Integrand size = 44, antiderivative size = 177 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)^3}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 g (d+e x)}{c^2 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {g \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{5/2} e^2} \]
2/3*(-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)^(3/2)+g*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^(5/2)/e^2-2*g*(e*x+d)/c^2/e^2/(d*(-b*e+c*d)-b*e^2*x-c* e^2*x^2)^(1/2)
Time = 0.37 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.06 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \left (-\frac {\sqrt {c} (d+e x)^3 (-b e+c (d-e x)) \left (3 b^2 e^2 g+4 b c e g (-2 d+e x)+c^2 \left (5 d^2 g-e^2 f x-d e (f+7 g x)\right )\right )}{2 c d-b e}-3 g (d+e x)^{5/2} (-b e+c (d-e x))^{5/2} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )\right )}{3 c^{5/2} e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}} \]
(2*(-((Sqrt[c]*(d + e*x)^3*(-(b*e) + c*(d - e*x))*(3*b^2*e^2*g + 4*b*c*e*g *(-2*d + e*x) + c^2*(5*d^2*g - e^2*f*x - d*e*(f + 7*g*x))))/(2*c*d - b*e)) - 3*g*(d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(5/2)*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])]))/(3*c^(5/2)*e^2*((d + e*x)*(-(b*e) + c* (d - e*x)))^(5/2))
Time = 0.39 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1218, 1124, 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.
| \(\displaystyle \int \frac {(d+e x)^3 (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {g \int \frac {(d+e x)^2}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{c e}\) |
| \(\Big \downarrow \) 1124 |
| \(\displaystyle \frac {2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {g \left (\frac {2 (d+e x)}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c}\right )}{c e}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {g \left (\frac {2 (d+e x)}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{c}\right )}{c e}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 (d+e x)^3 (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {g \left (\frac {2 (d+e x)}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{c^{3/2} e}\right )}{c e}\) |
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^3)/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b *e) - b*e^2*x - c*e^2*x^2)^(3/2)) - (g*((2*(d + e*x))/(c*e*Sqrt[d*(c*d - b *e) - b*e^2*x - c*e^2*x^2]) - ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c* d - b*e) - b*e^2*x - c*e^2*x^2])]/(c^(3/2)*e)))/(c*e)
3.23.25.3.1 Defintions of rubi rules used
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_))^(m_.)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x _Symbol] :> Simp[-2*e*(2*c*d - b*e)^(m - 2)*((d + e*x)/(c^(m - 1)*Sqrt[a + b*x + c*x^2])), x] + Simp[e^2/c^(m - 1) Int[(1/Sqrt[a + b*x + c*x^2])*Exp andToSum[((2*c*d - b*e)^(m - 1) - c^(m - 1)*(d + e*x)^(m - 1))/(c*d - b*e - c*e*x), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e ^2, 0] && IGtQ[m, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d , e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2425\) vs. \(2(163)=326\).
Time = 1.31 (sec) , antiderivative size = 2426, normalized size of antiderivative = 13.71
d^3*f*(2/3*(-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)^(3/2)-16/3*c*e^2/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)^ 2*(-2*c*e^2*x-b*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))+e^3*g*(1/3*x^ 3/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1/2*b/c*(x^2/c/e^2/(-c*e^2* x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+1/2*b/c*(1/2*x/c/e^2/(-c*e^2*x^2-b*e^2*x-b* d*e+c*d^2)^(3/2)-1/4*b/c*(1/3/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2) -1/2*b/c*(2/3*(-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)^(3/2)-16/3*c*e^2/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^ 4)^2*(-2*c*e^2*x-b*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))-1/2*(-b*d *e+c*d^2)/c/e^2*(2/3*(-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)^(3/2)-16/3*c*e^2/(-4*c*e^2*(-b*d*e+c*d^2) -b^2*e^4)^2*(-2*c*e^2*x-b*e^2)/(-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*(1/3/c/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-1/2 *b/c*(2/3*(-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)^(3/2)-16/3*c*e^2/(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)^2 *(-2*c*e^2*x-b*e^2)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))-1/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)*arc tan((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))))...
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (163) = 326\).
