Integrand size = 46, antiderivative size = 288 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {(3 c e f-7 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(3 c e f-7 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (2 c d-b e) (d+e x)^{3/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (2 c d-b e) (d+e x)^{7/2}}+\frac {\sqrt {2 c d-b e} (3 c e f-7 c d g+2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \]
-1/3*(2*b*e*g-7*c*d*g+3*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)^(3/2)-(-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)^(7/2)+(2*b*e*g-7*c*d*g+3*c*e*f)*arctanh((d*(-b *e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2*c*d)^(1/2)/(e*x+d)^(1/2))*(-b*e+2 *c*d)^(1/2)/e^2-(2*b*e*g-7*c*d*g+3*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2) ^(1/2)/e^2/(e*x+d)^(1/2)
Time = 0.40 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.67 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {\sqrt {(d+e x) (-b e+c (d-e x))} \left (\sqrt {-b e+c (d-e x)} \left (b e (-3 e f+11 d g+8 e g x)+2 c \left (-13 d^2 g+d e (6 f-9 g x)+e^2 x (3 f+g x)\right )\right )+3 \sqrt {-2 c d+b e} (-3 c e f+7 c d g-2 b e g) (d+e x) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )\right )}{3 e^2 (d+e x)^{3/2} \sqrt {-b e+c (d-e x)}} \]
-1/3*(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(Sqrt[-(b*e) + c*(d - e*x)]*( b*e*(-3*e*f + 11*d*g + 8*e*g*x) + 2*c*(-13*d^2*g + d*e*(6*f - 9*g*x) + e^2 *x*(3*f + g*x))) + 3*Sqrt[-2*c*d + b*e]*(-3*c*e*f + 7*c*d*g - 2*b*e*g)*(d + e*x)*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]]))/(e^2*(d + e *x)^(3/2)*Sqrt[-(b*e) + c*(d - e*x)])
Time = 0.53 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1220, 1131, 1131, 1136, 221}
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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {(2 b e g-7 c d g+3 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{5/2}}dx}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(2 b e g-7 c d g+3 c e f) \left ((2 c d-b e) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{3/2}}dx+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(2 b e g-7 c d g+3 c e f) \left ((2 c d-b e) \left ((2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle -\frac {(2 b e g-7 c d g+3 c e f) \left ((2 c d-b e) \left (2 e (2 c d-b e) \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\left ((2 c d-b e) \left (\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2 c d-b e} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right ) (2 b e g-7 c d g+3 c e f)}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}\) |
-(((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)^(7/2))) - ((3*c*e*f - 7*c*d*g + 2*b*e*g)*((2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) + (2*c*d - b*e)*((2*S qrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e*Sqrt[d + e*x]) - (2*Sqrt[2*c* d - b*e]*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b *e]*Sqrt[d + e*x])])/e)))/(2*e*(2*c*d - b*e))
3.23.46.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x _Symbol] :> Simp[2*e Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
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. \(686\) vs. \(2(266)=532\).
Time = 0.37 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.39
| method | result | size |
| default | \(\frac {\sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (6 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} e^{3} g x -33 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x +9 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} f x +42 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -18 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x -2 c \,e^{2} g \,x^{2} \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+6 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} d \,e^{2} g -33 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g +9 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} f +42 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -18 \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f -8 b \,e^{2} g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+18 c d e g x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-6 c \,e^{2} f x \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-11 b d e g \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+3 b \,e^{2} f \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}+26 c \,d^{2} g \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}-12 c d e f \sqrt {-x c e -b e +c d}\, \sqrt {b e -2 c d}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \sqrt {-x c e -b e +c d}\, e^{2} \sqrt {b e -2 c d}}\) | \(687\) |
