Integrand size = 38, antiderivative size = 215 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^3} \, dx=-\frac {(2 a c+(4 b c+a e) x) \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{4 c x^2 (a+b x)}+\frac {b \sqrt {d} \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {e+2 d x}{2 \sqrt {d} \sqrt {c+e x+d x^2}}\right )}{a+b x}-\frac {\left (4 a c d+4 b c e-a e^2\right ) \sqrt {a^2+2 a b x+b^2 x^2} \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+e x+d x^2}}\right )}{8 c^{3/2} (a+b x)} \]
-1/8*(4*a*c*d-a*e^2+4*b*c*e)*arctanh(1/2*(e*x+2*c)/c^(1/2)/(d*x^2+e*x+c)^( 1/2))*((b*x+a)^2)^(1/2)/c^(3/2)/(b*x+a)+b*arctanh(1/2*(2*d*x+e)/d^(1/2)/(d *x^2+e*x+c)^(1/2))*d^(1/2)*((b*x+a)^2)^(1/2)/(b*x+a)-1/4*(2*a*c+(a*e+4*b*c )*x)*((b*x+a)^2)^(1/2)*(d*x^2+e*x+c)^(1/2)/c/x^2/(b*x+a)
Time = 0.53 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^3} \, dx=-\frac {\sqrt {(a+b x)^2} \left (\left (4 a c d+4 b c e-a e^2\right ) x^2 \text {arctanh}\left (\frac {-\sqrt {d} x+\sqrt {c+x (e+d x)}}{\sqrt {c}}\right )+\sqrt {c} \left ((2 a c+4 b c x+a e x) \sqrt {c+x (e+d x)}+4 b c \sqrt {d} x^2 \log \left (e+2 d x-2 \sqrt {d} \sqrt {c+x (e+d x)}\right )\right )\right )}{4 c^{3/2} x^2 (a+b x)} \]
-1/4*(Sqrt[(a + b*x)^2]*((4*a*c*d + 4*b*c*e - a*e^2)*x^2*ArcTanh[(-(Sqrt[d ]*x) + Sqrt[c + x*(e + d*x)])/Sqrt[c]] + Sqrt[c]*((2*a*c + 4*b*c*x + a*e*x )*Sqrt[c + x*(e + d*x)] + 4*b*c*Sqrt[d]*x^2*Log[e + 2*d*x - 2*Sqrt[d]*Sqrt [c + x*(e + d*x)]])))/(c^(3/2)*x^2*(a + b*x))
Time = 0.40 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.79, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {1333, 27, 1229, 27, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+d x^2+e x}}{x^3} \, dx\) |
| \(\Big \downarrow \) 1333 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {2 b (a+b x) \sqrt {d x^2+e x+c}}{x^3}dx}{2 b (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \sqrt {d x^2+e x+c}}{x^3}dx}{a+b x}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {\int -\frac {4 b c e+a \left (4 c d-e^2\right )+8 b c d x}{2 x \sqrt {d x^2+e x+c}}dx}{4 c}-\frac {\sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2}\right )}{a+b x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {\int \frac {-a e^2+4 b c e+4 a c d+8 b c d x}{x \sqrt {d x^2+e x+c}}dx}{8 c}-\frac {\sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2}\right )}{a+b x}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {\left (4 a c d-a e^2+4 b c e\right ) \int \frac {1}{x \sqrt {d x^2+e x+c}}dx+8 b c d \int \frac {1}{\sqrt {d x^2+e x+c}}dx}{8 c}-\frac {\sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2}\right )}{a+b x}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {\left (4 a c d-a e^2+4 b c e\right ) \int \frac {1}{x \sqrt {d x^2+e x+c}}dx+16 b c d \int \frac {1}{4 d-\frac {(e+2 d x)^2}{d x^2+e x+c}}d\frac {e+2 d x}{\sqrt {d x^2+e x+c}}}{8 c}-\frac {\sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2}\right )}{a+b x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {\left (4 a c d-a e^2+4 b c e\right ) \int \frac {1}{x \sqrt {d x^2+e x+c}}dx+8 b c \sqrt {d} \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )}{8 c}-\frac {\sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2}\right )}{a+b x}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {8 b c \sqrt {d} \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )-2 \left (4 a c d-a e^2+4 b c