Integrand size = 35, antiderivative size = 424 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {35 e^2 (8 b B d-9 A b e+a B e)}{192 b (b d-a e)^4 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{4 b (b d-a e) (a+b x)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {8 b B d-9 A b e+a B e}{24 b (b d-a e)^2 (a+b x)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 e (8 b B d-9 A b e+a B e)}{96 b (b d-a e)^3 (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x)}{64 b (b d-a e)^5 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d-9 A b e+a B e) (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
35/64*e^3*(-9*A*b*e+B*a*e+8*B*b*d)*(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^(1/2)/((b*x+a)^2)^(1/2)-35/192*e^2*(- 9*A*b*e+B*a*e+8*B*b*d)/b/(-a*e+b*d)^4/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)+1/4* (-A*b+B*a)/b/(-a*e+b*d)/(b*x+a)^3/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)+1/24*(9* A*b*e-B*a*e-8*B*b*d)/b/(-a*e+b*d)^2/(b*x+a)^2/(e*x+d)^(1/2)/((b*x+a)^2)^(1 /2)+7/96*e*(-9*A*b*e+B*a*e+8*B*b*d)/b/(-a*e+b*d)^3/(b*x+a)/(e*x+d)^(1/2)/( (b*x+a)^2)^(1/2)-35/64*e^3*(-9*A*b*e+B*a*e+8*B*b*d)*(b*x+a)/b/(-a*e+b*d)^5 /(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)
Time = 2.43 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^3 (a+b x) \left (\frac {3 A \left (-128 a^4 e^4-a^3 b e^3 (325 d+837 e x)-3 a^2 b^2 e^2 \left (-70 d^2+185 d e x+511 e^2 x^2\right )-a b^3 e \left (88 d^3-156 d^2 e x+399 d e^2 x^2+1155 e^3 x^3\right )+b^4 \left (16 d^4-24 d^3 e x+42 d^2 e^2 x^2-105 d e^3 x^3-315 e^4 x^4\right )\right )+B \left (3 a^4 e^3 (221 d+93 e x)+a^3 b e^2 \left (370 d^2+2417 d e x+511 e^2 x^2\right )+8 b^4 d x \left (8 d^3-14 d^2 e x+35 d e^2 x^2+105 e^3 x^3\right )+a^2 b^2 e \left (-104 d^3+1428 d^2 e x+4221 d e^2 x^2+385 e^3 x^3\right )+a b^3 \left (16 d^4-408 d^3 e x+1050 d^2 e^2 x^2+3115 d e^3 x^3+105 e^4 x^4\right )\right )}{e^3 (-b d+a e)^5 (a+b x)^4 \sqrt {d+e x}}+\frac {105 (8 b B d-9 A b e+a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{11/2}}\right )}{192 \sqrt {(a+b x)^2}} \]
(e^3*(a + b*x)*((3*A*(-128*a^4*e^4 - a^3*b*e^3*(325*d + 837*e*x) - 3*a^2*b ^2*e^2*(-70*d^2 + 185*d*e*x + 511*e^2*x^2) - a*b^3*e*(88*d^3 - 156*d^2*e*x + 399*d*e^2*x^2 + 1155*e^3*x^3) + b^4*(16*d^4 - 24*d^3*e*x + 42*d^2*e^2*x ^2 - 105*d*e^3*x^3 - 315*e^4*x^4)) + B*(3*a^4*e^3*(221*d + 93*e*x) + a^3*b *e^2*(370*d^2 + 2417*d*e*x + 511*e^2*x^2) + 8*b^4*d*x*(8*d^3 - 14*d^2*e*x + 35*d*e^2*x^2 + 105*e^3*x^3) + a^2*b^2*e*(-104*d^3 + 1428*d^2*e*x + 4221* d*e^2*x^2 + 385*e^3*x^3) + a*b^3*(16*d^4 - 408*d^3*e*x + 1050*d^2*e^2*x^2 + 3115*d*e^3*x^3 + 105*e^4*x^4)))/(e^3*(-(b*d) + a*e)^5*(a + b*x)^4*Sqrt[d + e*x]) + (105*(8*b*B*d - 9*A*b*e + a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x]) /Sqrt[-(b*d) + a*e]])/(Sqrt[b]*(-(b*d) + a*e)^(11/2))))/(192*Sqrt[(a + b*x )^2])
Time = 0.42 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.72, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {1187, 27, 87, 52, 52, 52, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {A+B x}{b^5 (a+b x)^5 (d+e x)^{3/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {A+B x}{(a+b x)^5 (d+e x)^{3/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(a B e-9 A b e+8 b B d) \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}}dx}{8 b (b d-a e)}-\frac {A b-a B}{4 b (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(a B e-9 A b e+8 b B d) \left (-\frac {7 e \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}}dx}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 b (b d-a e)}-\frac {A b-a B}{4 b (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(a B e-9 A b e+8 b B d) \left (-\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 b (b d-a e)}-\frac {A b-a B}{4 b (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(a B e-9 A b e+8 b B d) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 b (b d-a e)}-\frac {A b-a B}{4 b (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(a B e-9 A b e+8 b B d) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 b (b d-a e)}-\frac {A b-a B}{4 b (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(a B e-9 A b e+8 b B d) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 b (b d-a e)}-\frac {A b-a B}{4 b (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(a B e-9 A b e+8 b B d) \left (-\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{6 (b d-a e)}-\frac {1}{3 (a+b x)^3 \sqrt {d+e x} (b d-a e)}\right )}{8 b (b d-a e)}-\frac {A b-a B}{4 b (a+b x)^4 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/4*(A*b - a*B)/(b*(b*d - a*e)*(a + b*x)^4*Sqrt[d + e*x]) + ( (8*b*B*d - 9*A*b*e + a*B*e)*(-1/3*1/((b*d - a*e)*(a + b*x)^3*Sqrt[d + e*x] ) - (7*e*(-1/2*1/((b*d - a*e)*(a + b*x)^2*Sqrt[d + e*x]) - (5*e*(-(1/((b*d - a*e)*(a + b*x)*Sqrt[d + e*x])) - (3*e*(2/((b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e)^( 3/2)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/(6*(b*d - a*e))))/(8*b*(b*d - a *e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.19.81.3.1 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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. \(1492\) vs. \(2(327)=654\).
Time = 0.28 (sec) , antiderivative size = 1493, normalized size of antiderivative = 3.52
-1/192*(945*A*(e*x+d)^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^ 4*b*e^4-1050*B*((a*e-b*d)*b)^(1/2)*a*b^3*d^2*e^2*x^2+1665*A*((a*e-b*d)*b)^ (1/2)*a^2*b^2*d*e^3*x-468*A*((a*e-b*d)*b)^(1/2)*a*b^3*d^2*e^2*x-2417*B*((a *e-b*d)*b)^(1/2)*a^3*b*d*e^3*x-1428*B*((a*e-b*d)*b)^(1/2)*a^2*b^2*d^2*e^2* x+408*B*((a*e-b*d)*b)^(1/2)*a*b^3*d^3*e*x+384*A*((a*e-b*d)*b)^(1/2)*a^4*e^ 4-48*A*((a*e-b*d)*b)^(1/2)*b^4*d^4-420*B*arctan(b*(e*x+d)^(1/2)/((a*e-b*d) *b)^(1/2))*a^2*b^3*e^4*x^3*(e*x+d)^(1/2)+5670*A*arctan(b*(e*x+d)^(1/2)/((a *e-b*d)*b)^(1/2))*a^2*b^3*e^4*x^2*(e*x+d)^(1/2)-630*B*arctan(b*(e*x+d)^(1/ 2)/((a*e-b*d)*b)^(1/2))*a^3*b^2*e^4*x^2*(e*x+d)^(1/2)+3780*A*arctan(b*(e*x +d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^3*b^2*e^4*x*(e*x+d)^(1/2)-420*B*arctan(b* (e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^4*b*e^4*x*(e*x+d)^(1/2)-105*B*(e*x+d) ^(1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^5*e^4-840*B*arctan(b* (e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a^4*b*d*e^3*(e*x+d)^(1/2)+945*A*arctan( b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^5*e^4*x^4*(e*x+d)^(1/2)+945*A*((a*e -b*d)*b)^(1/2)*b^4*e^4*x^4-279*B*((a*e-b*d)*b)^(1/2)*a^4*e^4*x-64*B*((a*e- b*d)*b)^(1/2)*b^4*d^4*x-663*B*((a*e-b*d)*b)^(1/2)*a^4*d*e^3-16*B*((a*e-b*d )*b)^(1/2)*a*b^3*d^4-3115*B*((a*e-b*d)*b)^(1/2)*a*b^3*d*e^3*x^3+1197*A*((a *e-b*d)*b)^(1/2)*a*b^3*d*e^3*x^2-4221*B*((a*e-b*d)*b)^(1/2)*a^2*b^2*d*e^3* x^2-105*B*((a*e-b*d)*b)^(1/2)*a*b^3*e^4*x^4-840*B*((a*e-b*d)*b)^(1/2)*b^4* d*e^3*x^4+3465*A*((a*e-b*d)*b)^(1/2)*a*b^3*e^4*x^3+315*A*((a*e-b*d)*b)^...
Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (326) = 652\).
Time = 0.66 (sec) , antiderivative size = 2968, normalized size of antiderivative = 7.00 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/384*(105*(8*B*a^4*b*d^2*e^3 + (B*a^5 - 9*A*a^4*b)*d*e^4 + (8*B*b^5*d*e^ 4 + (B*a*b^4 - 9*A*b^5)*e^5)*x^5 + (8*B*b^5*d^2*e^3 + 3*(11*B*a*b^4 - 3*A* b^5)*d*e^4 + 4*(B*a^2*b^3 - 9*A*a*b^4)*e^5)*x^4 + 2*(16*B*a*b^4*d^2*e^3 + 2*(13*B*a^2*b^3 - 9*A*a*b^4)*d*e^4 + 3*(B*a^3*b^2 - 9*A*a^2*b^3)*e^5)*x^3 + 2*(24*B*a^2*b^3*d^2*e^3 + (19*B*a^3*b^2 - 27*A*a^2*b^3)*d*e^4 + 2*(B*a^4 *b - 9*A*a^3*b^2)*e^5)*x^2 + (32*B*a^3*b^2*d^2*e^3 + 12*(B*a^4*b - 3*A*a^3 *b^2)*d*e^4 + (B*a^5 - 9*A*a^4*b)*e^5)*x)*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*A* a^5*b*e^5 + 16*(B*a*b^5 + 3*A*b^6)*d^5 - 24*(5*B*a^2*b^4 + 13*A*a*b^5)*d^4 *e + 6*(79*B*a^3*b^3 + 149*A*a^2*b^4)*d^3*e^2 + (293*B*a^4*b^2 - 1605*A*a^ 3*b^3)*d^2*e^3 - 3*(221*B*a^5*b - 197*A*a^4*b^2)*d*e^4 + 105*(8*B*b^6*d^2* e^3 - (7*B*a*b^5 + 9*A*b^6)*d*e^4 - (B*a^2*b^4 - 9*A*a*b^5)*e^5)*x^4 + 35* (8*B*b^6*d^3*e^2 + 9*(9*B*a*b^5 - A*b^6)*d^2*e^3 - 6*(13*B*a^2*b^4 + 15*A* a*b^5)*d*e^4 - 11*(B*a^3*b^3 - 9*A*a^2*b^4)*e^5)*x^3 - 7*(16*B*b^6*d^4*e - 2*(83*B*a*b^5 + 9*A*b^6)*d^3*e^2 - 3*(151*B*a^2*b^4 - 63*A*a*b^5)*d^2*e^3 + 2*(265*B*a^3*b^3 + 243*A*a^2*b^4)*d*e^4 + 73*(B*a^4*b^2 - 9*A*a^3*b^3)* e^5)*x^2 + (64*B*b^6*d^5 - 8*(59*B*a*b^5 + 9*A*b^6)*d^4*e + 108*(17*B*a^2* b^4 + 5*A*a*b^5)*d^3*e^2 + (989*B*a^3*b^3 - 2133*A*a^2*b^4)*d^2*e^3 - 2*(1 069*B*a^4*b^2 + 423*A*a^3*b^3)*d*e^4 - 279*(B*a^5*b - 9*A*a^4*b^2)*e^5)*x) *sqrt(e*x + d))/(a^4*b^7*d^7 - 6*a^5*b^6*d^6*e + 15*a^6*b^5*d^5*e^2 - 2...
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (326) = 652\).
