Integrand size = 26, antiderivative size = 216 \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=\frac {(B d-A e) \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {(3 A e (2 c d-b e)-B d (2 c d+b e)) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}+\frac {\left (8 A c^2 d^2-4 b c d (B d+2 A e)+b^2 e (B d+3 A e)\right ) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 d^{5/2} (c d-b e)^{5/2}} \]
1/8*(8*A*c^2*d^2-4*b*c*d*(2*A*e+B*d)+b^2*e*(3*A*e+B*d))*arctanh(1/2*(b*d+( -b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/d^(5/2)/(-b*e+c *d)^(5/2)+1/2*(-A*e+B*d)*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)^2-1/4*(3*A *e*(-b*e+2*c*d)-B*d*(b*e+2*c*d))*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e*x+d )
Time = 10.26 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {x} \left (\frac {(-B d+A e) \sqrt {x} (b+c x)}{(d+e x)^2}-\frac {(3 A e (-2 c d+b e)+B d (2 c d+b e)) \sqrt {x} (b+c x)}{2 d (c d-b e) (d+e x)}-\frac {\left (8 A c^2 d^2-4 b c d (B d+2 A e)+b^2 e (B d+3 A e)\right ) \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}\right )}{2 d (-c d+b e) \sqrt {x (b+c x)}} \]
(Sqrt[x]*(((-(B*d) + A*e)*Sqrt[x]*(b + c*x))/(d + e*x)^2 - ((3*A*e*(-2*c*d + b*e) + B*d*(2*c*d + b*e))*Sqrt[x]*(b + c*x))/(2*d*(c*d - b*e)*(d + e*x) ) - ((8*A*c^2*d^2 - 4*b*c*d*(B*d + 2*A*e) + b^2*e*(B*d + 3*A*e))*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(2*d^(3/2 )*(c*d - b*e)^(3/2))))/(2*d*(-(c*d) + b*e)*Sqrt[x*(b + c*x)])
Time = 0.43 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1237, 27, 1228, 1154, 219}
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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {b x+c x^2} (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\sqrt {b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)}-\frac {\int \frac {b B d-4 A c d+3 A b e-2 c (B d-A e) x}{2 (d+e x)^2 \sqrt {c x^2+b x}}dx}{2 d (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)}-\frac {\int \frac {b B d-4 A c d+3 A b e-2 c (B d-A e) x}{(d+e x)^2 \sqrt {c x^2+b x}}dx}{4 d (c d-b e)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\sqrt {b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} (3 A e (2 c d-b e)-B d (b e+2 c d))}{d (d+e x) (c d-b e)}-\frac {\left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{2 d (c d-b e)}}{4 d (c d-b e)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\sqrt {b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)}-\frac {\frac {\left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A c^2 d^2\right ) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{d (c d-b e)}+\frac {\sqrt {b x+c x^2} (3 A e (2 c d-b e)-B d (b e+2 c d))}{d (d+e x) (c d-b e)}}{4 d (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {b x+c x^2} (B d-A e)}{2 d (d+e x)^2 (c d-b e)}-\frac {\frac {\sqrt {b x+c x^2} (3 A e (2 c d-b e)-B d (b e+2 c d))}{d (d+e x) (c d-b e)}-\frac {\left (b^2 e (3 A e+B d)-4 b c d (2 A e+B d)+8 A c^2 d^2\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}}{4 d (c d-b e)}\) |
((B*d - A*e)*Sqrt[b*x + c*x^2])/(2*d*(c*d - b*e)*(d + e*x)^2) - (((3*A*e*( 2*c*d - b*e) - B*d*(2*c*d + b*e))*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*(d + e *x)) - ((8*A*c^2*d^2 - 4*b*c*d*(B*d + 2*A*e) + b^2*e*(B*d + 3*A*e))*ArcTan h[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/ (2*d^(3/2)*(c*d - b*e)^(3/2)))/(4*d*(c*d - b*e))
3.12.97.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.46 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {3 \left (\frac {4 \left (2 A \,c^{2}-B b c \right ) d^{2}}{3}-\frac {8 \left (A c -\frac {B b}{8}\right ) e b d}{3}+A \,b^{2} e^{2}\right ) \left (e x +d \right )^{2} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{4}+\frac {5 \left (\frac {4 B c \,d^{3}}{5}-\frac {8 \left (\frac {B b}{8}+c \left (-\frac {B x}{4}+A \right )\right ) e \,d^{2}}{5}+\left (\left (\frac {B x}{5}+A \right ) b -\frac {6 A c x}{5}\right ) e^{2} d +\frac {3 A b \,e^{3} x}{5}\right ) \sqrt {x \left (c x +b \right )}\, \sqrt {d \left (b e -c d \right )}}{4}}{\sqrt {d \left (b e -c d \right )}\, \left (b e -c d \right )^{2} d^{2} \left (e x +d \right )^{2}}\) | \(187\) |
| default | \(\frac {B \left (\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{3}}+\frac {\left (A e -B d \right ) \left (\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{2 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 \left (b e -2 c d \right ) e \left (\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{4 d \left (b e -c d \right )}-\frac {c \,e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{4}}\) | \(711\) |
5/4/(d*(b*e-c*d))^(1/2)*(-3/5*(4/3*(2*A*c^2-B*b*c)*d^2-8/3*(A*c-1/8*B*b)*e *b*d+A*b^2*e^2)*(e*x+d)^2*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2) )+(4/5*B*c*d^3-8/5*(1/8*B*b+c*(-1/4*B*x+A))*e*d^2+((1/5*B*x+A)*b-6/5*A*c*x )*e^2*d+3/5*A*b*e^3*x)*(x*(c*x+b))^(1/2)*(d*(b*e-c*d))^(1/2))/(b*e-c*d)^2/ d^2/(e*x+d)^2
Leaf count of result is larger than twice the leaf count of optimal. 471 vs. \(2 (193) = 386\).
