Integrand size = 46, antiderivative size = 360 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {(2 c d-b e) (5 c e f-9 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}-\frac {(5 c e f-9 c d g+2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^{5/2}}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (2 c d-b e) (d+e x)^{9/2}}+\frac {(2 c d-b e)^{3/2} (5 c e f-9 c d g+2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \]
-1/3*(2*b*e*g-9*c*d*g+5*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/ (e*x+d)^(3/2)-1/5*(2*b*e*g-9*c*d*g+5*c*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)^(5/2)-(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c *e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)/(e*x+d)^(9/2)+(-b*e+2*c*d)^(3/2)*(2*b*e*g -9*c*d*g+5*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))/e^2-(-b*e+2*c*d)*(2*b*e*g-9*c*d*g+5*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 = 0.60 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.71 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {b^2 e^2 (-15 e f+61 d g+46 e g x)+2 b c e \left (-146 d^2 g+5 d e (13 f-21 g x)+e^2 x (35 f+11 g x)\right )+2 c^2 \left (168 d^3 g+e^3 x^2 (5 f+3 g x)-6 d e^2 x (10 f+3 g x)+d^2 e (-95 f+117 g x)\right )}{(d+e x) (-c d+b e+c e x)^2}+\frac {15 (-2 c d+b e)^{3/2} (-5 c e f+9 c d g-2 b e g) \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-b e+c (d-e x))^{5/2}}\right )}{15 e^2 (d+e x)^{5/2}} \]
(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((b^2*e^2*(-15*e*f + 61*d*g + 46 *e*g*x) + 2*b*c*e*(-146*d^2*g + 5*d*e*(13*f - 21*g*x) + e^2*x*(35*f + 11*g *x)) + 2*c^2*(168*d^3*g + e^3*x^2*(5*f + 3*g*x) - 6*d*e^2*x*(10*f + 3*g*x) + d^2*e*(-95*f + 117*g*x)))/((d + e*x)*(-(c*d) + b*e + c*e*x)^2) + (15*(- 2*c*d + b*e)^(3/2)*(-5*c*e*f + 9*c*d*g - 2*b*e*g)*ArcTan[Sqrt[-(b*e) + c*( d - e*x)]/Sqrt[-2*c*d + b*e]])/(-(b*e) + c*(d - e*x))^(5/2)))/(15*e^2*(d + e*x)^(5/2))
Time = 0.62 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1220, 1131, 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.
| \(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle -\frac {(2 b e g-9 c d g+5 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^{7/2}}dx}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(2 b e g-9 c d g+5 c e f) \left ((2 c d-b e) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^{5/2}}dx+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(2 b e g-9 c d g+5 c e f) \left ((2 c d-b e) \left ((2 c d-b e) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{3/2}}dx+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1131 |
| \(\displaystyle -\frac {(2 b e g-9 c d g+5 c e f) \left ((2 c d-b e) \left ((2 c d-b e) \left ((2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle -\frac {(2 b e g-9 c d g+5 c e f) \left ((2 c d-b e) \left ((2 c d-b e) \left (2 e (2 c d-b e) \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\left ((2 c d-b e) \left ((2 c d-b e) \left (\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2 c d-b e} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\right ) (2 b e g-9 c d g+5 c e f)}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{e^2 (d+e x)^{9/2} (2 c d-b e)}\) |
-(((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(e^2*(2*c*d - b*e)*(d + e*x)^(9/2))) - ((5*c*e*f - 9*c*d*g + 2*b*e*g)*((2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e*(d + e*x)^(5/2)) + (2*c*d - b*e)*((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*Sqrt[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]/(S qrt[2*c*d - b*e]*Sqrt[d + e*x])])/e))))/(2*e*(2*c*d - b*e))
3.23.57.3.1 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. \(1127\) vs. \(2(332)=664\).
Time = 0.36 (sec) , antiderivative size = 1128, normalized size of antiderivative = 3.13
-1/15*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-6*c^2*e^3*g*x^3*(-c*e*x-b*e+c*d)^ (1/2)*(b*e-2*c*d)^(1/2)+15*b^2*e^3*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1 /2)-336*c^2*d^3*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+300*arctan((-c* e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*f-10*c^2*e^3*f*x^2*(-c*e*x -b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-46*b^2*e^3*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b *e-2*c*d)^(1/2)-61*b^2*d*e^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+19 0*c^2*d^2*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-540*arctan((-c*e*x- b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^3*e*g*x+300*arctan((-c*e*x-b*e+c*d )^(1/2)/(b*e-2*c*d)^(1/2))*c^3*d^2*e^2*f*x+660*arctan((-c*e*x-b*e+c*d)^(1/ 2)/(b*e-2*c*d)^(1/2))*b*c^2*d^3*e*g-300*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e -2*c*d)^(1/2))*b*c^2*d^2*e^2*f+75*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d )^(1/2))*b^2*c*e^4*f*x-540*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2) )*c^3*d^4*g-234*c^2*d^2*e*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+120 *c^2*d*e^2*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+292*b*c*d^2*e*g*(- c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-130*b*c*d*e^2*f*(-c*e*x-b*e+c*d)^(1 /2)*(b*e-2*c*d)^(1/2)+660*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2)) *b*c^2*d^2*e^2*g*x-22*b*c*e^3*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/ 2)+36*c^2*d*e^2*g*x^2*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-70*b*c*e^3* f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+30*arctan((-c*e*x-b*e+c*d)^(1 /2)/(b*e-2*c*d)^(1/2))*b^3*d*e^3*g+210*b*c*d*e^2*g*x*(-c*e*x-b*e+c*d)^(...
