a^{4}\right ) \sqrt {\left (a e -b d \right ) b}}{128}}{\left (a e -b d \right ) b^{4} \left (b x +a \right )^{5} \sqrt {\left (a e -b d \right ) b}}\) | \(221\) |
2*e^5*((7/256/(a*e-b*d)*(e*x+d)^(9/2)-79/384/b*(e*x+d)^(7/2)-7/30*(a*e-b*d )/b^2*(e*x+d)^(5/2)-49/384*(a^2*e^2-2*a*b*d*e+b^2*d^2)/b^3*(e*x+d)^(3/2)-7 /256*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/b^4*(e*x+d)^(1/2))/(b*( e*x+d)+a*e-b*d)^5+7/256/(a*e-b*d)/b^4/((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. 572 vs. \(2 (156) = 312\).
Time = 0.34 (sec) , antiderivative size = 1158, normalized size of antiderivative = 6.16 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [-\frac {105 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left (384 \, b^{6} d^{5} - 432 \, a b^{5} d^{4} e - 8 \, a^{2} b^{4} d^{3} e^{2} - 14 \, a^{3} b^{3} d^{2} e^{3} - 35 \, a^{4} b^{2} d e^{4} + 105 \, a^{5} b e^{5} + 105 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{4} + 10 \, {\left (121 \, b^{6} d^{2} e^{3} - 200 \, a b^{5} d e^{4} + 79 \, a^{2} b^{4} e^{5}\right )} x^{3} + 2 \, {\left (1052 \, b^{6} d^{3} e^{2} - 1341 \, a b^{5} d^{2} e^{3} - 159 \, a^{2} b^{4} d e^{4} + 448 \, a^{3} b^{3} e^{5}\right )} x^{2} + 2 \, {\left (744 \, b^{6} d^{4} e - 872 \, a b^{5} d^{3} e^{2} - 33 \, a^{2} b^{4} d^{2} e^{3} - 84 \, a^{3} b^{3} d e^{4} + 245 \, a^{4} b^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{3840 \, {\left (a^{5} b^{7} d^{2} - 2 \, a^{6} b^{6} d e + a^{7} b^{5} e^{2} + {\left (b^{12} d^{2} - 2 \, a b^{11} d e + a^{2} b^{10} e^{2}\right )} x^{5} + 5 \, {\left (a b^{11} d^{2} - 2 \, a^{2} b^{10} d e + a^{3} b^{9} e^{2}\right )} x^{4} + 10 \, {\left (a^{2} b^{10} d^{2} - 2 \, a^{3} b^{9} d e + a^{4} b^{8} e^{2}\right )} x^{3} + 10 \, {\left (a^{3} b^{9} d^{2} - 2 \, a^{4} b^{8} d e + a^{5} b^{7} e^{2}\right )} x^{2} + 5 \, {\left (a^{4} b^{8} d^{2} - 2 \, a^{5} b^{7} d e + a^{6} b^{6} e^{2}\right )} x\right )}}, -\frac {105 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (384 \, b^{6} d^{5} - 432 \, a b^{5} d^{4} e - 8 \, a^{2} b^{4} d^{3} e^{2} - 14 \, a^{3} b^{3} d^{2} e^{3} - 35 \, a^{4} b^{2} d e^{4} + 105 \, a^{5} b e^{5} + 105 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{4} + 10 \, {\left (121 \, b^{6} d^{2} e^{3} - 200 \, a b^{5} d e^{4} + 79 \, a^{2} b^{4} e^{5}\right )} x^{3} + 2 \, {\left (1052 \, b^{6} d^{3} e^{2} - 1341 \, a b^{5} d^{2} e^{3} - 159 \, a^{2} b^{4} d e^{4} + 448 \, a^{3} b^{3} e^{5}\right )} x^{2} + 2 \, {\left (744 \, b^{6} d^{4} e - 872 \, a b^{5} d^{3} e^{2} - 33 \, a^{2} b^{4} d^{2} e^{3} - 84 \, a^{3} b^{3} d e^{4} + 245 \, a^{4} b^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{1920 \, {\left (a^{5} b^{7} d^{2} - 2 \, a^{6} b^{6} d e + a^{7} b^{5} e^{2} + {\left (b^{12} d^{2} - 2 \, a b^{11} d e + a^{2} b^{10} e^{2}\right )} x^{5} + 5 \, {\left (a b^{11} d^{2} - 2 \, a^{2} b^{10} d e + a^{3} b^{9} e^{2}\right )} x^{4} + 10 \, {\left (a^{2} b^{10} d^{2} - 2 \, a^{3} b^{9} d e + a^{4} b^{8} e^{2}\right )} x^{3} + 10 \, {\left (a^{3} b^{9} d^{2} - 2 \, a^{4} b^{8} d e + a^{5} b^{7} e^{2}\right )} x^{2} + 5 \, {\left (a^{4} b^{8} d^{2} - 2 \, a^{5} b^{7} d e + a^{6} b^{6} e^{2}\right )} x\right )}}\right ] \]
[-1/3840*(105*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3 *b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2 *b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) + 2*(384*b^6* d^5 - 432*a*b^5*d^4*e - 8*a^2*b^4*d^3*e^2 - 14*a^3*b^3*d^2*e^3 - 35*a^4*b^ 2*d*e^4 + 105*a^5*b*e^5 + 105*(b^6*d*e^4 - a*b^5*e^5)*x^4 + 10*(121*b^6*d^ 2*e^3 - 200*a*b^5*d*e^4 + 79*a^2*b^4*e^5)*x^3 + 2*(1052*b^6*d^3*e^2 - 1341 *a*b^5*d^2*e^3 - 159*a^2*b^4*d*e^4 + 448*a^3*b^3*e^5)*x^2 + 2*(744*b^6*d^4 *e - 872*a*b^5*d^3*e^2 - 33*a^2*b^4*d^2*e^3 - 84*a^3*b^3*d*e^4 + 245*a^4*b ^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d^2 - 2*a^6*b^6*d*e + a^7*b^5*e^2 + (b^ 12*d^2 - 2*a*b^11*d*e + a^2*b^10*e^2)*x^5 + 5*(a*b^11*d^2 - 