Integrand size = 44, antiderivative size = 271 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=-\frac {3 (4 c e f-6 c d g+b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2}-\frac {(4 c e f-6 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (2 c d-b e) (d+e x)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^3}-\frac {3 (2 c d-b e) (4 c e f-6 c d g+b e g) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 \sqrt {c} e^2} \]
-1/2*(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)^(3/2)/e^2/(- b*e+2*c*d)/(e*x+d)-2*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2 /(-b*e+2*c*d)/(e*x+d)^3-3/8*(-b*e+2*c*d)*(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))/e^2/c^(1/2)- 3/4*(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)/e^2
Time = 0.91 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.80 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{3/2} \left (\frac {\sqrt {-b e+c (d-e x)} \left (b e (8 e f-13 d g-5 e g x)+2 c \left (14 d^2 g+5 d e (-2 f+g x)-e^2 x (2 f+g x)\right )\right )}{(d+e x)^2}-\frac {6 (2 c d-b e) (-4 c e f+6 c d g-b e g) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e}-\sqrt {c d-b e-c e x}}\right )}{\sqrt {c} (d+e x)^{3/2}}\right )}{4 e^2 (-b e+c (d-e x))^{3/2}} \]
(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((Sqrt[-(b*e) + c*(d - e*x)]*(b* e*(8*e*f - 13*d*g - 5*e*g*x) + 2*c*(14*d^2*g + 5*d*e*(-2*f + g*x) - e^2*x* (2*f + g*x))))/(d + e*x)^2 - (6*(2*c*d - b*e)*(-4*c*e*f + 6*c*d*g - b*e*g) *ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(Sqrt[2*c*d - b*e] - Sqrt[c*d - b*e - c*e* x])])/(Sqrt[c]*(d + e*x)^(3/2))))/(4*e^2*(-(b*e) + c*(d - e*x))^(3/2))
Time = 0.50 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1220, 1131, 1131, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {(b e g-6 c d g+4 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^2}dx}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(b e g-6 c d g+4 c e f) \left (\frac {3}{4} (2 c d-b e) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{d+e x}dx+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e (d+e x)}\right )}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(b e g-6 c d g+4 c e f) \left (\frac {3}{4} (2 c d-b e) \left (\frac {1}{2} (2 c d-b e) \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e (d+e x)}\right )}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {(b e g-6 c d g+4 c e f) \left (\frac {3}{4} (2 c d-b e) \left ((2 c d-b e) \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e (d+e x)}\right )}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\left (\frac {3}{4} (2 c d-b e) \left (\frac {(2 c d-b e) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt {c} e}+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e (d+e x)}\right ) (b e g-6 c d g+4 c e f)}{e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^3 (2 c d-b e)}\) |
(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - b*e)*(d + e*x)^3) - ((4*c*e*f - 6*c*d*g + b*e*g)*((d*(c*d - b*e) - b*e^2* x - c*e^2*x^2)^(3/2)/(2*e*(d + e*x)) + (3*(2*c*d - b*e)*(Sqrt[d*(c*d - b*e ) - b*e^2*x - c*e^2*x^2]/e + ((2*c*d - b*e)*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))/(e*(2 *c*d - b*e))
3.22.88.3.1 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)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1))) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b *d*e + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne Q[m + 2*p + 1, 0] && IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x ^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e *f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1)) 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[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0 ]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 ]
Leaf count of result is larger than twice the leaf count of optimal. \(730\) vs. \(2(251)=502\).
