Integrand size = 30, antiderivative size = 308 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {63 e^2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 e^2 (b d-a e)^{5/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
21/4*e^2*(-a*e+b*d)*(b*x+a)*(e*x+d)^(3/2)/b^4/((b*x+a)^2)^(1/2)+63/20*e^2* (b*x+a)*(e*x+d)^(5/2)/b^3/((b*x+a)^2)^(1/2)-9/4*e*(e*x+d)^(7/2)/b^2/((b*x+ a)^2)^(1/2)-1/2*(e*x+d)^(9/2)/b/(b*x+a)/((b*x+a)^2)^(1/2)-63/4*e^2*(-a*e+b *d)^(5/2)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(11/2) /((b*x+a)^2)^(1/2)+63/4*e^2*(-a*e+b*d)^2*(b*x+a)*(e*x+d)^(1/2)/b^5/((b*x+a )^2)^(1/2)
Time = 0.66 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^2 (a+b x)^3 \left (\frac {\sqrt {b} \sqrt {d+e x} \left (315 a^4 e^4+105 a^3 b e^3 (-7 d+5 e x)+21 a^2 b^2 e^2 \left (23 d^2-59 d e x+8 e^2 x^2\right )-3 a b^3 e \left (15 d^3-277 d^2 e x+136 d e^2 x^2+8 e^3 x^3\right )+b^4 \left (-10 d^4-85 d^3 e x+288 d^2 e^2 x^2+56 d e^3 x^3+8 e^4 x^4\right )\right )}{e^2 (a+b x)^2}-315 (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{20 b^{11/2} \left ((a+b x)^2\right )^{3/2}} \]
(e^2*(a + b*x)^3*((Sqrt[b]*Sqrt[d + e*x]*(315*a^4*e^4 + 105*a^3*b*e^3*(-7* d + 5*e*x) + 21*a^2*b^2*e^2*(23*d^2 - 59*d*e*x + 8*e^2*x^2) - 3*a*b^3*e*(1 5*d^3 - 277*d^2*e*x + 136*d*e^2*x^2 + 8*e^3*x^3) + b^4*(-10*d^4 - 85*d^3*e *x + 288*d^2*e^2*x^2 + 56*d*e^3*x^3 + 8*e^4*x^4)))/(e^2*(a + b*x)^2) - 315 *(-(b*d) + a*e)^(5/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])) /(20*b^(11/2)*((a + b*x)^2)^(3/2))
Time = 0.30 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.68, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1102, 27, 51, 51, 60, 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.
| \(\displaystyle \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {b^3 (a+b x) \int \frac {(d+e x)^{9/2}}{b^3 (a+b x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(d+e x)^{9/2}}{(a+b x)^3}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {9 e \int \frac {(d+e x)^{7/2}}{(a+b x)^2}dx}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {(a+b x) \left (\frac {9 e \left (\frac {7 e \int \frac {(d+e x)^{5/2}}{a+b x}dx}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \int \frac {(d+e x)^{3/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (\frac {9 e \left (\frac {7 e \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{2 b}-\frac {(d+e x)^{7/2}}{b (a+b x)}\right )}{4 b}-\frac {(d+e x)^{9/2}}{2 b (a+b x)^2}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/2*(d + e*x)^(9/2)/(b*(a + b*x)^2) + (9*e*(-((d + e*x)^(7/2) /(b*(a + b*x))) + (7*e*((2*(d + e*x)^(5/2))/(5*b) + ((b*d - a*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))/b))/(2*b) ))/(4*b)))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.11.3.1 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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 2.31 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {2 e^{2} \left (x^{2} b^{2} e^{2}-5 x a b \,e^{2}+7 b^{2} d e x +30 a^{2} e^{2}-65 a b d e +36 b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{5 b^{5} \left (b x +a \right )}-\frac {\left (2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) e^{2} \left (\frac {-\frac {17 \left (e x +d \right )^{\frac {3}{2}} b}{8}+\left (-\frac {15 a e}{8}+\frac {15 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {63 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}\) | \(226\) |
| default | \(\text {Expression too large to display}\) | \(1115\) |
