Integrand size = 46, antiderivative size = 308 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {-e f+d g}{2 e^2 (2 c d-b e) (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c e f+3 c d g-4 b e g}{4 e^2 (2 c d-b e)^2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {3 c (5 c e f+3 c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {3 c (5 c e f+3 c d g-4 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{7/2}} \]
-3/4*c*(-4*b*e*g+3*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)^(7/2)+1/2*(d*g-e *f)/e^2/(-b*e+2*c*d)/(e*x+d)^(3/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+ 1/4*(4*b*e*g-3*c*d*g-5*c*e*f)/e^2/(-b*e+2*c*d)^2/(e*x+d)^(1/2)/(d*(-b*e+c* d)-b*e^2*x-c*e^2*x^2)^(1/2)+3/4*c*(-4*b*e*g+3*c*d*g+5*c*e*f)*(e*x+d)^(1/2) /e^2/(-b*e+2*c*d)^3/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)
Time = 0.88 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.83 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {c (d+e x)^{3/2} \left (\frac {(-b e+c (d-e x)) \left (b c e \left (-9 d^2 g+e^2 x (5 f-12 g x)+13 d e (f-g x)\right )-2 b^2 e^2 (d g+e (f+2 g x))+c^2 \left (11 d^3 g+15 e^3 f x^2-3 d^2 e (f-4 g x)+d e^2 x (20 f+9 g x)\right )\right )}{c (2 c d-b e)^3 (d+e x)^2}-\frac {3 (5 c e f+3 c d g-4 b e g) (-b e+c (d-e x))^{3/2} \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{7/2}}\right )}{4 e^2 ((d+e x) (-b e+c (d-e x)))^{3/2}} \]
(c*(d + e*x)^(3/2)*(((-(b*e) + c*(d - e*x))*(b*c*e*(-9*d^2*g + e^2*x*(5*f - 12*g*x) + 13*d*e*(f - g*x)) - 2*b^2*e^2*(d*g + e*(f + 2*g*x)) + c^2*(11* d^3*g + 15*e^3*f*x^2 - 3*d^2*e*(f - 4*g*x) + d*e^2*x*(20*f + 9*g*x))))/(c* (2*c*d - b*e)^3*(d + e*x)^2) - (3*(5*c*e*f + 3*c*d*g - 4*b*e*g)*(-(b*e) + c*(d - e*x))^(3/2)*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]])/ (-2*c*d + b*e)^(7/2)))/(4*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2))
Time = 0.54 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1220, 1135, 1132, 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.
| \(\displaystyle \int \frac {f+g x}{(d+e x)^{3/2} \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-4 b e g+3 c d g+5 c e f) \int \frac {1}{\sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {(-4 b e g+3 c d g+5 c e f) \left (\frac {3 c \int \frac {\sqrt {d+e x}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {(-4 b e g+3 c d g+5 c e f) \left (\frac {3 c \left (\frac {\int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {(-4 b e g+3 c d g+5 c e f) \left (\frac {3 c \left (\frac {2 e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {3 c \left (\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-4 b e g+3 c d g+5 c e f)}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 (d+e x)^{3/2} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
-1/2*(e*f - d*g)/(e^2*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b *e^2*x - c*e^2*x^2]) + ((5*c*e*f + 3*c*d*g - 4*b*e*g)*(-(1/(e*(2*c*d - b*e )*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])) + (3*c*((2*Sqr t[d + e*x])/(e*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*ArcTanh[Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]/(Sqrt[2*c*d - b*e]*Sq rt[d + e*x])])/(e*(2*c*d - b*e)^(3/2))))/(2*(2*c*d - b*e))))/(4*e*(2*c*d - b*e))
3.23.74.3.1 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[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ [0, m, 1] && IntegerQ[2*p]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))) Int [(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && I ntegerQ[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. \(815\) vs. \(2(280)=560\).
