Integrand size = 46, antiderivative size = 337 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 (2 c d-b e)^3 (c e f+c d g-b e g) \sqrt {d+e x}}{c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (2 c d-b e)^2 (3 c e f+5 c d g-4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^5 e^2 \sqrt {d+e x}}-\frac {2 (2 c d-b e) (c e f+3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{c^5 e^2 (d+e x)^{3/2}}+\frac {2 (c e f+7 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^5 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{7 c^5 e^2 (d+e x)^{7/2}} \] Output:
2*(-b*e+2*c*d)^3*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(1/2)/c^5/e^2/(d*(-b*e+c*d)- b*e^2*x-c*e^2*x^2)^(1/2)+2*(-b*e+2*c*d)^2*(-4*b*e*g+5*c*d*g+3*c*e*f)*(d*(- b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c^5/e^2/(e*x+d)^(1/2)-2*(-b*e+2*c*d)*(-2 *b*e*g+3*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^5/e^2/(e*x+ d)^(3/2)+2/5*(-4*b*e*g+7*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/ 2)/c^5/e^2/(e*x+d)^(5/2)-2/7*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/c^5/ e^2/(e*x+d)^(7/2)
Time = 0.29 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (-128 b^4 e^4 g+16 b^3 c e^3 (7 e f+53 d g-4 e g x)-8 b^2 c^2 e^2 \left (257 d^2 g+d e (77 f-45 g x)-e^2 x (7 f+2 g x)\right )-2 b c^3 e \left (-1075 d^3 g+e^3 x^2 (7 f+4 g x)+d e^2 x (126 f+37 g x)+d^2 e (-553 f+334 g x)\right )+c^4 \left (-814 d^4 g+e^4 x^3 (7 f+5 g x)+d e^3 x^2 (49 f+29 g x)+d^2 e^2 x (301 f+93 g x)+d^3 e (-637 f+407 g x)\right )\right )}{35 c^5 e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:
Integrate[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2 )^(3/2),x]
Output:
(-2*Sqrt[d + e*x]*(-128*b^4*e^4*g + 16*b^3*c*e^3*(7*e*f + 53*d*g - 4*e*g*x ) - 8*b^2*c^2*e^2*(257*d^2*g + d*e*(77*f - 45*g*x) - e^2*x*(7*f + 2*g*x)) - 2*b*c^3*e*(-1075*d^3*g + e^3*x^2*(7*f + 4*g*x) + d*e^2*x*(126*f + 37*g*x ) + d^2*e*(-553*f + 334*g*x)) + c^4*(-814*d^4*g + e^4*x^3*(7*f + 5*g*x) + d*e^3*x^2*(49*f + 29*g*x) + d^2*e^2*x*(301*f + 93*g*x) + d^3*e*(-637*f + 4 07*g*x))))/(35*c^5*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
Time = 1.01 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1218, 1128, 1128, 1128, 1122}
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.
| \(\displaystyle \int \frac {(d+e x)^{9/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1218 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-8 b e g+9 c d g+7 c e f) \int \frac {(d+e x)^{7/2}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-8 b e g+9 c d g+7 c e f) \left (\frac {6 (2 c d-b e) \int \frac {(d+e x)^{5/2}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{7 c}-\frac {2 (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e}\right )}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-8 b e g+9 c d g+7 c e f) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \frac {(d+e x)^{3/2}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right )}{7 c}-\frac {2 (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e}\right )}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {(-8 b e g+9 c d g+7 c e f) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{3 c}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right )}{7 c}-\frac {2 (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e}\right )}{c e (2 c d-b e)}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (d+e x)^{9/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {\left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (-\frac {4 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c^2 e \sqrt {d+e x}}-\frac {2 \sqrt {d+e x} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e}\right )}{5 c}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c e}\right )}{7 c}-\frac {2 (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e}\right ) (-8 b e g+9 c d g+7 c e f)}{c e (2 c d-b e)}\) |
Input:
Int[((d + e*x)^(9/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2 ),x]
Output:
