Integrand size = 44, antiderivative size = 209 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {8 c (6 c e f+4 c d g-5 b e g) (b+2 c x)}{15 e (2 c d-b e)^4 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (e f-d g)}{5 e^2 (2 c d-b e) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (6 c e f+4 c d g-5 b e g)}{15 e^2 (2 c d-b e)^2 (d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]
8/15*c*(-5*b*e*g+4*c*d*g+6*c*e*f)*(2*c*x+b)/e/(-b*e+2*c*d)^4/(d*(-b*e+c*d) -b*e^2*x-c*e^2*x^2)^(1/2)-2/5*(-d*g+e*f)/e^2/(-b*e+2*c*d)/(e*x+d)^2/(d*(-b *e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-2/15*(-5*b*e*g+4*c*d*g+6*c*e*f)/e^2/(-b*e +2*c*d)^2/(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)
Time = 0.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.11 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \left (b^3 e^3 (3 e f+2 d g+5 e g x)+4 b c^2 e \left (4 d^3 g+2 d e^2 x (9 f-8 g x)+2 e^3 x^2 (3 f-5 g x)+7 d^2 e (3 f+g x)\right )+8 c^3 \left (d^4 g+6 e^4 f x^3+4 d e^3 x^2 (3 f+g x)+d^3 e (-6 f+2 g x)+d^2 e^2 x (3 f+8 g x)\right )-2 b^2 c e^2 \left (13 d^2 g+4 d e (3 f+8 g x)+e^2 x (3 f+10 g x)\right )\right )}{15 e^2 (-2 c d+b e)^4 (d+e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \]
(2*(b^3*e^3*(3*e*f + 2*d*g + 5*e*g*x) + 4*b*c^2*e*(4*d^3*g + 2*d*e^2*x*(9* f - 8*g*x) + 2*e^3*x^2*(3*f - 5*g*x) + 7*d^2*e*(3*f + g*x)) + 8*c^3*(d^4*g + 6*e^4*f*x^3 + 4*d*e^3*x^2*(3*f + g*x) + d^3*e*(-6*f + 2*g*x) + d^2*e^2* x*(3*f + 8*g*x)) - 2*b^2*c*e^2*(13*d^2*g + 4*d*e*(3*f + 8*g*x) + e^2*x*(3* f + 10*g*x))))/(15*e^2*(-2*c*d + b*e)^4*(d + e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 0.39 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1220, 1129, 1088}
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.
| \(\displaystyle \int \frac {f+g x}{(d+e x)^2 \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {(-5 b e g+4 c d g+6 c e f) \int \frac {1}{(d+e x) \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{5 e (2 c d-b e)}-\frac {2 (e f-d g)}{5 e^2 (d+e x)^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {(-5 b e g+4 c d g+6 c e f) \left (\frac {4 c \int \frac {1}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 (2 c d-b e)}-\frac {2}{3 e (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 e (2 c d-b e)}-\frac {2 (e f-d g)}{5 e^2 (d+e x)^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {\left (\frac {8 c (b+2 c x)}{3 (2 c d-b e)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2}{3 e (d+e x) (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-5 b e g+4 c d g+6 c e f)}{5 e (2 c d-b e)}-\frac {2 (e f-d g)}{5 e^2 (d+e x)^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\) |
(-2*(e*f - d*g))/(5*e^2*(2*c*d - b*e)*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e ^2*x - c*e^2*x^2]) + ((6*c*e*f + 4*c*d*g - 5*b*e*g)*((8*c*(b + 2*c*x))/(3* (2*c*d - b*e)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - 2/(3*e*(2*c*d - b*e)*(d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])))/(5*e*(2*c*d - b*e))
3.23.21.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
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*(Simplify[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, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 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 ]
