Integrand size = 28, antiderivative size = 178 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}-\frac {63 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} \sqrt {b d-a e}} \]
-21/64*e^3*(e*x+d)^(3/2)/b^4/(b*x+a)^2-21/80*e^2*(e*x+d)^(5/2)/b^3/(b*x+a) ^3-9/40*e*(e*x+d)^(7/2)/b^2/(b*x+a)^4-1/5*(e*x+d)^(9/2)/b/(b*x+a)^5-63/128 *e^5*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(11/2)/(-a*e+b*d)^( 1/2)-63/128*e^4*(e*x+d)^(1/2)/b^5/(b*x+a)
Time = 1.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (315 a^4 e^4+210 a^3 b e^3 (d+7 e x)+42 a^2 b^2 e^2 \left (4 d^2+23 d e x+64 e^2 x^2\right )+6 a b^3 e \left (24 d^3+128 d^2 e x+289 d e^2 x^2+395 e^3 x^3\right )+b^4 \left (128 d^4+656 d^3 e x+1368 d^2 e^2 x^2+1490 d e^3 x^3+965 e^4 x^4\right )\right )}{640 b^5 (a+b x)^5}+\frac {63 e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{11/2} \sqrt {-b d+a e}} \]
-1/640*(Sqrt[d + e*x]*(315*a^4*e^4 + 210*a^3*b*e^3*(d + 7*e*x) + 42*a^2*b^ 2*e^2*(4*d^2 + 23*d*e*x + 64*e^2*x^2) + 6*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*(128*d^4 + 656*d^3*e*x + 1368*d^2*e^2*x ^2 + 1490*d*e^3*x^3 + 965*e^4*x^4)))/(b^5*(a + b*x)^5) + (63*e^5*ArcTan[(S qrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(128*b^(11/2)*Sqrt[-(b*d) + a*e ])
Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11, 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, 51, 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.
| \(\displaystyle \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle b^6 \int \frac {(d+e x)^{9/2}}{b^6 (a+b x)^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d+e x)^{9/2}}{(a+b x)^6}dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \int \frac {(d+e x)^{7/2}}{(a+b x)^5}dx}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{(a+b x)^4}dx}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \int \frac {(d+e x)^{3/2}}{(a+b x)^3}dx}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \int \frac {\sqrt {d+e x}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {9 e \left (\frac {7 e \left (\frac {5 e \left (\frac {3 e \left (-\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{3/2}}{2 b (a+b x)^2}\right )}{6 b}-\frac {(d+e x)^{5/2}}{3 b (a+b x)^3}\right )}{8 b}-\frac {(d+e x)^{7/2}}{4 b (a+b x)^4}\right )}{10 b}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}\) |
-1/5*(d + e*x)^(9/2)/(b*(a + b*x)^5) + (9*e*(-1/4*(d + e*x)^(7/2)/(b*(a + b*x)^4) + (7*e*(-1/3*(d + e*x)^(5/2)/(b*(a + b*x)^3) + (5*e*(-1/2*(d + e*x )^(3/2)/(b*(a + b*x)^2) + (3*e*(-(Sqrt[d + e*x]/(b*(a + b*x))) - (e*ArcTan h[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b^(3/2)*Sqrt[b*d - a*e])))/(4 *b)))/(6*b)))/(8*b)))/(10*b)
3.17.67.3.1 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] :> 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 = 3.53 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.19
| method | result | size |
| pseudoelliptic | \(\frac {\frac {63 e^{5} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}-\frac {63 \left (\left (\frac {193}{63} e^{4} x^{4}+\frac {298}{63} d \,e^{3} x^{3}+\frac {152}{35} d^{2} e^{2} x^{2}+\frac {656}{315} d^{3} e x +\frac {128}{315} d^{4}\right ) b^{4}+\frac {16 \left (\frac {395}{24} e^{3} x^{3}+\frac {289}{24} d \,e^{2} x^{2}+\frac {16}{3} d^{2} e x +d^{3}\right ) e a \,b^{3}}{35}+\frac {8 e^{2} \left (16 x^{2} e^{2}+\frac {23}{4} d e x +d^{2}\right ) a^{2} b^{2}}{15}+\frac {2 a^{3} e^{3} \left (7 e x +d \right ) b}{3}+e^{4} a^{4}\right ) \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}}{128}}{b^{5} \left (b x +a \right )^{5} \sqrt {\left (a e -b d \right ) b}}\) | \(211\) |
| derivativedivides | \(2 e^{5} \left (\frac {-\frac {193 \left (e x +d \right )^{\frac {9}{2}}}{256 b}-\frac {237 \left (a e -b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{128 b^{2}}-\frac {21 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{10 b^{3}}-\frac {147 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{4}}-\frac {63 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {e x +d}}{256 b^{5}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {63 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{5} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(239\) |
| default | \(2 e^{5} \left (\frac {-\frac {193 \left (e x +d \right )^{\frac {9}{2}}}{256 b}-\frac {237 \left (a e -b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{128 b^{2}}-\frac {21 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{10 b^{3}}-\frac {147 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{4}}-\frac {63 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {e x +d}}{256 b^{5}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {63 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{5} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(239\) |
63/128/((a*e-b*d)*b)^(1/2)*(e^5*(b*x+a)^5*arctan(b*(e*x+d)^(1/2)/((a*e-b*d )*b)^(1/2))-((193/63*e^4*x^4+298/63*d*e^3*x^3+152/35*d^2*e^2*x^2+656/315*d ^3*e*x+128/315*d^4)*b^4+16/35*(395/24*e^3*x^3+289/24*d*e^2*x^2+16/3*d^2*e* x+d^3)*e*a*b^3+8/15*e^2*(16*x^2*e^2+23/4*d*e*x+d^2)*a^2*b^2+2/3*a^3*e^3*(7 *e*x+d)*b+e^4*a^4)*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2))/b^5/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (146) = 292\).