Time = 0.44 (sec) , antiderivative size = 954, normalized size of antiderivative = 4.42 \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=\left [\frac {{\left (3 \, A b^{2} d^{2} e^{2} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{4} + {\left (B b^{2} - 8 \, A b c\right )} d^{3} e + {\left (3 \, A b^{2} e^{4} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} e^{2} + {\left (B b^{2} - 8 \, A b c\right )} d e^{3}\right )} x^{2} + 2 \, {\left (3 \, A b^{2} d e^{3} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{3} e + {\left (B b^{2} - 8 \, A b c\right )} d^{2} e^{2}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + 2 \, {\left (4 \, B c^{2} d^{5} - 5 \, A b^{2} d^{2} e^{3} - {\left (5 \, B b c + 8 \, A c^{2}\right )} d^{4} e + {\left (B b^{2} + 13 \, A b c\right )} d^{3} e^{2} + {\left (2 \, B c^{2} d^{4} e - 3 \, A b^{2} d e^{4} - {\left (B b c + 6 \, A c^{2}\right )} d^{3} e^{2} - {\left (B b^{2} - 9 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c^{3} d^{8} - 3 \, b c^{2} d^{7} e + 3 \, b^{2} c d^{6} e^{2} - b^{3} d^{5} e^{3} + {\left (c^{3} d^{6} e^{2} - 3 \, b c^{2} d^{5} e^{3} + 3 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e - 3 \, b c^{2} d^{6} e^{2} + 3 \, b^{2} c d^{5} e^{3} - b^{3} d^{4} e^{4}\right )} x\right )}}, \frac {{\left (3 \, A b^{2} d^{2} e^{2} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{4} + {\left (B b^{2} - 8 \, A b c\right )} d^{3} e + {\left (3 \, A b^{2} e^{4} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{2} e^{2} + {\left (B b^{2} - 8 \, A b c\right )} d e^{3}\right )} x^{2} + 2 \, {\left (3 \, A b^{2} d e^{3} - 4 \, {\left (B b c - 2 \, A c^{2}\right )} d^{3} e + {\left (B b^{2} - 8 \, A b c\right )} d^{2} e^{2}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + {\left (4 \, B c^{2} d^{5} - 5 \, A b^{2} d^{2} e^{3} - {\left (5 \, B b c + 8 \, A c^{2}\right )} d^{4} e + {\left (B b^{2} + 13 \, A b c\right )} d^{3} e^{2} + {\left (2 \, B c^{2} d^{4} e - 3 \, A b^{2} d e^{4} - {\left (B b c + 6 \, A c^{2}\right )} d^{3} e^{2} - {\left (B b^{2} - 9 \, A b c\right )} d^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c^{3} d^{8} - 3 \, b c^{2} d^{7} e + 3 \, b^{2} c d^{6} e^{2} - b^{3} d^{5} e^{3} + {\left (c^{3} d^{6} e^{2} - 3 \, b c^{2} d^{5} e^{3} + 3 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e - 3 \, b c^{2} d^{6} e^{2} + 3 \, b^{2} c d^{5} e^{3} - b^{3} d^{4} e^{4}\right )} x\right )}}\right ] \]
[1/8*((3*A*b^2*d^2*e^2 - 4*(B*b*c - 2*A*c^2)*d^4 + (B*b^2 - 8*A*b*c)*d^3*e + (3*A*b^2*e^4 - 4*(B*b*c - 2*A*c^2)*d^2*e^2 + (B*b^2 - 8*A*b*c)*d*e^3)*x ^2 + 2*(3*A*b^2*d*e^3 - 4*(B*b*c - 2*A*c^2)*d^3*e + (B*b^2 - 8*A*b*c)*d^2* e^2)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b* d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(4*B*c^2*d^5 - 5*A*b^2*d^2*e^3 - (5 *B*b*c + 8*A*c^2)*d^4*e + (B*b^2 + 13*A*b*c)*d^3*e^2 + (2*B*c^2*d^4*e - 3* A*b^2*d*e^4 - (B*b*c + 6*A*c^2)*d^3*e^2 - (B*b^2 - 9*A*b*c)*d^2*e^3)*x)*sq rt(c*x^2 + b*x))/(c^3*d^8 - 3*b*c^2*d^7*e + 3*b^2*c*d^6*e^2 - b^3*d^5*e^3 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 + 3*b^2*c*d^4*e^4 - b^3*d^3*e^5)*x^2 + 2* (c^3*d^7*e - 3*b*c^2*d^6*e^2 + 3*b^2*c*d^5*e^3 - b^3*d^4*e^4)*x), 1/4*((3* A*b^2*d^2*e^2 - 4*(B*b*c - 2*A*c^2)*d^4 + (B*b^2 - 8*A*b*c)*d^3*e + (3*A*b ^2*e^4 - 4*(B*b*c - 2*A*c^2)*d^2*e^2 + (B*b^2 - 8*A*b*c)*d*e^3)*x^2 + 2*(3 *A*b^2*d*e^3 - 4*(B*b*c - 2*A*c^2)*d^3*e + (B*b^2 - 8*A*b*c)*d^2*e^2)*x)*s qrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + (4*B*c^2*d^5 - 5*A*b^2*d^2*e^3 - (5*B*b*c + 8*A*c^2)*d^4*e + ( B*b^2 + 13*A*b*c)*d^3*e^2 + (2*B*c^2*d^4*e - 3*A*b^2*d*e^4 - (B*b*c + 6*A* c^2)*d^3*e^2 - (B*b^2 - 9*A*b*c)*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/(c^3*d^8 - 3*b*c^2*d^7*e + 3*b^2*c*d^6*e^2 - b^3*d^5*e^3 + (c^3*d^6*e^2 - 3*b*c^2*d^ 5*e^3 + 3*b^2*c*d^4*e^4 - b^3*d^3*e^5)*x^2 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^ 2 + 3*b^2*c*d^5*e^3 - b^3*d^4*e^4)*x)]
