Integrand size = 46, antiderivative size = 378 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {e f-d g}{2 e^2 (2 c d-b e) \sqrt {d+e x} \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac {(7 c e f+c d g-4 b e g) \sqrt {d+e x}}{6 e^2 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {5 (7 c e f+c d g-4 b e g)}{12 e^2 (2 c d-b e)^3 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {5 c (7 c e f+c d g-4 b e g) \sqrt {d+e x}}{4 e^2 (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {5 c (7 c e f+c d g-4 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{9/2}} \]
-5/4*c*(-4*b*e*g+c*d*g+7*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)^(9/2)+1/2*(d*g-e*f )/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/(e*x+d)^(1/2)+1/ 6*(-4*b*e*g+c*d*g+7*c*e*f)*(e*x+d)^(1/2)/e^2/(-b*e+2*c*d)^2/(d*(-b*e+c*d)- b*e^2*x-c*e^2*x^2)^(3/2)-5/12*(-4*b*e*g+c*d*g+7*c*e*f)/e^2/(-b*e+2*c*d)^3/ (e*x+d)^(1/2)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+5/4*c*(-4*b*e*g+c*d*g +7*c*e*f)*(e*x+d)^(1/2)/e^2/(-b*e+2*c*d)^4/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2 )^(1/2)
Time = 1.07 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.87 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {c (d+e x)^{5/2} \left (\frac {(-b e+c (d-e x)) \left (6 b^3 e^3 (d g+e (f+2 g x))+c^3 \left (61 d^4 g-105 e^4 f x^3+d^2 e^2 x (161 f-5 g x)-5 d e^3 x^2 (7 f+3 g x)+d^3 e (43 f+23 g x)\right )-4 b c^2 e \left (33 d^3 g+49 d e^2 f x+5 e^3 x^2 (7 f-3 g x)+d^2 e (-4 f+30 g x)\right )+b^2 c e^2 \left (65 d^2 g+e^2 x (-21 f+80 g x)+d e (-57 f+109 g x)\right )\right )}{c (-2 c d+b e)^4 (d+e x)^2}+\frac {15 (7 c e f+c d g-4 b e g) (-b e+c (d-e x))^{5/2} \arctan \left (\frac {\sqrt {-b e+c (d-e x)}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{9/2}}\right )}{12 e^2 ((d+e x) (-b e+c (d-e x)))^{5/2}} \]
(c*(d + e*x)^(5/2)*(((-(b*e) + c*(d - e*x))*(6*b^3*e^3*(d*g + e*(f + 2*g*x )) + c^3*(61*d^4*g - 105*e^4*f*x^3 + d^2*e^2*x*(161*f - 5*g*x) - 5*d*e^3*x ^2*(7*f + 3*g*x) + d^3*e*(43*f + 23*g*x)) - 4*b*c^2*e*(33*d^3*g + 49*d*e^2 *f*x + 5*e^3*x^2*(7*f - 3*g*x) + d^2*e*(-4*f + 30*g*x)) + b^2*c*e^2*(65*d^ 2*g + e^2*x*(-21*f + 80*g*x) + d*e*(-57*f + 109*g*x))))/(c*(-2*c*d + b*e)^ 4*(d + e*x)^2) + (15*(7*c*e*f + c*d*g - 4*b*e*g)*(-(b*e) + c*(d - e*x))^(5 /2)*ArcTan[Sqrt[-(b*e) + c*(d - e*x)]/Sqrt[-2*c*d + b*e]])/(-2*c*d + b*e)^ (9/2)))/(12*e^2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))
Time = 0.64 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1220, 1132, 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.