Time = 0.36 (sec) , antiderivative size = 1003, normalized size of antiderivative = 5.63 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [\frac {315 \, {\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 (128 \, b^{6} d^{5} + 16 \, a b^{5} d^{4} e + 24 \, a^{2} b^{4} d^{3} e^{2} + 42 \, a^{3} b^{3} d^{2} e^{3} + 105 \, a^{4} b^{2} d e^{4} - 315 \, a^{5} b e^{5} + 965 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{4} + 10 \, {\left (149 \, b^{6} d^{2} e^{3} + 88 \, a b^{5} d e^{4} - 237 \, a^{2} b^{4} e^{5}\right )} x^{3} + 6 \, {\left (228 \, b^{6} d^{3} e^{2} + 61 \, 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 (328 \, b^{6} d^{4} e + 56 \, a b^{5} d^{3} e^{2} + 99 \, a^{2} b^{4} d^{2} e^{3} + 252 \, a^{3} b^{3} d e^{4} - 735 \, a^{4} b^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{1280 \, {\left (a^{5} b^{7} d - a^{6} b^{6} e + {\left (b^{12} d - a b^{11} e\right )} x^{5} + 5 \, {\left (a b^{11} d - a^{2} b^{10} e\right )} x^{4} + 10 \, {\left (a^{2} b^{10} d - a^{3} b^{9} e\right )} x^{3} + 10 \, {\left (a^{3} b^{9} d - a^{4} b^{8} e\right )} x^{2} + 5 \, {\left (a^{4} b^{8} d - a^{5} b^{7} e\right )} x\right )}}, \frac {315 \, {\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 (128 \, b^{6} d^{5} + 16 \, a b^{5} d^{4} e + 24 \, a^{2} b^{4} d^{3} e^{2} + 42 \, a^{3} b^{3} d^{2} e^{3} + 105 \, a^{4} b^{2} d e^{4} - 315 \, a^{5} b e^{5} + 965 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{4} + 10 \, {\left (149 \, b^{6} d^{2} e^{3} + 88 \, a b^{5} d e^{4} - 237 \, a^{2} b^{4} e^{5}\right )} x^{3} + 6 \, {\left (228 \, b^{6} d^{3} e^{2} + 61 \, 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 (328 \, b^{6} d^{4} e + 56 \, a b^{5} d^{3} e^{2} + 99 \, a^{2} b^{4} d^{2} e^{3} + 252 \, a^{3} b^{3} d e^{4} - 735 \, a^{4} b^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{640 \, {\left (a^{5} b^{7} d - a^{6} b^{6} e + {\left (b^{12} d - a b^{11} e\right )} x^{5} + 5 \, {\left (a b^{11} d - a^{2} b^{10} e\right )} x^{4} + 10 \, {\left (a^{2} b^{10} d - a^{3} b^{9} e\right )} x^{3} + 10 \, {\left (a^{3} b^{9} d - a^{4} b^{8} e\right )} x^{2} + 5 \, {\left (a^{4} b^{8} d - a^{5} b^{7} e\right )} x\right )}}\right ] \]
[1/1280*(315*(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*(128*b^6*d ^5 + 16*a*b^5*d^4*e + 24*a^2*b^4*d^3*e^2 + 42*a^3*b^3*d^2*e^3 + 105*a^4*b^ 2*d*e^4 - 315*a^5*b*e^5 + 965*(b^6*d*e^4 - a*b^5*e^5)*x^4 + 10*(149*b^6*d^ 2*e^3 + 88*a*b^5*d*e^4 - 237*a^2*b^4*e^5)*x^3 + 6*(228*b^6*d^3*e^2 + 61*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*(328*b^6*d^4*e + 56*a*b^5*d^3*e^2 + 99*a^2*b^4*d^2*e^3 + 252*a^3*b^3*d*e^4 - 735*a^4*b^2* e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d - a^6*b^6*e + (b^12*d - a*b^11*e)*x^5 + 5*(a*b^11*d - a^2*b^10*e)*x^4 + 10*(a^2*b^10*d - a^3*b^9*e)*x^3 + 10*(a^3* b^9*d - a^4*b^8*e)*x^2 + 5*(a^4*b^8*d - a^5*b^7*e)*x), 1/640*(315*(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)) - (128*b^6*d^5 + 16*a*b^5*d^4*e + 24*a^2*b^4*d^3*e^2 + 42*a^3*b^3*d^2*e^3 + 105*a^4*b^2*d*e^4 - 315*a^5*b*e^5 + 965*(b^6*d*e^4 - a*b^5*e^5)*x^4 + 10*(149*b^6*d^2*e^3 + 88*a*b^5*d*e^4 - 237*a^2*b^4*e^5 )*x^3 + 6*(228*b^6*d^3*e^2 + 61*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*(328*b^6*d^4*e + 56*a*b^5*d^3*e^2 + 99*a^2*b^4*d^2*e^3 + 252*a^3*b^3*d*e^4 - 735*a^4*b^2*e^5)*x)*sqrt(e*x + d))/(a^5*b^7*d - a^6* b^6*e + (b^12*d - a*b^11*e)*x^5 + 5*(a*b^11*d - a^2*b^10*e)*x^4 + 10*(a...
Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 330 vs. \(2 (146) = 292\).
Time = 0.31 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63 \, e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {965 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 2370 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 2688 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} + 315 \, \sqrt {e x + d} b^{4} d^{4} e^{5} + 2370 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 5376 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 4410 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} - 1260 \, \sqrt {e x + d} a b^{3} d^{3} e^{6} + 2688 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 4410 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} + 1890 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{7} + 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{8} - 1260 \, \sqrt {e x + d} a^{3} b d e^{8} + 315 \, \sqrt {e x + d} a^{4} e^{9}}{640 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5} b^{5}} \]
63/128*e^5*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b *e)*b^5) - 1/640*(965*(e*x + d)^(9/2)*b^4*e^5 - 2370*(e*x + d)^(7/2)*b^4*d *e^5 + 2688*(e*x + d)^(5/2)*b^4*d^2*e^5 - 1470*(e*x + d)^(3/2)*b^4*d^3*e^5 + 315*sqrt(e*x + d)*b^4*d^4*e^5 + 2370*(e*x + d)^(7/2)*a*b^3*e^6 - 5376*( e*x + d)^(5/2)*a*b^3*d*e^6 + 4410*(e*x + d)^(3/2)*a*b^3*d^2*e^6 - 1260*sqr t(e*x + d)*a*b^3*d^3*e^6 + 2688*(e*x + d)^(5/2)*a^2*b^2*e^7 - 4410*(e*x + d)^(3/2)*a^2*b^2*d*e^7 + 1890*sqrt(e*x + d)*a^2*b^2*d^2*e^7 + 1470*(e*x + d)^(3/2)*a^3*b*e^8 - 1260*sqrt(e*x + d)*a^3*b*d*e^8 + 315*sqrt(e*x + d)*a^ 4*e^9)/(((e*x + d)*b - b*d + a*e)^5*b^5)
Time = 9.56 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.70 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{11/2}\,\sqrt {a\,e-b\,d}}-\frac {\frac {193\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,b}+\frac {63\,e^5\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{128\,b^5}+\frac {21\,e^5\,{\left (d+e\,x\right )}^{5/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{5\,b^3}+\frac {147\,e^5\,{\left (d+e\,x\right )}^{3/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{64\,b^4}+\frac {237\,e^5\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{64\,b^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \]
(63*e^5*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(128*b^(11/2)*( a*e - b*d)^(1/2)) - ((193*e^5*(d + e*x)^(9/2))/(128*b) + (63*e^5*(d + e*x) ^(1/2)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3*b*d* e^3))/(128*b^5) + (21*e^5*(d + e*x)^(5/2)*(a^2*e^2 + b^2*d^2 - 2*a*b*d*e)) /(5*b^3) + (147*e^5*(d + e*x)^(3/2)*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3 *a^2*b*d*e^2))/(64*b^4) + (237*e^5*(a*e - b*d)*(d + e*x)^(7/2))/(64*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)