2/5*e^2*(b^2*e^2*x^2-5*a*b*e^2*x+7*b^2*d*e*x+30*a^2*e^2-65*a*b*d*e+36*b^2* d^2)*(e*x+d)^(1/2)/b^5*((b*x+a)^2)^(1/2)/(b*x+a)-1/b^5*(2*a^3*e^3-6*a^2*b* d*e^2+6*a*b^2*d^2*e-2*b^3*d^3)*e^2*((-17/8*(e*x+d)^(3/2)*b+(-15/8*a*e+15/8 *b*d)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^2+63/8/((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 = 0.29 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.37 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\left [\frac {315 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \, {\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \, {\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{40 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac {315 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \, {\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \, {\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{20 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]
[1/40*(315*(a^2*b^2*d^2*e^2 - 2*a^3*b*d*e^3 + a^4*e^4 + (b^4*d^2*e^2 - 2*a *b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 - 2*a^2*b^2*d*e^3 + a^3*b *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*(8*b^4*e^4*x^4 - 10*b^4*d^4 - 45*a*b^3 *d^3*e + 483*a^2*b^2*d^2*e^2 - 735*a^3*b*d*e^3 + 315*a^4*e^4 + 8*(7*b^4*d* e^3 - 3*a*b^3*e^4)*x^3 + 24*(12*b^4*d^2*e^2 - 17*a*b^3*d*e^3 + 7*a^2*b^2*e ^4)*x^2 - (85*b^4*d^3*e - 831*a*b^3*d^2*e^2 + 1239*a^2*b^2*d*e^3 - 525*a^3 *b*e^4)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/20*(315*(a^2 *b^2*d^2*e^2 - 2*a^3*b*d*e^3 + a^4*e^4 + (b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^ 2*b^2*e^4)*x^2 + 2*(a*b^3*d^2*e^2 - 2*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*sqrt(- (b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (8*b^4*e^4*x^4 - 10*b^4*d^4 - 45*a*b^3*d^3*e + 483*a^2*b^2*d^2*e^2 - 735* a^3*b*d*e^3 + 315*a^4*e^4 + 8*(7*b^4*d*e^3 - 3*a*b^3*e^4)*x^3 + 24*(12*b^4 *d^2*e^2 - 17*a*b^3*d*e^3 + 7*a^2*b^2*e^4)*x^2 - (85*b^4*d^3*e - 831*a*b^3 *d^2*e^2 + 1239*a^2*b^2*d*e^3 - 525*a^3*b*e^4)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5)]
\[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {9}{2}}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {63 \, {\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {17 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{2} - 15 \, \sqrt {e x + d} b^{4} d^{4} e^{2} - 51 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{3} + 60 \, \sqrt {e x + d} a b^{3} d^{3} e^{3} + 51 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{4} - 90 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{4} - 17 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{5} + 60 \, \sqrt {e x + d} a^{3} b d e^{5} - 15 \, \sqrt {e x + d} a^{4} e^{6}}{4 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2} b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {5}{2}} b^{12} e^{2} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{12} d e^{2} + 30 \, \sqrt {e x + d} b^{12} d^{2} e^{2} - 5 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{11} e^{3} - 60 \, \sqrt {e x + d} a b^{11} d e^{3} + 30 \, \sqrt {e x + d} a^{2} b^{10} e^{4}\right )}}{5 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \]
63/4*(b^3*d^3*e^2 - 3*a*b^2*d^2*e^3 + 3*a^2*b*d*e^4 - a^3*e^5)*arctan(sqrt (e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5*sgn(b*x + a)) - 1/4*(17*(e*x + d)^(3/2)*b^4*d^3*e^2 - 15*sqrt(e*x + d)*b^4*d^4*e^2 - 51* (e*x + d)^(3/2)*a*b^3*d^2*e^3 + 60*sqrt(e*x + d)*a*b^3*d^3*e^3 + 51*(e*x + d)^(3/2)*a^2*b^2*d*e^4 - 90*sqrt(e*x + d)*a^2*b^2*d^2*e^4 - 17*(e*x + d)^ (3/2)*a^3*b*e^5 + 60*sqrt(e*x + d)*a^3*b*d*e^5 - 15*sqrt(e*x + d)*a^4*e^6) /(((e*x + d)*b - b*d + a*e)^2*b^5*sgn(b*x + a)) + 2/5*((e*x + d)^(5/2)*b^1 2*e^2 + 5*(e*x + d)^(3/2)*b^12*d*e^2 + 30*sqrt(e*x + d)*b^12*d^2*e^2 - 5*( e*x + d)^(3/2)*a*b^11*e^3 - 60*sqrt(e*x + d)*a*b^11*d*e^3 + 30*sqrt(e*x + d)*a^2*b^10*e^4)/(b^15*sgn(b*x + a))
Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]