e\right ) \int \frac {1}{4 c-\frac {(2 c+e x)^2}{d x^2+e x+c}}d\frac {2 c+e x}{\sqrt {d x^2+e x+c}}}{8 c}-\frac {\sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2}\right )}{a+b x}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {8 b c \sqrt {d} \text {arctanh}\left (\frac {2 d x+e}{2 \sqrt {d} \sqrt {c+d x^2+e x}}\right )-\frac {\left (4 a c d-a e^2+4 b c e\right ) \text {arctanh}\left (\frac {2 c+e x}{2 \sqrt {c} \sqrt {c+d x^2+e x}}\right )}{\sqrt {c}}}{8 c}-\frac {\sqrt {c+d x^2+e x} (x (a e+4 b c)+2 a c)}{4 c x^2}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1/4*((2*a*c + (4*b*c + a*e)*x)*Sqrt[c + e *x + d*x^2])/(c*x^2) + (8*b*c*Sqrt[d]*ArcTanh[(e + 2*d*x)/(2*Sqrt[d]*Sqrt[ c + e*x + d*x^2])] - ((4*a*c*d + 4*b*c*e - a*e^2)*ArcTanh[(2*c + e*x)/(2*S qrt[c]*Sqrt[c + e*x + d*x^2])])/Sqrt[c])/(8*c)))/(a + b*x)
3.1.51.3.1 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/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_ ) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^Fr acPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])) Int[(g + h*x)^m* (b + 2*c*x)^(2*p)*(d + e*x + f*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g , h, m, p, q}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.61 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(-\frac {\sqrt {d \,x^{2}+e x +c}\, \left (a e x +4 b c x +2 a c \right ) \sqrt {\left (b x +a \right )^{2}}}{4 x^{2} c \left (b x +a \right )}+\frac {\left (8 b c \sqrt {d}\, \ln \left (\frac {\frac {e}{2}+d x}{\sqrt {d}}+\sqrt {d \,x^{2}+e x +c}\right )-\frac {\left (4 a c d -e^{2} a +4 b c e \right ) \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right )}{\sqrt {c}}\right ) \sqrt {\left (b x +a \right )^{2}}}{8 c \left (b x +a \right )}\) | \(157\) |
| default | \(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (4 d^{\frac {5}{2}} c^{\frac {3}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a \,x^{2}+2 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {5}{2}} a e \,x^{3}-8 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {5}{2}} b c \,x^{3}+4 d^{\frac {3}{2}} c^{\frac {3}{2}} \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) b e \,x^{2}-4 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {5}{2}} a c \,x^{2}-d^{\frac {3}{2}} \sqrt {c}\, \ln \left (\frac {2 c +e x +2 \sqrt {c}\, \sqrt {d \,x^{2}+e x +c}}{x}\right ) a \,e^{2} x^{2}-2 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} d^{\frac {3}{2}} a e x +8 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} d^{\frac {3}{2}} b c x +2 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {3}{2}} a \,e^{2} x^{2}-8 \sqrt {d \,x^{2}+e x +c}\, d^{\frac {3}{2}} b c e \,x^{2}+4 \left (d \,x^{2}+e x +c \right )^{\frac {3}{2}} d^{\frac {3}{2}} a c -8 \ln \left (\frac {2 \sqrt {d \,x^{2}+e x +c}\, \sqrt {d}+2 d x +e}{2 \sqrt {d}}\right ) b \,c^{2} d^{2} x^{2}\right )}{8 c^{2} x^{2} d^{\frac {3}{2}}}\) | \(359\) |
-1/4*(d*x^2+e*x+c)^(1/2)*(a*e*x+4*b*c*x+2*a*c)/x^2/c*((b*x+a)^2)^(1/2)/(b* x+a)+1/8/c*(8*b*c*d^(1/2)*ln((1/2*e+d*x)/d^(1/2)+(d*x^2+e*x+c)^(1/2))-(4*a *c*d-a*e^2+4*b*c*e)/c^(1/2)*ln((2*c+e*x+2*c^(1/2)*(d*x^2+e*x+c)^(1/2))/x)) *((b*x+a)^2)^(1/2)/(b*x+a)
