Integrand size = 35, antiderivative size = 328 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {21 e^2}{8 (b d-a e)^3 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {3 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (a+b x)}{8 (b d-a e)^4 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 b e^3 (a+b x)}{8 (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 b^{3/2} e^3 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
-21/8*e^2/(-a*e+b*d)^3/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)-1/3/(-a*e+b*d)/(b*x +a)^2/(e*x+d)^(3/2)/((b*x+a)^2)^(1/2)+3/4*e/(-a*e+b*d)^2/(b*x+a)/(e*x+d)^( 3/2)/((b*x+a)^2)^(1/2)-35/8*e^3*(b*x+a)/(-a*e+b*d)^4/(e*x+d)^(3/2)/((b*x+a )^2)^(1/2)+105/8*b^(3/2)*e^3*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b *d)^(1/2))/(-a*e+b*d)^(11/2)/((b*x+a)^2)^(1/2)-105/8*b*e^3*(b*x+a)/(-a*e+b *d)^5/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)
Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.73 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3 (a+b x) \left (\frac {-16 a^4 e^4+16 a^3 b e^3 (13 d+9 e x)+3 a^2 b^2 e^2 \left (55 d^2+318 d e x+231 e^2 x^2\right )+2 a b^3 e \left (-25 d^3+90 d^2 e x+567 d e^2 x^2+420 e^3 x^3\right )+b^4 \left (8 d^4-18 d^3 e x+63 d^2 e^2 x^2+420 d e^3 x^3+315 e^4 x^4\right )}{e^3 (-b d+a e)^5 (a+b x)^3 (d+e x)^{3/2}}+\frac {315 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{11/2}}\right )}{24 \sqrt {(a+b x)^2}} \]
(e^3*(a + b*x)*((-16*a^4*e^4 + 16*a^3*b*e^3*(13*d + 9*e*x) + 3*a^2*b^2*e^2 *(55*d^2 + 318*d*e*x + 231*e^2*x^2) + 2*a*b^3*e*(-25*d^3 + 90*d^2*e*x + 56 7*d*e^2*x^2 + 420*e^3*x^3) + b^4*(8*d^4 - 18*d^3*e*x + 63*d^2*e^2*x^2 + 42 0*d*e^3*x^3 + 315*e^4*x^4))/(e^3*(-(b*d) + a*e)^5*(a + b*x)^3*(d + e*x)^(3 /2)) + (315*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(- (b*d) + a*e)^(11/2)))/(24*Sqrt[(a + b*x)^2])
Time = 0.35 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.81, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1187, 27, 52, 52, 52, 61, 61, 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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {1}{b^5 (a+b x)^4 (d+e x)^{5/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^4 (d+e x)^{5/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {3 e \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {3 e \left (-\frac {7 e \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {3 e \left (-\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{3 (a+b x)^3 (d+e x)^{3/2} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/3*1/((b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)) - (3*e*(-1/2* 1/((b*d - a*e)*(a + b*x)^2*(d + e*x)^(3/2)) - (7*e*(-(1/((b*d - a*e)*(a + b*x)*(d + e*x)^(3/2))) - (5*e*(2/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (b*(2/( (b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sq rt[b*d - a*e]])/(b*d - a*e)^(3/2)))/(b*d - a*e)))/(2*(b*d - a*e))))/(4*(b* d - a*e))))/(2*(b*d - a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.22.44.3.1 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 + 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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(562\) vs. \(2(230)=460\).