Time = 0.36 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.65
| method | result | size |
| default | \(-\frac {\sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (12 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}-9 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}-15 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}+24 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x -18 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -30 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x +12 \sqrt {b e -2 c d}\, b c \,e^{3} g \,x^{2}-9 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} g \,x^{2}-15 \sqrt {b e -2 c d}\, c^{2} e^{3} f \,x^{2}+12 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -9 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -15 \sqrt {-x c e -b e +c d}\, \arctan \left (\frac {\sqrt {-x c e -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +4 \sqrt {b e -2 c d}\, b^{2} e^{3} g x +13 \sqrt {b e -2 c d}\, b c d \,e^{2} g x -5 \sqrt {b e -2 c d}\, b c \,e^{3} f x -12 \sqrt {b e -2 c d}\, c^{2} d^{2} e g x -20 \sqrt {b e -2 c d}\, c^{2} d \,e^{2} f x +2 \sqrt {b e -2 c d}\, b^{2} d \,e^{2} g +2 \sqrt {b e -2 c d}\, b^{2} e^{3} f +9 \sqrt {b e -2 c d}\, b c \,d^{2} e g -13 \sqrt {b e -2 c d}\, b c d \,e^{2} f -11 \sqrt {b e -2 c d}\, c^{2} d^{3} g +3 \sqrt {b e -2 c d}\, c^{2} d^{2} e f \right )}{4 \left (e x +d \right )^{\frac {5}{2}} \left (x c e +b e -c d \right ) e^{2} \left (b e -2 c d \right )^{\frac {7}{2}}}\) | \(816\) |
-1/4/(e*x+d)^(5/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(12*(-c*e*x-b*e+c*d)^( 1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*e^3*g*x^2-9*(-c* e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d* e^2*g*x^2-15*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c *d)^(1/2))*c^2*e^3*f*x^2+24*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c*d) ^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d*e^2*g*x-18*(-c*e*x-b*e+c*d)^(1/2)*arctan(( -c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*g*x-30*(-c*e*x-b*e+c*d) ^(1/2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e^2*f*x+12*( b*e-2*c*d)^(1/2)*b*c*e^3*g*x^2-9*(b*e-2*c*d)^(1/2)*c^2*d*e^2*g*x^2-15*(b*e -2*c*d)^(1/2)*c^2*e^3*f*x^2+12*(-c*e*x-b*e+c*d)^(1/2)*arctan((-c*e*x-b*e+c *d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d^2*e*g-9*(-c*e*x-b*e+c*d)^(1/2)*arctan(( -c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3*g-15*(-c*e*x-b*e+c*d)^(1/ 2)*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f+4*(b*e-2*c *d)^(1/2)*b^2*e^3*g*x+13*(b*e-2*c*d)^(1/2)*b*c*d*e^2*g*x-5*(b*e-2*c*d)^(1/ 2)*b*c*e^3*f*x-12*(b*e-2*c*d)^(1/2)*c^2*d^2*e*g*x-20*(b*e-2*c*d)^(1/2)*c^2 *d*e^2*f*x+2*(b*e-2*c*d)^(1/2)*b^2*d*e^2*g+2*(b*e-2*c*d)^(1/2)*b^2*e^3*f+9 *(b*e-2*c*d)^(1/2)*b*c*d^2*e*g-13*(b*e-2*c*d)^(1/2)*b*c*d*e^2*f-11*(b*e-2* c*d)^(1/2)*c^2*d^3*g+3*(b*e-2*c*d)^(1/2)*c^2*d^2*e*f)/(c*e*x+b*e-c*d)/e^2/ (b*e-2*c*d)^(7/2)
Leaf count of result is larger than twice the leaf count of optimal. 972 vs. \(2 (280) = 560\).