| \(\displaystyle \int \frac {f+g x}{\sqrt {d+e x} \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-4 b e g+c d g+7 c e f) \int \frac {\sqrt {d+e x}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}dx}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {(-4 b e g+c d g+7 c e f) \left (\frac {5 \int \frac {1}{\sqrt {d+e x} \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1135 |
| \(\displaystyle \frac {(-4 b e g+c d g+7 c e f) \left (\frac {5 \left (\frac {3 c \int \frac {\sqrt {d+e x}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1132 |
| \(\displaystyle \frac {(-4 b e g+c d g+7 c e f) \left (\frac {5 \left (\frac {3 c \left (\frac {\int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1136 |
| \(\displaystyle \frac {(-4 b e g+c d g+7 c e f) \left (\frac {5 \left (\frac {3 c \left (\frac {2 e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}}{2 c d-b e}+\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right )}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {5 \left (\frac {3 c \left (\frac {2 \sqrt {d+e x}}{e (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e (2 c d-b e)^{3/2}}\right )}{2 (2 c d-b e)}-\frac {1}{e \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 (2 c d-b e)}+\frac {2 \sqrt {d+e x}}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\right ) (-4 b e g+c d g+7 c e f)}{4 e (2 c d-b e)}-\frac {e f-d g}{2 e^2 \sqrt {d+e x} (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}\) |
-1/2*(e*f - d*g)/(e^2*(2*c*d - b*e)*Sqrt[d + e*x]*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + ((7*c*e*f + c*d*g - 4*b*e*g)*((2*Sqrt[d + e*x])/(3* e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) + (5*(-(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*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]) - (2*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*c*d - b*e)^(3/2))))/(2*(2*c*d - b*e))))/( 3*(2*c*d - b*e))))/(4*e*(2*c*d - b*e))
3.23.83.3.1 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. \(1519\) vs. \(2(344)=688\).
Time = 0.37 (sec) , antiderivative size = 1520, normalized size of antiderivative = 4.02
1/12*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-105*(-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^3*e^4*f*x^3+60*(b*e-2*c*d)^(1/ 2)*b*c^2*e^4*g*x^3-15*(b*e-2*c*d)^(1/2)*c^3*d*e^3*g*x^3+80*(b*e-2*c*d)^(1/ 2)*b^2*c*e^4*g*x^2-140*(b*e-2*c*d)^(1/2)*b*c^2*e^4*f*x^2-5*(b*e-2*c*d)^(1/ 2)*c^3*d^2*e^2*g*x^2-35*(b*e-2*c*d)^(1/2)*c^3*d*e^3*f*x^2+105*(-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^3*d^3*e*f-21 *(b*e-2*c*d)^(1/2)*b^2*c*e^4*f*x+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^3*d^4*g+12*(b*e-2*c*d)^(1/2)*b^3*e^4*g *x+6*(b*e-2*c*d)^(1/2)*b^3*d*e^3*g+43*(b*e-2*c*d)^(1/2)*c^3*d^3*e*f+60*arc tan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c*d^2*e^2*g*(-c*e*x-b*e+ c*d)^(1/2)-105*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*(-c*e*x-b*e+c*d)^(1/2)+6*(b*e-2*c*d)^(1/2)*b^3*e^4*f+61*(b*e-2*c*d)^ (1/2)*c^3*d^4*g+60*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b^2*c* e^4*g*x^2*(-c*e*x-b*e+c*d)^(1/2)-105*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2* c*d)^(1/2))*b*c^2*e^4*f*x^2*(-c*e*x-b*e+c*d)^(1/2)+45*(-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^2*d*e^3*g*x^2-90*( -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 ^2*d^2*e^2*g*x+60*(-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^2*e^4*g*x^3-15*(-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^3*d*e^3*g*x^3-15*(-c*e*x-b*e+c*d)^(1...
Leaf count of result is larger than twice the leaf count of optimal. 1532 vs. \(2 (344) = 688\).
