Integrand size = 44, antiderivative size = 287 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 (4 c e f+6 c d g-5 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^3 e^2}+\frac {(4 c e f+6 c d g-5 b e g) (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c^2 e^2 (2 c d-b e)}-\frac {3 (2 c d-b e) (4 c e f+6 c d g-5 b e g) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{7/2} e^2} \]
-3/8*(-b*e+2*c*d)*(-5*b*e*g+6*c*d*g+4*c*e*f)*arctan(1/2*e*(2*c*x+b)/c^(1/2 )/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(7/2)/e^2+2*(-b*e*g+c*d*g+c*e* f)*(e*x+d)^3/c/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+3/4 *(-5*b*e*g+6*c*d*g+4*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^3/e^2 +1/2*(-5*b*e*g+6*c*d*g+4*c*e*f)*(e*x+d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^( 1/2)/c^2/e^2/(-b*e+2*c*d)
Time = 1.18 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {(-b e+c (d-e x))^{3/2} \left (\frac {\sqrt {c} (d+e x)^2 \left (15 b^2 e^2 g+b c e (-12 e f-43 d g+5 e g x)+2 c^2 \left (14 d^2 g+5 d e (2 f-g x)-e^2 x (2 f+g x)\right )\right )}{\sqrt {-b e+c (d-e x)}}+6 (2 c d-b e) (4 c e f+6 c d g-5 b e g) (d+e x)^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e}-\sqrt {c d-b e-c e x}}\right )\right )}{4 c^{7/2} e^2 ((d+e x) (-b e+c (d-e x)))^{3/2}} \]
((-(b*e) + c*(d - e*x))^(3/2)*((Sqrt[c]*(d + e*x)^2*(15*b^2*e^2*g + b*c*e* (-12*e*f - 43*d*g + 5*e*g*x) + 2*c^2*(14*d^2*g + 5*d*e*(2*f - g*x) - e^2*x *(2*f + g*x))))/Sqrt[-(b*e) + c*(d - e*x)] + 6*(2*c*d - b*e)*(4*c*e*f + 6* c*d*g - 5*b*e*g)*(d + e*x)^(3/2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(Sqrt[2*c* d - b*e] - Sqrt[c*d - b*e - c*e*x])]))/(4*c^(7/2)*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))
Time = 0.64 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1211, 2192, 27, 1160, 1092, 217}
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.
| \(\displaystyle \int \frac {(d+e x)^3 (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1211 |
| \(\displaystyle \frac {2 (d+e x) (2 c d-b e) (-b e g+c d g+c e f)}{c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\int \frac {c^2 g x^2 e^6+c (c e f+3 c d g-b e g) x e^5+\left (d (3 e f+4 d g) c^2-b e (e f+4 d g) c+b^2 e^2 g\right ) e^4}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c^3 e^5}\) |
| \(\Big \downarrow \) 2192 |
| \(\displaystyle \frac {2 (d+e x) (2 c d-b e) (-b e g+c d g+c e f)}{c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {-\frac {\int -\frac {c e^6 (2 (3 c d-b e) (2 c e f+3 c d g-2 b e g)-c e (7 b e g-4 c (e f+3 d g)) x)}{2 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c e^2}-\frac {1}{2} c e^4 g x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d+e x) (2 c d-b e) (-b e g+c d g+c e f)}{c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\frac {1}{4} e^4 \int \frac {2 (3 c d-b e) (2 c e f+3 c d g-2 b e g)-c e (7 b e g-4 c (e f+3 d g)) x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {1}{2} c e^4 g x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^5}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {2 (d+e x) (2 c d-b e) (-b e g+c d g+c e f)}{c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\frac {1}{4} e^4 \left (\frac {3}{2} (2 c d-b e) (-5 b e g+6 c d g+4 c e f) \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+12 c d g+4 c e f)}{e}\right )-\frac {1}{2} c e^4 g x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^5}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 (d+e x) (2 c d-b e) (-b e g+c d g+c e f)}{c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\frac {1}{4} e^4 \left (3 (2 c d-b e) (-5 b e g+6 c d g+4 c e f) \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+12 