Time = 0.60 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.82
| method | result | size |
| trager | \(-\frac {2 \left (-40 b \,c^{2} e^{4} g \,x^{3}+32 c^{3} d \,e^{3} g \,x^{3}+48 c^{3} e^{4} f \,x^{3}-20 b^{2} c \,e^{4} g \,x^{2}-64 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}+64 c^{3} d^{2} e^{2} g \,x^{2}+96 c^{3} d \,e^{3} f \,x^{2}+5 b^{3} e^{4} g x -64 b^{2} c d \,e^{3} g x -6 b^{2} c \,e^{4} f x +28 b \,c^{2} d^{2} e^{2} g x +72 b \,c^{2} d \,e^{3} f x +16 c^{3} d^{3} e g x +24 c^{3} d^{2} e^{2} f x +2 b^{3} d \,e^{3} g +3 b^{3} e^{4} f -26 b^{2} c \,d^{2} e^{2} g -24 b^{2} c d \,e^{3} f +16 b \,c^{2} d^{3} e g +84 b \,c^{2} d^{2} e^{2} f +8 c^{3} d^{4} g -48 c^{3} d^{3} e f \right ) \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}}{15 \left (x c e +b e -c d \right ) \left (b e -2 c d \right ) \left (b^{3} e^{3}-6 b^{2} c d \,e^{2}+12 b \,c^{2} d^{2} e -8 c^{3} d^{3}\right ) e^{2} \left (e x +d \right )^{3}}\) | \(380\) |
| gosper | \(-\frac {2 \left (x c e +b e -c d \right ) \left (-40 b \,c^{2} e^{4} g \,x^{3}+32 c^{3} d \,e^{3} g \,x^{3}+48 c^{3} e^{4} f \,x^{3}-20 b^{2} c \,e^{4} g \,x^{2}-64 b \,c^{2} d \,e^{3} g \,x^{2}+24 b \,c^{2} e^{4} f \,x^{2}+64 c^{3} d^{2} e^{2} g \,x^{2}+96 c^{3} d \,e^{3} f \,x^{2}+5 b^{3} e^{4} g x -64 b^{2} c d \,e^{3} g x -6 b^{2} c \,e^{4} f x +28 b \,c^{2} d^{2} e^{2} g x +72 b \,c^{2} d \,e^{3} f x +16 c^{3} d^{3} e g x +24 c^{3} d^{2} e^{2} f x +2 b^{3} d \,e^{3} g +3 b^{3} e^{4} f -26 b^{2} c \,d^{2} e^{2} g -24 b^{2} c d \,e^{3} f +16 b \,c^{2} d^{3} e g +84 b \,c^{2} d^{2} e^{2} f +8 c^{3} d^{4} g -48 c^{3} d^{3} e f \right )}{15 \left (e x +d \right ) e^{2} \left (b^{4} e^{4}-8 b^{3} c d \,e^{3}+24 b^{2} c^{2} d^{2} e^{2}-32 b \,c^{3} d^{3} e +16 c^{4} d^{4}\right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(382\) |
| default | \(\frac {g \left (-\frac {2}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c \,e^{2} \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right )}{3 \left (-b \,e^{2}+2 c d e \right )^{3} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{e^{2}}+\frac {\left (-d g +e f \right ) \left (-\frac {2}{5 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )^{2} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}+\frac {6 c \,e^{2} \left (-\frac {2}{3 \left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}-\frac {8 c \,e^{2} \left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 c d e \right )}{3 \left (-b \,e^{2}+2 c d e \right )^{3} \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 c d e \right ) \left (x +\frac {d}{e}\right )}}\right )}{5 \left (-b \,e^{2}+2 c d e \right )}\right )}{e^{3}}\) | \(397\) |
-2/15*(-40*b*c^2*e^4*g*x^3+32*c^3*d*e^3*g*x^3+48*c^3*e^4*f*x^3-20*b^2*c*e^ 4*g*x^2-64*b*c^2*d*e^3*g*x^2+24*b*c^2*e^4*f*x^2+64*c^3*d^2*e^2*g*x^2+96*c^ 3*d*e^3*f*x^2+5*b^3*e^4*g*x-64*b^2*c*d*e^3*g*x-6*b^2*c*e^4*f*x+28*b*c^2*d^ 2*e^2*g*x+72*b*c^2*d*e^3*f*x+16*c^3*d^3*e*g*x+24*c^3*d^2*e^2*f*x+2*b^3*d*e ^3*g+3*b^3*e^4*f-26*b^2*c*d^2*e^2*g-24*b^2*c*d*e^3*f+16*b*c^2*d^3*e*g+84*b *c^2*d^2*e^2*f+8*c^3*d^4*g-48*c^3*d^3*e*f)/(c*e*x+b*e-c*d)/(b*e-2*c*d)/(b^ 3*e^3-6*b^2*c*d*e^2+12*b*c^2*d^2*e-8*c^3*d^3)/e^2/(e*x+d)^3*(-c*e^2*x^2-b* e^2*x-b*d*e+c*d^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (197) = 394\).
