Integrand size = 35, antiderivative size = 215 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {(4 b B d-3 A b e-a B e) \sqrt {d+e x}}{4 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) \sqrt {d+e x}}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (4 b B d-3 A b e-a B e) (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
1/4*e*(-3*A*b*e-B*a*e+4*B*b*d)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e +b*d)^(1/2))/b^(3/2)/(-a*e+b*d)^(5/2)/((b*x+a)^2)^(1/2)-1/4*(-3*A*b*e-B*a* e+4*B*b*d)*(e*x+d)^(1/2)/b/(-a*e+b*d)^2/((b*x+a)^2)^(1/2)-1/2*(A*b-B*a)*(e *x+d)^(1/2)/b/(-a*e+b*d)/(b*x+a)/((b*x+a)^2)^(1/2)
Time = 0.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e (a+b x)^3 \left (\frac {\sqrt {b} \sqrt {d+e x} \left (A b (-2 b d+5 a e+3 b e x)-B \left (a^2 e+4 b^2 d x+a b (2 d-e x)\right )\right )}{e (b d-a e)^2 (a+b x)^2}-\frac {(4 b B d-3 A b e-a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}}\right )}{4 b^{3/2} \left ((a+b x)^2\right )^{3/2}} \]
(e*(a + b*x)^3*((Sqrt[b]*Sqrt[d + e*x]*(A*b*(-2*b*d + 5*a*e + 3*b*e*x) - B *(a^2*e + 4*b^2*d*x + a*b*(2*d - e*x))))/(e*(b*d - a*e)^2*(a + b*x)^2) - ( (4*b*B*d - 3*A*b*e - a*B*e)*ArcTan[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a *e]])/(-(b*d) + a*e)^(5/2)))/(4*b^(3/2)*((a + b*x)^2)^(3/2))
Time = 0.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {1187, 27, 87, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^3 (a+b x) \int \frac {A+B x}{b^3 (a+b x)^3 \sqrt {d+e x}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {A+B x}{(a+b x)^3 \sqrt {d+e x}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-a B e-3 A b e+4 b B d) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}}dx}{4 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-a B e-3 A b e+4 b B d) \left (-\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 (b d-a e)}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-a B e-3 A b e+4 b B d) \left (-\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b d-a e}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(-a B e-3 A b e+4 b B d) \left (\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 b (b d-a e)}-\frac {\sqrt {d+e x} (A b-a B)}{2 b (a+b x)^2 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/2*((A*b - a*B)*Sqrt[d + e*x])/(b*(b*d - a*e)*(a + b*x)^2) + ((4*b*B*d - 3*A*b*e - a*B*e)*(-(Sqrt[d + e*x]/((b*d - a*e)*(a + b*x))) + (e*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^ (3/2))))/(4*b*(b*d - a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.19.70.3.1 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] :> 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. \(555\) vs. \(2(162)=324\).
Time = 0.30 (sec) , antiderivative size = 556, normalized size of antiderivative = 2.59
| method | result | size |
| default | \(\frac {\left (b x +a \right ) \left (3 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{3} e^{3} x^{2}+B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{2} e^{3} x^{2}-4 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{3} d \,e^{2} x^{2}+6 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{2} e^{3} x +2 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b \,e^{3} x -8 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{2} d \,e^{2} x +3 A \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{2} e +3 A \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b \,e^{3}+B \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a b e -4 B \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{2} d +B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} e^{3}-4 B \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b d \,e^{2}+5 A \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a b \,e^{2}-5 A \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{2} d e -B \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{2} e^{2}-3 B \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a b d e +4 B \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b^{2} d^{2}\right )}{4 e \sqrt {\left (a e -b d \right ) b}\, b \left (a e -b d \right )^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(556\) |
1/4*(b*x+a)/e*(3*A*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^3*e^3*x^2 +B*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a*b^2*e^3*x^2-4*B*arctan(b* (e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^3*d*e^2*x^2+6*A*arctan(b*(e*x+d)^(1/2 )/((a*e-b*d)*b)^(1/2))*a*b^2*e^3*x+2*B*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b )^(1/2))*a^2*b*e^3*x-8*B*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a*b^2 *d*e^2*x+3*A*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*b^2*e+3*A*arctan(b*(e*x+d)^ (1/2)/((a*e-b*d)*b)^(1/2))*a^2*b*e^3+B*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*a *b*e-4*B*(e*x+d)^(3/2)*((a*e-b*d)*b)^(1/2)*b^2*d+B*arctan(b*(e*x+d)^(1/2)/ ((a*e-b*d)*b)^(1/2))*a^3*e^3-4*B*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2 ))*a^2*b*d*e^2+5*A*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a*b*e^2-5*A*(e*x+d)^( 1/2)*((a*e-b*d)*b)^(1/2)*b^2*d*e-B*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a^2*e ^2-3*B*(e*x+d)^(1/2)*((a*e-b*d)*b)^(1/2)*a*b*d*e+4*B*(e*x+d)^(1/2)*((a*e-b *d)*b)^(1/2)*b^2*d^2)/((a*e-b*d)*b)^(1/2)/b/(a*e-b*d)^2/((b*x+a)^2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (162) = 324\).