Time = 0.31 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.72
| method | result | size |
| default | \(-\frac {\left (-315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \left (e x +d \right )^{\frac {3}{2}} b^{5} e^{3} x^{3}-945 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \left (e x +d \right )^{\frac {3}{2}} a \,b^{4} e^{3} x^{2}-315 \sqrt {\left (a e -b d \right ) b}\, b^{4} e^{4} x^{4}-945 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \left (e x +d \right )^{\frac {3}{2}} a^{2} b^{3} e^{3} x -840 \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} e^{4} x^{3}-420 \sqrt {\left (a e -b d \right ) b}\, b^{4} d \,e^{3} x^{3}-315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \left (e x +d \right )^{\frac {3}{2}} a^{3} b^{2} e^{3}-693 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} e^{4} x^{2}-1134 \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d \,e^{3} x^{2}-63 \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{2} e^{2} x^{2}-144 \sqrt {\left (a e -b d \right ) b}\, a^{3} b \,e^{4} x -954 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} d \,e^{3} x -180 \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d^{2} e^{2} x +18 \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{3} e x +16 \sqrt {\left (a e -b d \right ) b}\, a^{4} e^{4}-208 \sqrt {\left (a e -b d \right ) b}\, a^{3} b d \,e^{3}-165 \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} d^{2} e^{2}+50 \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d^{3} e -8 \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{4}\right ) \left (b x +a \right )^{2}}{24 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} \left (a e -b d \right )^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(563\) |
-1/24*(-315*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*b^5* e^3*x^3-945*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*a*b^ 4*e^3*x^2-315*((a*e-b*d)*b)^(1/2)*b^4*e^4*x^4-945*arctan(b*(e*x+d)^(1/2)/( (a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*a^2*b^3*e^3*x-840*((a*e-b*d)*b)^(1/2)*a* b^3*e^4*x^3-420*((a*e-b*d)*b)^(1/2)*b^4*d*e^3*x^3-315*arctan(b*(e*x+d)^(1/ 2)/((a*e-b*d)*b)^(1/2))*(e*x+d)^(3/2)*a^3*b^2*e^3-693*((a*e-b*d)*b)^(1/2)* a^2*b^2*e^4*x^2-1134*((a*e-b*d)*b)^(1/2)*a*b^3*d*e^3*x^2-63*((a*e-b*d)*b)^ (1/2)*b^4*d^2*e^2*x^2-144*((a*e-b*d)*b)^(1/2)*a^3*b*e^4*x-954*((a*e-b*d)*b )^(1/2)*a^2*b^2*d*e^3*x-180*((a*e-b*d)*b)^(1/2)*a*b^3*d^2*e^2*x+18*((a*e-b *d)*b)^(1/2)*b^4*d^3*e*x+16*((a*e-b*d)*b)^(1/2)*a^4*e^4-208*((a*e-b*d)*b)^ (1/2)*a^3*b*d*e^3-165*((a*e-b*d)*b)^(1/2)*a^2*b^2*d^2*e^2+50*((a*e-b*d)*b) ^(1/2)*a*b^3*d^3*e-8*((a*e-b*d)*b)^(1/2)*b^4*d^4)*(b*x+a)^2/((a*e-b*d)*b)^ (1/2)/(e*x+d)^(3/2)/(a*e-b*d)^5/((b*x+a)^2)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (230) = 460\).
Time = 0.44 (sec) , antiderivative size = 1840, normalized size of antiderivative = 5.61 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[-1/48*(315*(b^4*e^5*x^5 + a^3*b*d^2*e^3 + (2*b^4*d*e^4 + 3*a*b^3*e^5)*x^4 + (b^4*d^2*e^3 + 6*a*b^3*d*e^4 + 3*a^2*b^2*e^5)*x^3 + (3*a*b^3*d^2*e^3 + 6*a^2*b^2*d*e^4 + a^3*b*e^5)*x^2 + (3*a^2*b^2*d^2*e^3 + 2*a^3*b*d*e^4)*x)* sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e - 2*(b*d - a*e)*sqrt(e*x + d) *sqrt(b/(b*d - a*e)))/(b*x + a)) + 2*(315*b^4*e^4*x^4 + 8*b^4*d^4 - 50*a*b ^3*d^3*e + 165*a^2*b^2*d^2*e^2 + 208*a^3*b*d*e^3 - 16*a^4*e^4 + 420*(b^4*d *e^3 + 2*a*b^3*e^4)*x^3 + 63*(b^4*d^2*e^2 + 18*a*b^3*d*e^3 + 11*a^2*b^2*e^ 4)*x^2 - 18*(b^4*d^3*e - 10*a*b^3*d^2*e^2 - 53*a^2*b^2*d*e^3 - 8*a^3*b*e^4 )*x)*sqrt(e*x + d))/(a^3*b^5*d^7 - 5*a^4*b^4*d^6*e + 10*a^5*b^3*d^5*e^2 - 10*a^6*b^2*d^4*e^3 + 5*a^7*b*d^3*e^4 - a^8*d^2*e^5 + (b^8*d^5*e^2 - 5*a*b^ 7*d^4*e^3 + 10*a^2*b^6*d^3*e^4 - 10*a^3*b^5*d^2*e^5 + 5*a^4*b^4*d*e^6 - a^ 5*b^3*e^7)*x^5 + (2*b^8*d^6*e - 7*a*b^7*d^5*e^2 + 5*a^2*b^6*d^4*e^3 + 10*a ^3*b^5*d^3*e^4 - 20*a^4*b^4*d^2*e^5 + 13*a^5*b^3*d*e^6 - 3*a^6*b^2*e^7)*x^ 4 + (b^8*d^7 + a*b^7*d^6*e - 17*a^2*b^6*d^5*e^2 + 35*a^3*b^5*d^4*e^3 - 25* a^4*b^4*d^3*e^4 - a^5*b^3*d^2*e^5 + 9*a^6*b^2*d*e^6 - 3*a^7*b*e^7)*x^3 + ( 3*a*b^7*d^7 - 9*a^2*b^6*d^6*e + a^3*b^5*d^5*e^2 + 25*a^4*b^4*d^4*e^3 - 35* a^5*b^3*d^3*e^4 + 17*a^6*b^2*d^2*e^5 - a^7*b*d*e^6 - a^8*e^7)*x^2 + (3*a^2 *b^6*d^7 - 13*a^3*b^5*d^6*e + 20*a^4*b^4*d^5*e^2 - 10*a^5*b^3*d^4*e^3 - 5* a^6*b^2*d^3*e^4 + 7*a^7*b*d^2*e^5 - 2*a^8*d*e^6)*x), 1/24*(315*(b^4*e^5*x^ 5 + a^3*b*d^2*e^3 + (2*b^4*d*e^4 + 3*a*b^3*e^5)*x^4 + (b^4*d^2*e^3 + 6*...
Timed out. \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 504 vs. \(2 (230) = 460\).
Time = 0.29 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.54 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {105 \, b^{2} e^{3} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {315 \, {\left (e x + d\right )}^{4} b^{4} e^{3} - 840 \, {\left (e x + d\right )}^{3} b^{4} d e^{3} + 693 \, {\left (e x + d\right )}^{2} b^{4} d^{2} e^{3} - 144 \, {\left (e x + d\right )} b^{4} d^{3} e^{3} - 16 \, b^{4} d^{4} e^{3} + 840 \, {\left (e x + d\right )}^{3} a b^{3} e^{4} - 1386 \, {\left (e x + d\right )}^{2} a b^{3} d e^{4} + 432 \, {\left (e x + d\right )} a b^{3} d^{2} e^{4} + 64 \, a b^{3} d^{3} e^{4} + 693 \, {\left (e x + d\right )}^{2} a^{2} b^{2} e^{5} - 432 \, {\left (e x + d\right )} a^{2} b^{2} d e^{5} - 96 \, a^{2} b^{2} d^{2} e^{5} + 144 \, {\left (e x + d\right )} a^{3} b e^{6} + 64 \, a^{3} b d e^{6} - 16 \, a^{4} e^{7}}{24 \, {\left (b^{5} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{2} b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 10 \, a^{3} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b d e^{4} \mathrm {sgn}\left (b x + a\right ) - a^{5} e^{5} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )}^{\frac {3}{2}} b - \sqrt {e x + d} b d + \sqrt {e x + d} a e\right )}^{3}} \]
-105/8*b^2*e^3*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^5*sgn( b*x + a) - 5*a*b^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn (b*x + a))*sqrt(-b^2*d + a*b*e)) - 1/24*(315*(e*x + d)^4*b^4*e^3 - 840*(e* x + d)^3*b^4*d*e^3 + 693*(e*x + d)^2*b^4*d^2*e^3 - 144*(e*x + d)*b^4*d^3*e ^3 - 16*b^4*d^4*e^3 + 840*(e*x + d)^3*a*b^3*e^4 - 1386*(e*x + d)^2*a*b^3*d *e^4 + 432*(e*x + d)*a*b^3*d^2*e^4 + 64*a*b^3*d^3*e^4 + 693*(e*x + d)^2*a^ 2*b^2*e^5 - 432*(e*x + d)*a^2*b^2*d*e^5 - 96*a^2*b^2*d^2*e^5 + 144*(e*x + d)*a^3*b*e^6 + 64*a^3*b*d*e^6 - 16*a^4*e^7)/((b^5*d^5*sgn(b*x + a) - 5*a*b ^4*d^4*e*sgn(b*x + a) + 10*a^2*b^3*d^3*e^2*sgn(b*x + a) - 10*a^3*b^2*d^2*e ^3*sgn(b*x + a) + 5*a^4*b*d*e^4*sgn(b*x + a) - a^5*e^5*sgn(b*x + a))*((e*x + d)^(3/2)*b - sqrt(e*x + d)*b*d + sqrt(e*x + d)*a*e)^3)
Timed out. \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {a+b\,x}{{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]