Time = 0.56 (sec) , antiderivative size = 1976, normalized size of antiderivative = 6.42 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
[-1/8*(3*((5*c^3*e^5*f + (3*c^3*d*e^4 - 4*b*c^2*e^5)*g)*x^4 + (5*(2*c^3*d* e^4 + b*c^2*e^5)*f + (6*c^3*d^2*e^3 - 5*b*c^2*d*e^4 - 4*b^2*c*e^5)*g)*x^3 + 3*(5*b*c^2*d*e^4*f + (3*b*c^2*d^2*e^3 - 4*b^2*c*d*e^4)*g)*x^2 - 5*(c^3*d ^4*e - b*c^2*d^3*e^2)*f - (3*c^3*d^5 - 7*b*c^2*d^4*e + 4*b^2*c*d^3*e^2)*g - (5*(2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3)*f + (6*c^3*d^4*e - 17*b*c^2*d^3*e^2 + 12*b^2*c*d^2*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*sqrt(-c* e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(3*(5*(2*c^3*d*e^3 - b*c^2*e^4)*f + (6* c^3*d^2*e^2 - 11*b*c^2*d*e^3 + 4*b^2*c*e^4)*g)*x^2 - (6*c^3*d^3*e - 29*b*c ^2*d^2*e^2 + 17*b^2*c*d*e^3 - 2*b^3*e^4)*f + (22*c^3*d^4 - 29*b*c^2*d^3*e + 5*b^2*c*d^2*e^2 + 2*b^3*d*e^3)*g + (5*(8*c^3*d^2*e^2 - 2*b*c^2*d*e^3 - b ^2*c*e^4)*f + (24*c^3*d^3*e - 38*b*c^2*d^2*e^2 + 5*b^2*c*d*e^3 + 4*b^3*e^4 )*g)*x)*sqrt(e*x + d))/(16*c^5*d^8*e^2 - 48*b*c^4*d^7*e^3 + 56*b^2*c^3*d^6 *e^4 - 32*b^3*c^2*d^5*e^5 + 9*b^4*c*d^4*e^6 - b^5*d^3*e^7 - (16*c^5*d^4*e^ 6 - 32*b*c^4*d^3*e^7 + 24*b^2*c^3*d^2*e^8 - 8*b^3*c^2*d*e^9 + b^4*c*e^10)* x^4 - (32*c^5*d^5*e^5 - 48*b*c^4*d^4*e^6 + 16*b^2*c^3*d^3*e^7 + 8*b^3*c^2* d^2*e^8 - 6*b^4*c*d*e^9 + b^5*e^10)*x^3 - 3*(16*b*c^4*d^5*e^5 - 32*b^2*c^3 *d^4*e^6 + 24*b^3*c^2*d^3*e^7 - 8*b^4*c*d^2*e^8 + b^5*d*e^9)*x^2 + (32*c^5 *d^7*e^3 - 112*b*c^4*d^6*e^4 + 144*b^2*c^3*d^5*e^5 - 88*b^3*c^2*d^4*e^6...
\[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.48 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.52 \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {3 \, {\left (5 \, c^{2} e f + 3 \, c^{2} d g - 4 \, b c e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{4 \, {\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} \sqrt {-2 \, c d + b e}} + \frac {2 \, {\left (c^{2} e f + c^{2} d g - b c e g\right )}}{{\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}} - \frac {18 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 9 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 7 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g - 7 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f - {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g}{4 \, {\left (8 \, c^{3} d^{3} e^{2} - 12 \, b c^{2} d^{2} e^{3} + 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} {\left (e x + d\right )}^{2} c^{2}} \]
3/4*(5*c^2*e*f + 3*c^2*d*g - 4*b*c*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d - b*e)/sqrt(-2*c*d + b*e))/((8*c^3*d^3*e^2 - 12*b*c^2*d^2*e^3 + 6*b^2*c*d*e ^4 - b^3*e^5)*sqrt(-2*c*d + b*e)) + 2*(c^2*e*f + c^2*d*g - b*c*e*g)/((8*c^ 3*d^3*e^2 - 12*b*c^2*d^2*e^3 + 6*b^2*c*d*e^4 - b^3*e^5)*sqrt(-(e*x + d)*c + 2*c*d - b*e)) - 1/4*(18*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d*e*f - 9*s qrt(-(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 - 7*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*d*e*g + 4*sqrt (-(e*x + d)*c + 2*c*d - b*e)*b^2*c*e^2*g - 7*(-(e*x + d)*c + 2*c*d - b*e)^ (3/2)*c^2*e*f - (-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d*g + 4*(-(e*x + d) *c + 2*c*d - b*e)^(3/2)*b*c*e*g)/((8*c^3*d^3*e^2 - 12*b*c^2*d^2*e^3 + 6*b^ 2*c*d*e^4 - b^3*e^5)*(e*x + d)^2*c^2)
Timed out. \[ \int \frac {f+g x}{(d+e x)^{3/2} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {f+g\,x}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}} \,d x \]