1/3*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e -2*c*d)^(1/2))*b^2*e^3*g*x-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1 /2))*b*c*d*e^2*g*x+9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c* e^3*f*x+42*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*g*x- 18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e^2*f*x-2*c*e^2* g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+6*arctan((-c*e*x-b*e+c*d)^( 1/2)/(b*e-2*c*d)^(1/2))*b^2*d*e^2*g-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e- 2*c*d)^(1/2))*b*c*d^2*e*g+9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2 ))*b*c*d*e^2*f+42*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3 *g-18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f-8*b*e^2 *g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+18*c*d*e*g*x*(-c*e*x-b*e+c*d )^(1/2)*(b*e-2*c*d)^(1/2)-6*c*e^2*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^( 1/2)-11*b*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+3*b*e^2*f*(-c*e*x -b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+26*c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2 *c*d)^(1/2)-12*c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))/(e*x+d)^( 3/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)
Time = 0.51 (sec) , antiderivative size = 639, normalized size of antiderivative = 2.22 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\left [\frac {3 \, {\left (3 \, c d^{2} e f + {\left (3 \, c e^{3} f - {\left (7 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (7 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (3 \, c d e^{2} f - {\left (7 \, c d^{2} e - 2 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (2 \, c e^{2} g x^{2} + 3 \, {\left (4 \, c d e - b e^{2}\right )} f - {\left (26 \, c d^{2} - 11 \, b d e\right )} g + 2 \, {\left (3 \, c e^{2} f - {\left (9 \, c d e - 4 \, b e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{6 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {3 \, {\left (3 \, c d^{2} e f + {\left (3 \, c e^{3} f - {\left (7 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (7 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (3 \, c d e^{2} f - {\left (7 \, c d^{2} e - 2 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{c e^{2} x^{2} + b e^{2} x - c d^{2} + b d e}\right ) - {\left (2 \, c e^{2} g x^{2} + 3 \, {\left (4 \, c d e - b e^{2}\right )} f - {\left (26 \, c d^{2} - 11 \, b d e\right )} g + 2 \, {\left (3 \, c e^{2} f - {\left (9 \, c d e - 4 \, b e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]
[1/6*(3*(3*c*d^2*e*f + (3*c*e^3*f - (7*c*d*e^2 - 2*b*e^3)*g)*x^2 - (7*c*d^ 3 - 2*b*d^2*e)*g + 2*(3*c*d*e^2*f - (7*c*d^2*e - 2*b*d*e^2)*g)*x)*sqrt(2*c *d - b*e)*log(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x - 2*sq rt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/ (e^2*x^2 + 2*d*e*x + d^2)) - 2*(2*c*e^2*g*x^2 + 3*(4*c*d*e - b*e^2)*f - (2 6*c*d^2 - 11*b*d*e)*g + 2*(3*c*e^2*f - (9*c*d*e - 4*b*e^2)*g)*x)*sqrt(-c*e ^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(e^4*x^2 + 2*d*e^3*x + d^ 2*e^2), 1/3*(3*(3*c*d^2*e*f + (3*c*e^3*f - (7*c*d*e^2 - 2*b*e^3)*g)*x^2 - (7*c*d^3 - 2*b*d^2*e)*g + 2*(3*c*d*e^2*f - (7*c*d^2*e - 2*b*d*e^2)*g)*x)*s qrt(-2*c*d + b*e)*arctan(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(- 2*c*d + b*e)*sqrt(e*x + d)/(c*e^2*x^2 + b*e^2*x - c*d^2 + b*d*e)) - (2*c*e ^2*g*x^2 + 3*(4*c*d*e - b*e^2)*f - (26*c*d^2 - 11*b*d*e)*g + 2*(3*c*e^2*f - (9*c*d*e - 4*b*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqr t(e*x + d))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Time = 0.44 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.12 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} e f - 18 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} d g + 6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c e g - 2 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c g + \frac {3 \, {\left (6 \, c^{3} d e f - 3 \, b c^{2} e^{2} f - 14 \, c^{3} d^{2} g + 11 \, b c^{2} d e g - 2 \, b^{2} c e^{2} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, {\left (2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g + \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g\right )}}{{\left (e x + d\right )} c}}{3 \, c e^{2}} \]
-1/3*(6*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*e*f - 18*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*d*g + 6*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c*e*g - 2*(-(e *x + d)*c + 2*c*d - b*e)^(3/2)*c*g + 3*(6*c^3*d*e*f - 3*b*c^2*e^2*f - 14*c ^3*d^2*g + 11*b*c^2*d*e*g - 2*b^2*c*e^2*g)*arctan(sqrt(-(e*x + d)*c + 2*c* d - b*e)/sqrt(-2*c*d + b*e))/sqrt(-2*c*d + b*e) + 3*(2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d*e*f - sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*e^2*f - 2 *sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d^2*g + sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d*e*g)/((e*x + d)*c))/(c*e^2)
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]