Time = 0.53 (sec) , antiderivative size = 693, normalized size of antiderivative = 3.22 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^3} \, dx=\left [\frac {8 \, b c^{2} \sqrt {d} x^{2} \log \left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) - {\left (4 \, a c d + 4 \, b c e - a e^{2}\right )} \sqrt {c} x^{2} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} + 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) - 4 \, {\left (2 \, a c^{2} + {\left (4 \, b c^{2} + a c e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{16 \, c^{2} x^{2}}, -\frac {16 \, b c^{2} \sqrt {-d} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) + {\left (4 \, a c d + 4 \, b c e - a e^{2}\right )} \sqrt {c} x^{2} \log \left (\frac {8 \, c e x + {\left (4 \, c d + e^{2}\right )} x^{2} + 4 \, \sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {c} + 8 \, c^{2}}{x^{2}}\right ) + 4 \, {\left (2 \, a c^{2} + {\left (4 \, b c^{2} + a c e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{16 \, c^{2} x^{2}}, \frac {4 \, b c^{2} \sqrt {d} x^{2} \log \left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, \sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {d} + 4 \, c d + e^{2}\right ) + {\left (4 \, a c d + 4 \, b c e - a e^{2}\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) - 2 \, {\left (2 \, a c^{2} + {\left (4 \, b c^{2} + a c e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{8 \, c^{2} x^{2}}, -\frac {8 \, b c^{2} \sqrt {-d} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (2 \, d x + e\right )} \sqrt {-d}}{2 \, {\left (d^{2} x^{2} + d e x + c d\right )}}\right ) - {\left (4 \, a c d + 4 \, b c e - a e^{2}\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {d x^{2} + e x + c} {\left (e x + 2 \, c\right )} \sqrt {-c}}{2 \, {\left (c d x^{2} + c e x + c^{2}\right )}}\right ) + 2 \, {\left (2 \, a c^{2} + {\left (4 \, b c^{2} + a c e\right )} x\right )} \sqrt {d x^{2} + e x + c}}{8 \, c^{2} x^{2}}\right ] \]
[1/16*(8*b*c^2*sqrt(d)*x^2*log(8*d^2*x^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(d) + 4*c*d + e^2) - (4*a*c*d + 4*b*c*e - a*e^2)*sqrt(c )*x^2*log((8*c*e*x + (4*c*d + e^2)*x^2 + 4*sqrt(d*x^2 + e*x + c)*(e*x + 2* c)*sqrt(c) + 8*c^2)/x^2) - 4*(2*a*c^2 + (4*b*c^2 + a*c*e)*x)*sqrt(d*x^2 + e*x + c))/(c^2*x^2), -1/16*(16*b*c^2*sqrt(-d)*x^2*arctan(1/2*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(-d)/(d^2*x^2 + d*e*x + c*d)) + (4*a*c*d + 4*b*c* e - a*e^2)*sqrt(c)*x^2*log((8*c*e*x + (4*c*d + e^2)*x^2 + 4*sqrt(d*x^2 + e *x + c)*(e*x + 2*c)*sqrt(c) + 8*c^2)/x^2) + 4*(2*a*c^2 + (4*b*c^2 + a*c*e) *x)*sqrt(d*x^2 + e*x + c))/(c^2*x^2), 1/8*(4*b*c^2*sqrt(d)*x^2*log(8*d^2*x ^2 + 8*d*e*x + 4*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(d) + 4*c*d + e^2) + (4*a*c*d + 4*b*c*e - a*e^2)*sqrt(-c)*x^2*arctan(1/2*sqrt(d*x^2 + e*x + c )*(e*x + 2*c)*sqrt(-c)/(c*d*x^2 + c*e*x + c^2)) - 2*(2*a*c^2 + (4*b*c^2 + a*c*e)*x)*sqrt(d*x^2 + e*x + c))/(c^2*x^2), -1/8*(8*b*c^2*sqrt(-d)*x^2*arc tan(1/2*sqrt(d*x^2 + e*x + c)*(2*d*x + e)*sqrt(-d)/(d^2*x^2 + d*e*x + c*d) ) - (4*a*c*d + 4*b*c*e - a*e^2)*sqrt(-c)*x^2*arctan(1/2*sqrt(d*x^2 + e*x + c)*(e*x + 2*c)*sqrt(-c)/(c*d*x^2 + c*e*x + c^2)) + 2*(2*a*c^2 + (4*b*c^2 + a*c*e)*x)*sqrt(d*x^2 + e*x + c))/(c^2*x^2)]
\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^3} \, dx=\int \frac {\sqrt {c + d x^{2} + e x} \sqrt {\left (a + b x\right )^{2}}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^3} \, dx=\int { \frac {\sqrt {d x^{2} + e x + c} \sqrt {{\left (b x + a\right )}^{2}}}{x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (160) = 320\).