c d g+4 c e f)}{e}\right )-\frac {1}{2} c e^4 g x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^5}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 (d+e x) (2 c d-b e) (-b e g+c d g+c e f)}{c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\frac {1}{4} e^4 \left (\frac {3 (2 c d-b e) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-5 b e g+6 c d g+4 c e f)}{2 \sqrt {c} e}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+12 c d g+4 c e f)}{e}\right )-\frac {1}{2} c e^4 g x \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^3 e^5}\) |
(2*(2*c*d - b*e)*(c*e*f + c*d*g - b*e*g)*(d + e*x))/(c^3*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (-1/2*(c*e^4*g*x*Sqrt[d*(c*d - b*e) - b*e^ 2*x - c*e^2*x^2]) + (e^4*(-(((4*c*e*f + 12*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/e) + (3*(2*c*d - b*e)*(4*c*e*f + 6*c*d*g - 5* b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c* e^2*x^2])])/(2*Sqrt[c]*e)))/4)/(c^3*e^5)
3.23.17.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[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[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*( e*f + d*g) - b*e*g)^n*((d + e*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c* x^2])), x] + Simp[1/(c^(m + n - 1)*e^(n - 2)) Int[ExpandToSum[((2*c*d - b *e)^(m - 1)*(c*(e*f + d*g) - b*e*g)^n - c^(m + n - 1)*e^n*(d + e*x)^(m - 1) *(f + g*x)^n)/(c*d - b*e - c*e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; Free Q[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1)) Int[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b *e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c , p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && !LeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1543\) vs. \(2(267)=534\).
Time = 1.45 (sec) , antiderivative size = 1544, normalized size of antiderivative = 5.38
2*d^3*f*(-2*c*e^2*x-b*e^2)/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c*e^2*x^2-b *e^2*x-b*d*e+c*d^2)^(1/2)+e^3*g*(-1/2*x^3/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+ c*d^2)^(1/2)-5/4*b/c*(-x^2/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-3/ 2*b/c*(x/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c*(1/c/e^2/(-c *e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-b/c*(-2*c*e^2*x-b*e^2)/(-4*c*e^2*(-b*d *e+c*d^2)-b^2*e^4)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-1/c/e^2/(c*e^2) ^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^( 1/2)))+2*(-b*d*e+c*d^2)/c/e^2*(1/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1 /2)-b/c*(-2*c*e^2*x-b*e^2)/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c*e^2*x^2-b *e^2*x-b*d*e+c*d^2)^(1/2)))+3/2*(-b*d*e+c*d^2)/c/e^2*(x/c/e^2/(-c*e^2*x^2- b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c*(1/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^ 2)^(1/2)-b/c*(-2*c*e^2*x-b*e^2)/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c*e^2* x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-1/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2) *(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+(3*d*e^2*g+e^3*f)*( -x^2/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-3/2*b/c*(x/c/e^2/(-c*e^2 *x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c*(1/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e +c*d^2)^(1/2)-b/c*(-2*c*e^2*x-b*e^2)/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/(-c *e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))-1/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^ (1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))+2*(-b*d*e+c*d^2 )/c/e^2*(1/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-b/c*(-2*c*e^2*x...