Time = 20.59 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.11 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (8 \, {\left (6 \, c^{3} e^{4} f + {\left (4 \, c^{3} d e^{3} - 5 \, b c^{2} e^{4}\right )} g\right )} x^{3} + 4 \, {\left (6 \, {\left (4 \, c^{3} d e^{3} + b c^{2} e^{4}\right )} f + {\left (16 \, c^{3} d^{2} e^{2} - 16 \, b c^{2} d e^{3} - 5 \, b^{2} c e^{4}\right )} g\right )} x^{2} - 3 \, {\left (16 \, c^{3} d^{3} e - 28 \, b c^{2} d^{2} e^{2} + 8 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} f + 2 \, {\left (4 \, c^{3} d^{4} + 8 \, b c^{2} d^{3} e - 13 \, b^{2} c d^{2} e^{2} + b^{3} d e^{3}\right )} g + {\left (6 \, {\left (4 \, c^{3} d^{2} e^{2} + 12 \, b c^{2} d e^{3} - b^{2} c e^{4}\right )} f + {\left (16 \, c^{3} d^{3} e + 28 \, b c^{2} d^{2} e^{2} - 64 \, b^{2} c d e^{3} + 5 \, b^{3} e^{4}\right )} g\right )} x\right )}}{15 \, {\left (16 \, c^{5} d^{8} e^{2} - 48 \, b c^{4} d^{7} e^{3} + 56 \, b^{2} c^{3} d^{6} e^{4} - 32 \, b^{3} c^{2} d^{5} e^{5} + 9 \, b^{4} c d^{4} e^{6} - b^{5} d^{3} e^{7} - {\left (16 \, c^{5} d^{4} e^{6} - 32 \, b c^{4} d^{3} e^{7} + 24 \, b^{2} c^{3} d^{2} e^{8} - 8 \, b^{3} c^{2} d e^{9} + b^{4} c e^{10}\right )} x^{4} - {\left (32 \, c^{5} d^{5} e^{5} - 48 \, b c^{4} d^{4} e^{6} + 16 \, b^{2} c^{3} d^{3} e^{7} + 8 \, b^{3} c^{2} d^{2} e^{8} - 6 \, b^{4} c d e^{9} + b^{5} e^{10}\right )} x^{3} - 3 \, {\left (16 \, b c^{4} d^{5} e^{5} - 32 \, b^{2} c^{3} d^{4} e^{6} + 24 \, b^{3} c^{2} d^{3} e^{7} - 8 \, b^{4} c d^{2} e^{8} + b^{5} d e^{9}\right )} x^{2} + {\left (32 \, c^{5} d^{7} e^{3} - 112 \, b c^{4} d^{6} e^{4} + 144 \, b^{2} c^{3} d^{5} e^{5} - 88 \, b^{3} c^{2} d^{4} e^{6} + 26 \, b^{4} c d^{3} e^{7} - 3 \, b^{5} d^{2} e^{8}\right )} x\right )}} \]
2/15*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(6*c^3*e^4*f + (4*c^3*d *e^3 - 5*b*c^2*e^4)*g)*x^3 + 4*(6*(4*c^3*d*e^3 + b*c^2*e^4)*f + (16*c^3*d^ 2*e^2 - 16*b*c^2*d*e^3 - 5*b^2*c*e^4)*g)*x^2 - 3*(16*c^3*d^3*e - 28*b*c^2* d^2*e^2 + 8*b^2*c*d*e^3 - b^3*e^4)*f + 2*(4*c^3*d^4 + 8*b*c^2*d^3*e - 13*b ^2*c*d^2*e^2 + b^3*d*e^3)*g + (6*(4*c^3*d^2*e^2 + 12*b*c^2*d*e^3 - b^2*c*e ^4)*f + (16*c^3*d^3*e + 28*b*c^2*d^2*e^2 - 64*b^2*c*d*e^3 + 5*b^3*e^4)*g)* x)/(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 + 26*b^4*c*d^3*e^7 - 3 *b^5*d^2*e^8)*x)
\[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\int \frac {f + g x}{\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 7557 vs. \(2 (197) = 394\).