Time = 0.34 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^3} \, dx=-b \sqrt {d} \log \left ({\left | 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )} \sqrt {d} + e \right |}\right ) \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (4 \, a c d \mathrm {sgn}\left (b x + a\right ) + 4 \, b c e \mathrm {sgn}\left (b x + a\right ) - a e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (-\frac {\sqrt {d} x - \sqrt {d x^{2} + e x + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c} c} + \frac {4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )}^{3} a c d \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )}^{3} b c e \mathrm {sgn}\left (b x + a\right ) + {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )}^{3} a e^{2} \mathrm {sgn}\left (b x + a\right ) + 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )}^{2} b c^{2} \sqrt {d} \mathrm {sgn}\left (b x + a\right ) + 8 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )}^{2} a c \sqrt {d} e \mathrm {sgn}\left (b x + a\right ) + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )} a c^{2} d \mathrm {sgn}\left (b x + a\right ) - 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )} b c^{2} e \mathrm {sgn}\left (b x + a\right ) + {\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )} a c e^{2} \mathrm {sgn}\left (b x + a\right ) - 8 \, b c^{3} \sqrt {d} \mathrm {sgn}\left (b x + a\right )}{4 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + e x + c}\right )}^{2} - c\right )}^{2} c} \]
-b*sqrt(d)*log(abs(2*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))*sqrt(d) + e))*sgn (b*x + a) + 1/4*(4*a*c*d*sgn(b*x + a) + 4*b*c*e*sgn(b*x + a) - a*e^2*sgn(b *x + a))*arctan(-(sqrt(d)*x - sqrt(d*x^2 + e*x + c))/sqrt(-c))/(sqrt(-c)*c ) + 1/4*(4*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^3*a*c*d*sgn(b*x + a) + 4*(s qrt(d)*x - sqrt(d*x^2 + e*x + c))^3*b*c*e*sgn(b*x + a) + (sqrt(d)*x - sqrt (d*x^2 + e*x + c))^3*a*e^2*sgn(b*x + a) + 8*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))^2*b*c^2*sqrt(d)*sgn(b*x + a) + 8*(sqrt(d)*x - sqrt(d*x^2 + e*x + c)) ^2*a*c*sqrt(d)*e*sgn(b*x + a) + 4*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))*a*c^ 2*d*sgn(b*x + a) - 4*(sqrt(d)*x - sqrt(d*x^2 + e*x + c))*b*c^2*e*sgn(b*x + a) + (sqrt(d)*x - sqrt(d*x^2 + e*x + c))*a*c*e^2*sgn(b*x + a) - 8*b*c^3*s qrt(d)*sgn(b*x + a))/(((sqrt(d)*x - sqrt(d*x^2 + e*x + c))^2 - c)^2*c)
Timed out. \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2} \sqrt {c+e x+d x^2}}{x^3} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d\,x^2+e\,x+c}}{x^3} \,d x \]