Time = 0.37 (sec) , antiderivative size = 808, normalized size of antiderivative = 3.76 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (4 \, B a^{2} b d e - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (B a b^{2} + 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (B a^{2} b + 3 \, A a b^{2}\right )} e^{2}\right )} x\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 (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (B a^{2} b^{2} + 7 \, A a b^{3}\right )} d e - {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} e^{2} + {\left (4 \, B b^{4} d^{2} - {\left (5 \, B a b^{3} + 3 \, A b^{4}\right )} d e + {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{5} d^{3} - 3 \, a^{3} b^{4} d^{2} e + 3 \, a^{4} b^{3} d e^{2} - a^{5} b^{2} e^{3} + {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} x^{2} + 2 \, {\left (a b^{6} d^{3} - 3 \, a^{2} b^{5} d^{2} e + 3 \, a^{3} b^{4} d e^{2} - a^{4} b^{3} e^{3}\right )} x\right )}}, -\frac {{\left (4 \, B a^{2} b d e - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2} + {\left (4 \, B b^{3} d e - {\left (B a b^{2} + 3 \, A b^{3}\right )} e^{2}\right )} x^{2} + 2 \, {\left (4 \, B a b^{2} d e - {\left (B a^{2} b + 3 \, A a b^{2}\right )} e^{2}\right )} x\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 (2 \, {\left (B a b^{3} + A b^{4}\right )} d^{2} - {\left (B a^{2} b^{2} + 7 \, A a b^{3}\right )} d e - {\left (B a^{3} b - 5 \, A a^{2} b^{2}\right )} e^{2} + {\left (4 \, B b^{4} d^{2} - {\left (5 \, B a b^{3} + 3 \, A b^{4}\right )} d e + {\left (B a^{2} b^{2} + 3 \, A a b^{3}\right )} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{5} d^{3} - 3 \, a^{3} b^{4} d^{2} e + 3 \, a^{4} b^{3} d e^{2} - a^{5} b^{2} e^{3} + {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} x^{2} + 2 \, {\left (a b^{6} d^{3} - 3 \, a^{2} b^{5} d^{2} e + 3 \, a^{3} b^{4} d e^{2} - a^{4} b^{3} e^{3}\right )} x\right )}}\right ] \]
[-1/8*((4*B*a^2*b*d*e - (B*a^3 + 3*A*a^2*b)*e^2 + (4*B*b^3*d*e - (B*a*b^2 + 3*A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (B*a^2*b + 3*A*a*b^2)*e^2)*x)*sqr t(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*(2*(B*a*b^3 + A*b^4)*d^2 - (B*a^2*b^2 + 7*A*a*b^3)*d *e - (B*a^3*b - 5*A*a^2*b^2)*e^2 + (4*B*b^4*d^2 - (5*B*a*b^3 + 3*A*b^4)*d* e + (B*a^2*b^2 + 3*A*a*b^3)*e^2)*x)*sqrt(e*x + d))/(a^2*b^5*d^3 - 3*a^3*b^ 4*d^2*e + 3*a^4*b^3*d*e^2 - a^5*b^2*e^3 + (b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2 *b^5*d*e^2 - a^3*b^4*e^3)*x^2 + 2*(a*b^6*d^3 - 3*a^2*b^5*d^2*e + 3*a^3*b^4 *d*e^2 - a^4*b^3*e^3)*x), -1/4*((4*B*a^2*b*d*e - (B*a^3 + 3*A*a^2*b)*e^2 + (4*B*b^3*d*e - (B*a*b^2 + 3*A*b^3)*e^2)*x^2 + 2*(4*B*a*b^2*d*e - (B*a^2*b + 3*A*a*b^2)*e^2)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqr t(e*x + d)/(b*e*x + b*d)) + (2*(B*a*b^3 + A*b^4)*d^2 - (B*a^2*b^2 + 7*A*a* b^3)*d*e - (B*a^3*b - 5*A*a^2*b^2)*e^2 + (4*B*b^4*d^2 - (5*B*a*b^3 + 3*A*b ^4)*d*e + (B*a^2*b^2 + 3*A*a*b^3)*e^2)*x)*sqrt(e*x + d))/(a^2*b^5*d^3 - 3* a^3*b^4*d^2*e + 3*a^4*b^3*d*e^2 - a^5*b^2*e^3 + (b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*x^2 + 2*(a*b^6*d^3 - 3*a^2*b^5*d^2*e + 3*a ^3*b^4*d*e^2 - a^4*b^3*e^3)*x)]
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} \sqrt {e x + d}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {{\left (4 \, B b d e - B a e^{2} - 3 \, A b e^{2}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {4 \, {\left (e x + d\right )}^{\frac {3}{2}} B b^{2} d e - 4 \, \sqrt {e x + d} B b^{2} d^{2} e - {\left (e x + d\right )}^{\frac {3}{2}} B a b e^{2} - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} A b^{2} e^{2} + 3 \, \sqrt {e x + d} B a b d e^{2} + 5 \, \sqrt {e x + d} A b^{2} d e^{2} + \sqrt {e x + d} B a^{2} e^{3} - 5 \, \sqrt {e x + d} A a b e^{3}}{4 \, {\left (b^{3} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b^{2} d e \mathrm {sgn}\left (b x + a\right ) + a^{2} b e^{2} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} \]
-1/4*(4*B*b*d*e - B*a*e^2 - 3*A*b*e^2)*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d^2*sgn(b*x + a) - 2*a*b^2*d*e*sgn(b*x + a) + a^2*b*e^2*sg n(b*x + a))*sqrt(-b^2*d + a*b*e)) - 1/4*(4*(e*x + d)^(3/2)*B*b^2*d*e - 4*s qrt(e*x + d)*B*b^2*d^2*e - (e*x + d)^(3/2)*B*a*b*e^2 - 3*(e*x + d)^(3/2)*A *b^2*e^2 + 3*sqrt(e*x + d)*B*a*b*d*e^2 + 5*sqrt(e*x + d)*A*b^2*d*e^2 + sqr t(e*x + d)*B*a^2*e^3 - 5*sqrt(e*x + d)*A*a*b*e^3)/((b^3*d^2*sgn(b*x + a) - 2*a*b^2*d*e*sgn(b*x + a) + a^2*b*e^2*sgn(b*x + a))*((e*x + d)*b - b*d + a *e)^2)
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]