Time = 2.04 (sec) , antiderivative size = 785, normalized size of antiderivative = 4.44 \[ \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 )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g x^{2} - 2 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g x + {\left (2 \, c^{3} d^{3} - 5 \, b c^{2} d^{2} e + 4 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g\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 (c^{3} d e f - {\left (5 \, c^{3} d^{2} - 8 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + {\left (c^{3} e^{2} f + {\left (7 \, c^{3} d e - 4 \, 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}}{6 \, {\left (2 \, c^{6} d^{3} e^{2} - 5 \, b c^{5} d^{2} e^{3} + 4 \, b^{2} c^{4} d e^{4} - b^{3} c^{3} e^{5} + {\left (2 \, c^{6} d e^{4} - b c^{5} e^{5}\right )} x^{2} - 2 \, {\left (2 \, c^{6} d^{2} e^{3} - 3 \, b c^{5} d e^{4} + b^{2} c^{4} e^{5}\right )} x\right )}}, -\frac {3 \, {\left ({\left (2 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} g x^{2} - 2 \, {\left (2 \, c^{3} d^{2} e - 3 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g x + {\left (2 \, c^{3} d^{3} - 5 \, b c^{2} d^{2} e + 4 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g\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 (c^{3} d e f - {\left (5 \, c^{3} d^{2} - 8 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + {\left (c^{3} e^{2} f + {\left (7 \, c^{3} d e - 4 \, 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}}{3 \, {\left (2 \, c^{6} d^{3} e^{2} - 5 \, b c^{5} d^{2} e^{3} + 4 \, b^{2} c^{4} d e^{4} - b^{3} c^{3} e^{5} + {\left (2 \, c^{6} d e^{4} - b c^{5} e^{5}\right )} x^{2} - 2 \, {\left (2 \, c^{6} d^{2} e^{3} - 3 \, b c^{5} d e^{4} + b^{2} c^{4} e^{5}\right )} x\right )}}\right ] \]
[-1/6*(3*((2*c^3*d*e^2 - b*c^2*e^3)*g*x^2 - 2*(2*c^3*d^2*e - 3*b*c^2*d*e^2 + b^2*c*e^3)*g*x + (2*c^3*d^3 - 5*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^3*e^3)* g)*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*(c^3*d*e*f - (5*c^3*d^2 - 8*b*c^2*d*e + 3*b^2*c*e^2)*g + (c^3*e^2*f + (7*c^3*d*e - 4*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d *e))/(2*c^6*d^3*e^2 - 5*b*c^5*d^2*e^3 + 4*b^2*c^4*d*e^4 - b^3*c^3*e^5 + (2 *c^6*d*e^4 - b*c^5*e^5)*x^2 - 2*(2*c^6*d^2*e^3 - 3*b*c^5*d*e^4 + b^2*c^4*e ^5)*x), -1/3*(3*((2*c^3*d*e^2 - b*c^2*e^3)*g*x^2 - 2*(2*c^3*d^2*e - 3*b*c^ 2*d*e^2 + b^2*c*e^3)*g*x + (2*c^3*d^3 - 5*b*c^2*d^2*e + 4*b^2*c*d*e^2 - b^ 3*e^3)*g)*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*( c^3*d*e*f - (5*c^3*d^2 - 8*b*c^2*d*e + 3*b^2*c*e^2)*g + (c^3*e^2*f + (7*c^ 3*d*e - 4*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(2* c^6*d^3*e^2 - 5*b*c^5*d^2*e^3 + 4*b^2*c^4*d*e^4 - b^3*c^3*e^5 + (2*c^6*d*e ^4 - b*c^5*e^5)*x^2 - 2*(2*c^6*d^2*e^3 - 3*b*c^5*d*e^4 + b^2*c^4*e^5)*x)]
\[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{3} \left (f + g x\right )}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 943 vs. \(2 (163) = 326\).
Time = 1.77 (sec) , antiderivative size = 943, normalized size of antiderivative = 5.33 \[ \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 )^{5/2}} \, dx=-\frac {g \log \left ({\left | b \sqrt {-c} c^{4} d^{4} e^{2} - 4 \, b^{2} \sqrt {-c} c^{3} d^{3} e^{3} + 6 \, b^{3} \sqrt {-c} c^{2} d^{2} e^{4} - 4 \, b^{4} \sqrt {-c} c d e^{5} + b^{5} \sqrt {-c} e^{6} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} c^{5} d^{4} {\left | e \right |} - 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b c^{4} d^{3} e {\left | e \right |} + 24 \, {\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} c^{3} d^{2} e^{2} {\left | e \right |} - 20 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b^{3} c^{2} d e^{3} {\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^{4} c e^{4} {\left | e \right |} + 8 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} \sqrt {-c} c^{4} d^{3} e - 30 \, {\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 \sqrt {-c} c^{3} d^{2} e^{2} + 36 \, {\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^{2} \sqrt {-c} c^{2} d e^{3} - 14 \, {\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^{3} \sqrt {-c} c e^{4} - 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} c^{4} d^{2} {\left | e \right |} + 28 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} b c^{3} d e {\left | e \right |} - 16 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} b^{2} c^{2} e^{2} {\left | e \right |} - 8 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{4} \sqrt {-c} c^{3} d e + 9 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{4} b \sqrt {-c} c^{2} 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 )}^{5} c^{3} {\left | e \right |} \right |}\right )}{5 \, \sqrt {-c} c^{2} e {\left | e \right |}} \]
-1/5*g*log(abs(b*sqrt(-c)*c^4*d^4*e^2 - 4*b^2*sqrt(-c)*c^3*d^3*e^3 + 6*b^3 *sqrt(-c)*c^2*d^2*e^4 - 4*b^4*sqrt(-c)*c*d*e^5 + b^5*sqrt(-c)*e^6 + 2*(sqr t(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*c^5*d^4*abs(e) - 12*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b*c^4*d^ 3*e*abs(e) + 24*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d* e))*b^2*c^3*d^2*e^2*abs(e) - 20*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2* x + c*d^2 - b*d*e))*b^3*c^2*d*e^3*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^4*c*e^4*abs(e) + 8*(sqrt(-c*e^2)*x - sq rt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*sqrt(-c)*c^4*d^3*e - 30*(sqrt( -c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*b*sqrt(-c)*c^3*d ^2*e^2 + 36*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^ 2*b^2*sqrt(-c)*c^2*d*e^3 - 14*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*b^3*sqrt(-c)*c*e^4 - 12*(sqrt(-c*e^2)*x - sqrt(-c*e^2* x^2 - b*e^2*x + c*d^2 - b*d*e))^3*c^4*d^2*abs(e) + 28*(sqrt(-c*e^2)*x - sq rt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*b*c^3*d*e*abs(e) - 16*(sqrt(-c *e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*b^2*c^2*e^2*abs(e) - 8*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^4*sqrt( -c)*c^3*d*e + 9*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d* e))^4*b*sqrt(-c)*c^2*e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^5*c^3*abs(e)))/(sqrt(-c)*c^2*e*abs(e))
Timed out. \[ \int \frac {(d+e x)^3 (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d x \]