\[ \int \frac {A+B x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (193) = 386\).
Time = 0.30 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.57 \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=-\frac {{\left (4 \, B b c d^{2} - 8 \, A c^{2} d^{2} - B b^{2} d e + 8 \, A b c d e - 3 \, A b^{2} e^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b c d^{2} e^{2} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{2} d^{2} e^{2} - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} d e^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c d e^{3} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} e^{4} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B c^{\frac {5}{2}} d^{4} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b c^{\frac {3}{2}} d^{3} e - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c^{\frac {5}{2}} d^{3} e + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} \sqrt {c} d^{2} e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c^{\frac {3}{2}} d^{2} e^{2} - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} \sqrt {c} d e^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b c^{2} d^{4} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b c^{2} d^{3} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{3} d^{2} e^{2} + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} c d^{2} e^{2} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} d e^{3} + 2 \, B b^{2} c^{\frac {3}{2}} d^{4} + B b^{3} \sqrt {c} d^{3} e - 6 \, A b^{2} c^{\frac {3}{2}} d^{3} e + 3 \, A b^{3} \sqrt {c} d^{2} e^{2}}{4 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2}} \]
-1/4*(4*B*b*c*d^2 - 8*A*c^2*d^2 - B*b^2*d*e + 8*A*b*c*d*e - 3*A*b^2*e^2)*a rctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e ))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) + 1/4*(4*( sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c*d^2*e^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^2*d^2*e^2 - (sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*d*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c*d*e^3 - 3*(sqrt(c)*x - sqrt(c*x ^2 + b*x))^3*A*b^2*e^4 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*c^(5/2)*d^4 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^(3/2)*d^3*e - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*c^(5/2)*d^3*e + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^ 2*B*b^2*sqrt(c)*d^2*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^(3/2) *d^2*e^2 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*sqrt(c)*d*e^3 + 8*(sq rt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^2*d^4 - 24*(sqrt(c)*x - sqrt(c*x^2 + b* x))*A*b*c^2*d^3*e + (sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*d^2*e^2 + 20*(sq rt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*c*d^2*e^2 - 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*d*e^3 + 2*B*b^2*c^(3/2)*d^4 + B*b^3*sqrt(c)*d^3*e - 6*A*b^2*c ^(3/2)*d^3*e + 3*A*b^3*sqrt(c)*d^2*e^2)/((c^2*d^4*e - 2*b*c*d^3*e^2 + b^2* d^2*e^3)*((sqrt(c)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2)
Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^3} \,d x \]