Time = 2.58 (sec) , antiderivative size = 3096, normalized size of antiderivative = 8.19 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[-1/24*(15*((7*c^4*e^6*f + (c^4*d*e^5 - 4*b*c^3*e^6)*g)*x^5 + (7*(c^4*d*e^ 5 + 2*b*c^3*e^6)*f + (c^4*d^2*e^4 - 2*b*c^3*d*e^5 - 8*b^2*c^2*e^6)*g)*x^4 - (7*(2*c^4*d^2*e^4 - 4*b*c^3*d*e^5 - b^2*c^2*e^6)*f + (2*c^4*d^3*e^3 - 12 *b*c^3*d^2*e^4 + 15*b^2*c^2*d*e^5 + 4*b^3*c*e^6)*g)*x^3 - (7*(2*c^4*d^3*e^ 3 - 3*b^2*c^2*d*e^5)*f + (2*c^4*d^4*e^2 - 8*b*c^3*d^3*e^3 - 3*b^2*c^2*d^2* e^4 + 12*b^3*c*d*e^5)*g)*x^2 + 7*(c^4*d^5*e - 2*b*c^3*d^4*e^2 + b^2*c^2*d^ 3*e^3)*f + (c^4*d^6 - 6*b*c^3*d^5*e + 9*b^2*c^2*d^4*e^2 - 4*b^3*c*d^3*e^3) *g + (7*(c^4*d^4*e^2 - 4*b*c^3*d^3*e^3 + 3*b^2*c^2*d^2*e^4)*f + (c^4*d^5*e - 8*b*c^3*d^4*e^2 + 19*b^2*c^2*d^3*e^3 - 12*b^3*c*d^2*e^4)*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*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)* (15*(7*(2*c^4*d*e^4 - b*c^3*e^5)*f + (2*c^4*d^2*e^3 - 9*b*c^3*d*e^4 + 4*b^ 2*c^2*e^5)*g)*x^3 + 5*(7*(2*c^4*d^2*e^3 + 7*b*c^3*d*e^4 - 4*b^2*c^2*e^5)*f + (2*c^4*d^3*e^2 - b*c^3*d^2*e^3 - 32*b^2*c^2*d*e^4 + 16*b^3*c*e^5)*g)*x^ 2 - (86*c^4*d^4*e - 11*b*c^3*d^3*e^2 - 130*b^2*c^2*d^2*e^3 + 69*b^3*c*d*e^ 4 - 6*b^4*e^5)*f - (122*c^4*d^5 - 325*b*c^3*d^4*e + 262*b^2*c^2*d^3*e^2 - 53*b^3*c*d^2*e^3 - 6*b^4*d*e^4)*g - (7*(46*c^4*d^3*e^2 - 79*b*c^3*d^2*e^3 + 22*b^2*c^2*d*e^4 + 3*b^3*c*e^5)*f + (46*c^4*d^4*e - 263*b*c^3*d^3*e^2 + 338*b^2*c^2*d^2*e^3 - 85*b^3*c*d*e^4 - 12*b^4*e^5)*g)*x)*sqrt(e*x + d))...
\[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \]
\[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int { \frac {g x + f}{{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
Time = 0.61 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.63 \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {5 \, {\left (7 \, c^{2} e f + c^{2} d g - 4 \, b c e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{4 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} \sqrt {-2 \, c d + b e}} - \frac {2 \, {\left (2 \, c^{3} d e f - b c^{2} e^{2} f + 2 \, c^{3} d^{2} g - 3 \, b c^{2} d e g + b^{2} c e^{2} g - 9 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{2} e f - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{2} d g + 6 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b c e g\right )}}{3 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}} - \frac {26 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - 13 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 10 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g - 3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c e^{2} g - 11 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} e f + 3 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c e g}{4 \, {\left (16 \, c^{4} d^{4} e^{2} - 32 \, b c^{3} d^{3} e^{3} + 24 \, b^{2} c^{2} d^{2} e^{4} - 8 \, b^{3} c d e^{5} + b^{4} e^{6}\right )} {\left (e x + d\right )}^{2} c^{2}} \]
5/4*(7*c^2*e*f + 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))/((16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2*c^2*d ^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*sqrt(-2*c*d + b*e)) - 2/3*(2*c^3*d*e*f - b*c^2*e^2*f + 2*c^3*d^2*g - 3*b*c^2*d*e*g + b^2*c*e^2*g - 9*((e*x + d)*c - 2*c*d + b*e)*c^2*e*f - 3*((e*x + d)*c - 2*c*d + b*e)*c^2*d*g + 6*((e*x + d)*c - 2*c*d + b*e)*b*c*e*g)/((16*c^4*d^4*e^2 - 32*b*c^3*d^3*e^3 + 24*b^2 *c^2*d^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*((e*x + d)*c - 2*c*d + b*e)*sqrt(- (e*x + d)*c + 2*c*d - b*e)) - 1/4*(26*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3 *d*e*f - 13*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*e^2*f - 10*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d^2*g - 3*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 - 11*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*e*f + 3*(-(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)/((16*c^4*d^4*e^2 - 32*b* c^3*d^3*e^3 + 24*b^2*c^2*d^2*e^4 - 8*b^3*c*d*e^5 + b^4*e^6)*(e*x + d)^2*c^ 2)
Timed out. \[ \int \frac {f+g x}{\sqrt {d+e x} \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\int \frac {f+g\,x}{\sqrt {d+e\,x}\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}} \,d x \]