Time = 0.38 (sec) , antiderivative size = 990, normalized size of antiderivative = 2.75 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\left [-\frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\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 (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, b^{2} e^{3}\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}}{30 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {15 \, {\left ({\left (5 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} f - {\left (18 \, c^{2} d^{2} e^{2} - 13 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} f - {\left (18 \, c^{2} d^{4} - 13 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2}\right )} g + 2 \, {\left (5 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} f - {\left (18 \, c^{2} d^{3} e - 13 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\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 (6 \, c^{2} e^{3} g x^{3} + 2 \, {\left (5 \, c^{2} e^{3} f - {\left (18 \, c^{2} d e^{2} - 11 \, b c e^{3}\right )} g\right )} x^{2} - 5 \, {\left (38 \, c^{2} d^{2} e - 26 \, b c d e^{2} + 3 \, b^{2} e^{3}\right )} f + {\left (336 \, c^{2} d^{3} - 292 \, b c d^{2} e + 61 \, b^{2} d e^{2}\right )} g - 2 \, {\left (5 \, {\left (12 \, c^{2} d e^{2} - 7 \, b c e^{3}\right )} f - {\left (117 \, c^{2} d^{2} e - 105 \, b c d e^{2} + 23 \, b^{2} e^{3}\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}}{15 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \]
[-1/30*(15*((5*(2*c^2*d*e^3 - b*c*e^4)*f - (18*c^2*d^2*e^2 - 13*b*c*d*e^3 + 2*b^2*e^4)*g)*x^2 + 5*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (18*c^2*d^4 - 13*b *c*d^3*e + 2*b^2*d^2*e^2)*g + 2*(5*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (18*c^2 *d^3*e - 13*b*c*d^2*e^2 + 2*b^2*d*e^3)*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*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)) - 2*(6*c^2*e^3*g*x^3 + 2*(5*c^2*e^3*f - (18*c^2*d*e^2 - 11*b*c*e^3)* g)*x^2 - 5*(38*c^2*d^2*e - 26*b*c*d*e^2 + 3*b^2*e^3)*f + (336*c^2*d^3 - 29 2*b*c*d^2*e + 61*b^2*d*e^2)*g - 2*(5*(12*c^2*d*e^2 - 7*b*c*e^3)*f - (117*c ^2*d^2*e - 105*b*c*d*e^2 + 23*b^2*e^3)*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), 1/15*(15*((5 *(2*c^2*d*e^3 - b*c*e^4)*f - (18*c^2*d^2*e^2 - 13*b*c*d*e^3 + 2*b^2*e^4)*g )*x^2 + 5*(2*c^2*d^3*e - b*c*d^2*e^2)*f - (18*c^2*d^4 - 13*b*c*d^3*e + 2*b ^2*d^2*e^2)*g + 2*(5*(2*c^2*d^2*e^2 - b*c*d*e^3)*f - (18*c^2*d^3*e - 13*b* c*d^2*e^2 + 2*b^2*d*e^3)*g)*x)*sqrt(-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)) + (6*c^2*e^3*g*x^3 + 2*(5*c^2*e^3*f - (18*c^2*d*e^ 2 - 11*b*c*e^3)*g)*x^2 - 5*(38*c^2*d^2*e - 26*b*c*d*e^2 + 3*b^2*e^3)*f + ( 336*c^2*d^3 - 292*b*c*d^2*e + 61*b^2*d*e^2)*g - 2*(5*(12*c^2*d*e^2 - 7*b*c *e^3)*f - (117*c^2*d^2*e - 105*b*c*d*e^2 + 23*b^2*e^3)*g)*x)*sqrt(-c*e^...
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]
\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {9}{2}}} \,d x } \]
Time = 0.54 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.61 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {120 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 60 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 240 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g + 180 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g - 30 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g + 10 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f - 30 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 10 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g - 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c g + \frac {15 \, {\left (20 \, c^{4} d^{2} e f - 20 \, b c^{3} d e^{2} f + 5 \, b^{2} c^{2} e^{3} f - 36 \, c^{4} d^{3} g + 44 \, b c^{3} d^{2} e g - 17 \, b^{2} c^{2} d e^{2} g + 2 \, b^{3} c e^{3} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {15 \, {\left (4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{2} e f - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d e^{2} f + \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} e^{3} f - 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{3} g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d^{2} e g - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} d e^{2} g\right )}}{{\left (e x + d\right )} c}}{15 \, c e^{2}} \]
-1/15*(120*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d*e*f - 60*sqrt(-(e*x + d) *c + 2*c*d - b*e)*b*c^2*e^2*f - 240*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d ^2*g + 180*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d*e*g - 30*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c*e^2*g + 10*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^ 2*e*f - 30*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d*g + 10*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c*e*g - 6*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*g + 15*(20*c^4*d^2*e*f - 20*b*c^3*d*e^2*f + 5*b^2*c ^2*e^3*f - 36*c^4*d^3*g + 44*b*c^3*d^2*e*g - 17*b^2*c^2*d*e^2*g + 2*b^3*c* e^3*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/sqrt(-2 *c*d + b*e) + 15*(4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^2*e*f - 4*sqrt( -(e*x + d)*c + 2*c*d - b*e)*b*c^3*d*e^2*f + sqrt(-(e*x + d)*c + 2*c*d - b* e)*b^2*c^2*e^3*f - 4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^3*g + 4*sqrt(- (e*x + d)*c + 2*c*d - b*e)*b*c^3*d^2*e*g - sqrt(-(e*x + d)*c + 2*c*d - b*e )*b^2*c^2*d*e^2*g)/((e*x + d)*c))/(c*e^2)
Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]