Time = 0.84 (sec) , antiderivative size = 745, normalized size of antiderivative = 2.60 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (4 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (12 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 21 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g - {\left (4 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} f + {\left (12 \, c^{3} d^{2} e - 16 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, {\left (2 \, c^{3} e^{2} g x^{2} - 4 \, {\left (5 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f - {\left (28 \, c^{3} d^{2} - 43 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + {\left (4 \, c^{3} e^{2} f + 5 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{16 \, {\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )}}, -\frac {3 \, {\left (4 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (12 \, c^{3} d^{3} - 28 \, b c^{2} d^{2} e + 21 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g - {\left (4 \, {\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} f + {\left (12 \, c^{3} d^{2} e - 16 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (2 \, c^{3} e^{2} g x^{2} - 4 \, {\left (5 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f - {\left (28 \, c^{3} d^{2} - 43 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + {\left (4 \, c^{3} e^{2} f + 5 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{8 \, {\left (c^{5} e^{3} x - c^{5} d e^{2} + b c^{4} e^{3}\right )}}\right ] \]
[1/16*(3*(4*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*f + (12*c^3*d^3 - 28 *b*c^2*d^2*e + 21*b^2*c*d*e^2 - 5*b^3*e^3)*g - (4*(2*c^3*d*e^2 - b*c^2*e^3 )*f + (12*c^3*d^2*e - 16*b*c^2*d*e^2 + 5*b^2*c*e^3)*g)*x)*sqrt(-c)*log(8*c ^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 4*sqrt(-c*e^2 *x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) + 4*(2*c^3*e^2*g *x^2 - 4*(5*c^3*d*e - 3*b*c^2*e^2)*f - (28*c^3*d^2 - 43*b*c^2*d*e + 15*b^2 *c*e^2)*g + (4*c^3*e^2*f + 5*(2*c^3*d*e - b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^5*e^3*x - c^5*d*e^2 + b*c^4*e^3), -1/8*(3* (4*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*f + (12*c^3*d^3 - 28*b*c^2*d^ 2*e + 21*b^2*c*d*e^2 - 5*b^3*e^3)*g - (4*(2*c^3*d*e^2 - b*c^2*e^3)*f + (12 *c^3*d^2*e - 16*b*c^2*d*e^2 + 5*b^2*c*e^3)*g)*x)*sqrt(c)*arctan(1/2*sqrt(- c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) - 2*(2*c^3*e^2*g*x^2 - 4*(5*c^3*d*e - 3* b*c^2*e^2)*f - (28*c^3*d^2 - 43*b*c^2*d*e + 15*b^2*c*e^2)*g + (4*c^3*e^2*f + 5*(2*c^3*d*e - b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d *e))/(c^5*e^3*x - c^5*d*e^2 + b*c^4*e^3)]
\[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{3} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(b*e-2*c*d)>0)', see `assume?` for more
Time = 0.46 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.72 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {1}{4} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (\frac {2 \, g x}{c^{2} e} + \frac {4 \, c^{6} e^{3} f + 12 \, c^{6} d e^{2} g - 7 \, b c^{5} e^{3} g}{c^{8} e^{4}}\right )} + \frac {{\left (8 \, c^{2} d e f - 4 \, b c e^{2} f + 12 \, c^{2} d^{2} g - 16 \, b c d e g + 5 \, b^{2} e^{2} g\right )} \log \left ({\left | b c^{2} d^{2} e^{2} - 2 \, b^{2} c d e^{3} + b^{3} e^{4} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} c^{2} d^{2} {\left | e \right |} + 6 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b \sqrt {-c} c d e {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b^{2} \sqrt {-c} e^{2} {\left | e \right |} + 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} c^{2} d e - 5 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b c e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} \sqrt {-c} c {\left | e \right |} \right |}\right )}{8 \, \sqrt {-c} c^{3} e {\left | e \right |}} \]
1/4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*g*x/(c^2*e) + (4*c^6*e^3 *f + 12*c^6*d*e^2*g - 7*b*c^5*e^3*g)/(c^8*e^4)) + 1/8*(8*c^2*d*e*f - 4*b*c *e^2*f + 12*c^2*d^2*g - 16*b*c*d*e*g + 5*b^2*e^2*g)*log(abs(b*c^2*d^2*e^2 - 2*b^2*c*d*e^3 + b^3*e^4 - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*c^2*d^2*abs(e) + 6*(sqrt(-c*e^2)*x - sqrt(-c*e^ 2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b*sqrt(-c)*c*d*e*abs(e) - 4*(sqrt(-c*e^2 )*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b^2*sqrt(-c)*e^2*abs(e) + 4*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*c^2*d* e - 5*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*b*c* e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*sq rt(-c)*c*abs(e)))/(sqrt(-c)*c^3*e*abs(e))
Timed out. \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}} \,d x \]