Time = 0.48 (sec) , antiderivative size = 7557, normalized size of antiderivative = 36.16 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
-2/15*(8*(6*c^3*e*f + 4*c^3*d*g - 5*b*c^2*e*g)*sgn(1/(e*x + d))*sgn(e)/(16 *sqrt(-c)*c^4*d^4*e - 32*b*sqrt(-c)*c^3*d^3*e^2 + 24*b^2*sqrt(-c)*c^2*d^2* e^3 - 8*b^3*sqrt(-c)*c*d*e^4 + b^4*sqrt(-c)*e^5) + (196608*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*c^16*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d ^16*e^5*f*sgn(1/(e*x + d))^4*sgn(e)^4 + 2949120*c^18*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^16*e^5*f*sgn(1/(e*x + d))^4*sgn(e)^4 + 983040*c^17 *(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d^16*e^5*f*sgn(1/(e*x + d))^ 4*sgn(e)^4 - 1572864*b*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*c^15*sqrt(- c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^15*e^6*f*sgn(1/(e*x + d))^4*sgn(e)^ 4 - 23592960*b*c^17*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^15*e^6*f* sgn(1/(e*x + d))^4*sgn(e)^4 - 7864320*b*c^16*(-c + 2*c*d/(e*x + d) - b*e/( e*x + d))^(3/2)*d^15*e^6*f*sgn(1/(e*x + d))^4*sgn(e)^4 + 5898240*b^2*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*c^14*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e *x + d))*d^14*e^7*f*sgn(1/(e*x + d))^4*sgn(e)^4 + 88473600*b^2*c^16*sqrt(- c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^14*e^7*f*sgn(1/(e*x + d))^4*sgn(e)^ 4 + 29491200*b^2*c^15*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d^14*e^ 7*f*sgn(1/(e*x + d))^4*sgn(e)^4 - 13762560*b^3*(c - 2*c*d/(e*x + d) + b*e/ (e*x + d))^2*c^13*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^13*e^8*f*sg n(1/(e*x + d))^4*sgn(e)^4 - 206438400*b^3*c^15*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^13*e^8*f*sgn(1/(e*x + d))^4*sgn(e)^4 - 68812800*b^3*c...
Time = 13.45 (sec) , antiderivative size = 2126, normalized size of antiderivative = 10.17 \[ \int \frac {f+g x}{(d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
(((d*((16*c^3*f - 16*b*c^2*g)/(15*(b*e - 2*c*d)^5) + (8*c^3*d*g)/(15*e*(b* e - 2*c*d)^5)))/e + (2*b*c*(3*b*g - 4*c*f))/(15*(b*e - 2*c*d)^5))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((4*b*c*g)/(15*e*(b*e - 2*c*d)^4) - (8*c^2*d*g)/(15*e^2*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d *e - b*e^2*x)^(1/2))/(d + e*x) + (((2*b*g)/(5*(3*b*e^2 - 6*c*d*e)*(b*e - 2 *c*d)^2) - (4*c*d*g)/(5*e*(3*b*e^2 - 6*c*d*e)*(b*e - 2*c*d)^2))*(c*d^2 - c *e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^2 + (((4*c*g*(3*b*e - 4*c*d)) /(15*e^2*(b*e - 2*c*d)^4) - (8*c^2*d*g)/(15*e^2*(b*e - 2*c*d)^4))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) + (((2*d*g)/(5*b^2*e^4 + 20 *c^2*d^2*e^2 - 20*b*c*d*e^3) - (2*e*f)/(5*b^2*e^4 + 20*c^2*d^2*e^2 - 20*b* c*d*e^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^3 + (((d* ((2*c*e*(3*b*e*g + 2*c*d*g - 4*c*e*f))/(5*(b*e - 2*c*d)^2*(3*b^2*e^4 + 12* c^2*d^2*e^2 - 12*b*c*d*e^3)) - (4*c^2*d*e*g)/(5*(b*e - 2*c*d)^2*(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3))))/e - (12*b^2*e^2*g + 12*c^2*d^2*g - 18 *b*c*e^2*f + 28*c^2*d*e*f - 24*b*c*d*e*g)/(5*(b*e - 2*c*d)^2*(3*b^2*e^4 + 12*c^2*d^2*e^2 - 12*b*c*d*e^3)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/ 2))/(d + e*x)^2 - ((x*((d*(b*e - c*d)*((16*c^5*g*(e*(b*e - c*d) + c*d*e))/ (15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b*c^2*d*e)) + (16*c^5*e*(c* d*g - 3*b*e*g + 2*c*e*f))/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2 - 4*b *c^2*d*e)) - (8*b*c^5*e^2*g)/(15*(b*e - 2*c*d)^4*(4*c^3*d^2 + b^2*c*e^2...