2*a^2*b^10*d*e + a^3*b^9*e^2)*x^4 + 10*(a^2*b^10*d^2 - 2*a^3*b^9*d*e + a^4*b^8*e^2)*x^3 + 10*(a^3*b^9*d^2 - 2*a^4*b^8*d*e + a^5*b^7*e^2)*x^2 + 5*(a^4*b^8*d^2 - 2* a^5*b^7*d*e + a^6*b^6*e^2)*x), -1/1920*(105*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*sqrt (-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (384*b^6*d^5 - 432*a*b^5*d^4*e - 8*a^2*b^4*d^3*e^2 - 14*a^3*b^3*d^2*e^3 - 35*a^4*b^2*d*e^4 + 105*a^5*b*e^5 + 105*(b^6*d*e^4 - a*b^5*e^5)*x^4 + 10* (121*b^6*d^2*e^3 - 200*a*b^5*d*e^4 + 79*a^2*b^4*e^5)*x^3 + 2*(1052*b^6*d^3 *e^2 - 1341*a*b^5*d^2*e^3 - 159*a^2*b^4*d*e^4 + 448*a^3*b^3*e^5)*x^2 + 2*( 744*b^6*d^4*e - 872*a*b^5*d^3*e^2 - 33*a^2*b^4*d^2*e^3 - 84*a^3*b^3*d*e...
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 354 vs. \(2 (156) = 312\).
Time = 0.29 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {7 \, e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{5} d - a b^{4} e\right )} \sqrt {-b^{2} d + a b e}} - \frac {105 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{4} e^{5} + 790 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{5} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} + 490 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} - 105 \, \sqrt {e x + d} b^{4} d^{4} e^{5} - 790 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{6} + 1792 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} - 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} + 420 \, \sqrt {e x + d} a b^{3} d^{3} e^{6} - 896 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} + 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} - 630 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{7} - 490 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{8} + 420 \, \sqrt {e x + d} a^{3} b d e^{8} - 105 \, \sqrt {e x + d} a^{4} e^{9}}{1920 \, {\left (b^{5} d - a b^{4} e\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]
-7/128*e^5*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d - a*b^4*e) *sqrt(-b^2*d + a*b*e)) - 1/1920*(105*(e*x + d)^(9/2)*b^4*e^5 + 790*(e*x + d)^(7/2)*b^4*d*e^5 - 896*(e*x + d)^(5/2)*b^4*d^2*e^5 + 490*(e*x + d)^(3/2) *b^4*d^3*e^5 - 105*sqrt(e*x + d)*b^4*d^4*e^5 - 790*(e*x + d)^(7/2)*a*b^3*e ^6 + 1792*(e*x + d)^(5/2)*a*b^3*d*e^6 - 1470*(e*x + d)^(3/2)*a*b^3*d^2*e^6 + 420*sqrt(e*x + d)*a*b^3*d^3*e^6 - 896*(e*x + d)^(5/2)*a^2*b^2*e^7 + 147 0*(e*x + d)^(3/2)*a^2*b^2*d*e^7 - 630*sqrt(e*x + d)*a^2*b^2*d^2*e^7 - 490* (e*x + d)^(3/2)*a^3*b*e^8 + 420*sqrt(e*x + d)*a^3*b*d*e^8 - 105*sqrt(e*x + d)*a^4*e^9)/((b^5*d - a*b^4*e)*((e*x + d)*b - b*d + a*e)^5)
Time = 9.54 (sec) , antiderivative size = 439, normalized size of antiderivative = 2.34 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {7\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{9/2}\,{\left (a\,e-b\,d\right )}^{3/2}}-\frac {\frac {79\,e^5\,{\left (d+e\,x\right )}^{7/2}}{192\,b}-\frac {7\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,\left (a\,e-b\,d\right )}+\frac {49\,e^5\,{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{192\,b^3}+\frac {7\,e^5\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{128\,b^4}+\frac {7\,e^5\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{5/2}}{15\,b^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \]
(7*e^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(128*b^(9/2)*(a* e - b*d)^(3/2)) - ((79*e^5*(d + e*x)^(7/2))/(192*b) - (7*e^5*(d + e*x)^(9/ 2))/(128*(a*e - b*d)) + (49*e^5*(d + e*x)^(3/2)*(a^2*e^2 + b^2*d^2 - 2*a*b *d*e))/(192*b^3) + (7*e^5*(d + e*x)^(1/2)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2 *e - 3*a^2*b*d*e^2))/(128*b^4) + (7*e^5*(a*e - b*d)*(d + e*x)^(5/2))/(15*b ^2))/((d + e*x)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d ^2*e^2 - 20*a*b^4*d^3*e) - (d + e*x)^2*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a ^2*b^3*d*e^2 - 30*a*b^4*d^2*e) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*( d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)