Time = 1.07 (sec) , antiderivative size = 731, normalized size of antiderivative = 2.70
| method | result | size |
| default | \(\frac {g \left (\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {6 c \,e^{2} \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (-b \,e^{2}+2 c d e \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 c d e \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2}\right )}{-b \,e^{2}+2 c d e}\right )}{e^{3}}+\frac {\left (-d g +e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {4 c \,e^{2} \left (\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {6 c \,e^{2} \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (-b \,e^{2}+2 c d e \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 c d e \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 c d e}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2}\right )}{-b \,e^{2}+2 c d e}\right )}{-b \,e^{2}+2 c d e}\right )}{e^{4}}\) | \(731\) |
g/e^3*(2/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+ d/e))^(5/2)+6*c*e^2/(-b*e^2+2*c*d*e)*(1/3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d* e)*(x+d/e))^(3/2)+1/2*(-b*e^2+2*c*d*e)*(-1/4*(-2*c*e^2*(x+d/e)-b*e^2+2*c*d *e)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+1/8*(-b*e^2+2* c*d*e)^2/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d *e)/c/e^2)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)))))+(-d*g+e*f )/e^4*(-2/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x +d/e))^(5/2)-4*c*e^2/(-b*e^2+2*c*d*e)*(2/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^ 2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+6*c*e^2/(-b*e^2+2*c*d*e)*(1/3* (-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(3/2)+1/2*(-b*e^2+2*c*d*e)*(-1 /4*(-2*c*e^2*(x+d/e)-b*e^2+2*c*d*e)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d* e)*(x+d/e))^(1/2)+1/8*(-b*e^2+2*c*d*e)^2/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2 )^(1/2)*(x+d/e-1/2*(-b*e^2+2*c*d*e)/c/e^2)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d *e)*(x+d/e))^(1/2))))))
Time = 0.91 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.34 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\left [\frac {3 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f - {\left (12 \, c^{2} d^{3} - 8 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left (4 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (12 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (2 \, c^{2} e^{2} g x^{2} + 4 \, {\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} f - {\left (28 \, c^{2} d^{2} - 13 \, b c d e\right )} g + {\left (4 \, c^{2} e^{2} f - 5 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{16 \, {\left (c e^{3} x + c d e^{2}\right )}}, \frac {3 \, {\left (4 \, {\left (2 \, c^{2} d^{2} e - b c d e^{2}\right )} f - {\left (12 \, c^{2} d^{3} - 8 \, b c d^{2} e + b^{2} d e^{2}\right )} g + {\left (4 \, {\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} f - {\left (12 \, c^{2} d^{2} e - 8 \, b c d e^{2} + b^{2} e^{3}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) - 2 \, {\left (2 \, c^{2} e^{2} g x^{2} + 4 \, {\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} f - {\left (28 \, c^{2} d^{2} - 13 \, b c d e\right )} g + {\left (4 \, c^{2} e^{2} f - 5 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{8 \, {\left (c e^{3} x + c d e^{2}\right )}}\right ] \]
[1/16*(3*(4*(2*c^2*d^2*e - b*c*d*e^2)*f - (12*c^2*d^3 - 8*b*c*d^2*e + b^2* d*e^2)*g + (4*(2*c^2*d*e^2 - b*c*e^3)*f - (12*c^2*d^2*e - 8*b*c*d*e^2 + b^ 2*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^2*e^2*g*x^2 + 4*(5*c^2*d*e - 2*b*c*e^2)*f - (28*c^2* d^2 - 13*b*c*d*e)*g + (4*c^2*e^2*f - 5*(2*c^2*d*e - b*c*e^2)*g)*x)*sqrt(-c *e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c*e^3*x + c*d*e^2), 1/8*(3*(4*(2*c^2 *d^2*e - b*c*d*e^2)*f - (12*c^2*d^3 - 8*b*c*d^2*e + b^2*d*e^2)*g + (4*(2*c ^2*d*e^2 - b*c*e^3)*f - (12*c^2*d^2*e - 8*b*c*d*e^2 + b^2*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)*s qrt(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) - 2*(2*c^2*e^2*g*x^2 + 4*(5*c^2*d*e - 2*b*c*e^2)*f - (28*c^2*d^2 - 13*b*c*d*e)*g + (4*c^2*e^2* f - 5*(2*c^2*d*e - b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d* e))/(c*e^3*x + c*d*e^2)]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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-2*c*d>0)', see `assume?` for more deta
Time = 0.45 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.49 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=-\frac {1}{4} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (\frac {2 \, c g x}{e} + \frac {4 \, c^{2} e^{3} f - 12 \, c^{2} d e^{2} g + 5 \, b c e^{3} g}{c e^{4}}\right )} + \frac {{\left (8 \, c^{2} d e f - 4 \, b c e^{2} f - 12 \, c^{2} d^{2} g + 8 \, b c d e g - b^{2} e^{2} g\right )} \log \left ({\left | b c d^{2} e^{2} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} c d^{2} {\left | e \right |} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b \sqrt {-c} d e {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} c d e - {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} \sqrt {-c} {\left | e \right |} \right |}\right )}{8 \, \sqrt {-c} e {\left | e \right |}} \]
-1/4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*g*x/e + (4*c^2*e^3*f - 12*c^2*d*e^2*g + 5*b*c*e^3*g)/(c*e^4)) + 1/8*(8*c^2*d*e*f - 4*b*c*e^2*f - 12*c^2*d^2*g + 8*b*c*d*e*g - b^2*e^2*g)*log(abs(b*c*d^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))*sqrt(-c)*c*d^2*abs(e ) - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b*sqrt (-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))^2*c*d*e - (sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b* d*e))^2*b*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*sqrt(-c)*abs(e)))/(sqrt(-c)*e*abs(e))
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^3} \,d x \]