Time = 0.35 (sec) , antiderivative size = 852, normalized size of antiderivative = 2.01 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {35 \, {\left (8 \, B b d e^{3} + B a e^{4} - 9 \, A b e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\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 {2 \, {\left (B d e^{3} - A e^{4}\right )}}{{\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 {e x + d}} - \frac {456 \, {\left (e x + d\right )}^{\frac {7}{2}} B b^{4} d e^{3} - 1544 \, {\left (e x + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{3} + 1784 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{3} - 696 \, \sqrt {e x + d} B b^{4} d^{4} e^{3} + 105 \, {\left (e x + d\right )}^{\frac {7}{2}} B a b^{3} e^{4} - 561 \, {\left (e x + d\right )}^{\frac {7}{2}} A b^{4} e^{4} + 1159 \, {\left (e x + d\right )}^{\frac {5}{2}} B a b^{3} d e^{4} + 1929 \, {\left (e x + d\right )}^{\frac {5}{2}} A b^{4} d e^{4} - 3057 \, {\left (e x + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{4} - 2295 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{4} + 1809 \, \sqrt {e x + d} B a b^{3} d^{3} e^{4} + 975 \, \sqrt {e x + d} A b^{4} d^{3} e^{4} + 385 \, {\left (e x + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{5} - 1929 \, {\left (e x + d\right )}^{\frac {5}{2}} A a b^{3} e^{5} + 762 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{5} + 4590 \, {\left (e x + d\right )}^{\frac {3}{2}} A a b^{3} d e^{5} - 1251 \, \sqrt {e x + d} B a^{2} b^{2} d^{2} e^{5} - 2925 \, \sqrt {e x + d} A a b^{3} d^{2} e^{5} + 511 \, {\left (e x + d\right )}^{\frac {3}{2}} B a^{3} b e^{6} - 2295 \, {\left (e x + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{6} - 141 \, \sqrt {e x + d} B a^{3} b d e^{6} + 2925 \, \sqrt {e x + d} A a^{2} b^{2} d e^{6} + 279 \, \sqrt {e x + d} B a^{4} e^{7} - 975 \, \sqrt {e x + d} A a^{3} b e^{7}}{192 \, {\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 )} b - b d + a e\right )}^{4}} \]
-35/64*(8*B*b*d*e^3 + B*a*e^4 - 9*A*b*e^4)*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)) - 2*(B*d*e^3 - A*e^4)/((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(e*x + d)) - 1/192*(456*(e*x + d)^(7 /2)*B*b^4*d*e^3 - 1544*(e*x + d)^(5/2)*B*b^4*d^2*e^3 + 1784*(e*x + d)^(3/2 )*B*b^4*d^3*e^3 - 696*sqrt(e*x + d)*B*b^4*d^4*e^3 + 105*(e*x + d)^(7/2)*B* a*b^3*e^4 - 561*(e*x + d)^(7/2)*A*b^4*e^4 + 1159*(e*x + d)^(5/2)*B*a*b^3*d *e^4 + 1929*(e*x + d)^(5/2)*A*b^4*d*e^4 - 3057*(e*x + d)^(3/2)*B*a*b^3*d^2 *e^4 - 2295*(e*x + d)^(3/2)*A*b^4*d^2*e^4 + 1809*sqrt(e*x + d)*B*a*b^3*d^3 *e^4 + 975*sqrt(e*x + d)*A*b^4*d^3*e^4 + 385*(e*x + d)^(5/2)*B*a^2*b^2*e^5 - 1929*(e*x + d)^(5/2)*A*a*b^3*e^5 + 762*(e*x + d)^(3/2)*B*a^2*b^2*d*e^5 + 4590*(e*x + d)^(3/2)*A*a*b^3*d*e^5 - 1251*sqrt(e*x + d)*B*a^2*b^2*d^2*e^ 5 - 2925*sqrt(e*x + d)*A*a*b^3*d^2*e^5 + 511*(e*x + d)^(3/2)*B*a^3*b*e^6 - 2295*(e*x + d)^(3/2)*A*a^2*b^2*e^6 - 141*sqrt(e*x + d)*B*a^3*b*d*e^6 + 29 25*sqrt(e*x + d)*A*a^2*b^2*d*e^6 + 279*sqrt(e*x + d)*B*a^4*e^7 - 975*sqrt( e*x + d)*A*a^3*b*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...
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]