(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(9/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c *d - b*e) - b*e^2*x - c*e^2*x^2]) - ((7*c*e*f + 9*c*d*g - 8*b*e*g)*((-2*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*c*e) + (6*(2*c *d - b*e)*((-2*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/ (5*c*e) + (4*(2*c*d - b*e)*((-4*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(3*c^2*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*Sqrt[d*(c*d - b* e) - b*e^2*x - c*e^2*x^2])/(3*c*e)))/(5*c)))/(7*c)))/(c*e*(2*c*d - b*e))
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) 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] && IGtQ[Simplify[m + p], 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + ( c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(( a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(2*c*d - b*e))), x] - Simp[e*((m*(g*(c* d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g))/(c*(p + 1)*(2*c*d - b*e))) I nt[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d , e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 1.57 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (-5 g \,e^{4} x^{4} c^{4}+8 b \,c^{3} e^{4} g \,x^{3}-29 c^{4} d \,e^{3} g \,x^{3}-7 c^{4} e^{4} f \,x^{3}-16 b^{2} c^{2} e^{4} g \,x^{2}+74 b \,c^{3} d \,e^{3} g \,x^{2}+14 b \,c^{3} e^{4} f \,x^{2}-93 c^{4} d^{2} e^{2} g \,x^{2}-49 c^{4} d \,e^{3} f \,x^{2}+64 b^{3} c \,e^{4} g x -360 b^{2} c^{2} d \,e^{3} g x -56 b^{2} c^{2} e^{4} f x +668 b \,c^{3} d^{2} e^{2} g x +252 b \,c^{3} d \,e^{3} f x -407 c^{4} d^{3} e g x -301 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -848 b^{3} c d \,e^{3} g -112 b^{3} c \,e^{4} f +2056 b^{2} c^{2} d^{2} e^{2} g +616 b^{2} c^{2} d \,e^{3} f -2150 b \,c^{3} d^{3} e g -1106 b \,c^{3} d^{2} e^{2} f +814 c^{4} d^{4} g +637 d^{3} f \,c^{4} e \right )}{35 \sqrt {e x +d}\, \left (c e x +b e -c d \right ) c^{5} e^{2}}\) | \(361\) |
| gosper | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-5 g \,e^{4} x^{4} c^{4}+8 b \,c^{3} e^{4} g \,x^{3}-29 c^{4} d \,e^{3} g \,x^{3}-7 c^{4} e^{4} f \,x^{3}-16 b^{2} c^{2} e^{4} g \,x^{2}+74 b \,c^{3} d \,e^{3} g \,x^{2}+14 b \,c^{3} e^{4} f \,x^{2}-93 c^{4} d^{2} e^{2} g \,x^{2}-49 c^{4} d \,e^{3} f \,x^{2}+64 b^{3} c \,e^{4} g x -360 b^{2} c^{2} d \,e^{3} g x -56 b^{2} c^{2} e^{4} f x +668 b \,c^{3} d^{2} e^{2} g x +252 b \,c^{3} d \,e^{3} f x -407 c^{4} d^{3} e g x -301 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -848 b^{3} c d \,e^{3} g -112 b^{3} c \,e^{4} f +2056 b^{2} c^{2} d^{2} e^{2} g +616 b^{2} c^{2} d \,e^{3} f -2150 b \,c^{3} d^{3} e g -1106 b \,c^{3} d^{2} e^{2} f +814 c^{4} d^{4} g +637 d^{3} f \,c^{4} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{35 c^{5} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(367\) |
| orering | \(-\frac {2 \left (c e x +b e -c d \right ) \left (-5 g \,e^{4} x^{4} c^{4}+8 b \,c^{3} e^{4} g \,x^{3}-29 c^{4} d \,e^{3} g \,x^{3}-7 c^{4} e^{4} f \,x^{3}-16 b^{2} c^{2} e^{4} g \,x^{2}+74 b \,c^{3} d \,e^{3} g \,x^{2}+14 b \,c^{3} e^{4} f \,x^{2}-93 c^{4} d^{2} e^{2} g \,x^{2}-49 c^{4} d \,e^{3} f \,x^{2}+64 b^{3} c \,e^{4} g x -360 b^{2} c^{2} d \,e^{3} g x -56 b^{2} c^{2} e^{4} f x +668 b \,c^{3} d^{2} e^{2} g x +252 b \,c^{3} d \,e^{3} f x -407 c^{4} d^{3} e g x -301 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -848 b^{3} c d \,e^{3} g -112 b^{3} c \,e^{4} f +2056 b^{2} c^{2} d^{2} e^{2} g +616 b^{2} c^{2} d \,e^{3} f -2150 b \,c^{3} d^{3} e g -1106 b \,c^{3} d^{2} e^{2} f +814 c^{4} d^{4} g +637 d^{3} f \,c^{4} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{35 c^{5} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}}}\) | \(367\) |
Input:
int((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x,method= _RETURNVERBOSE)
Output:
-2/35/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-5*c^4*e^4*g*x^4+8*b *c^3*e^4*g*x^3-29*c^4*d*e^3*g*x^3-7*c^4*e^4*f*x^3-16*b^2*c^2*e^4*g*x^2+74* b*c^3*d*e^3*g*x^2+14*b*c^3*e^4*f*x^2-93*c^4*d^2*e^2*g*x^2-49*c^4*d*e^3*f*x ^2+64*b^3*c*e^4*g*x-360*b^2*c^2*d*e^3*g*x-56*b^2*c^2*e^4*f*x+668*b*c^3*d^2 *e^2*g*x+252*b*c^3*d*e^3*f*x-407*c^4*d^3*e*g*x-301*c^4*d^2*e^2*f*x+128*b^4 *e^4*g-848*b^3*c*d*e^3*g-112*b^3*c*e^4*f+2056*b^2*c^2*d^2*e^2*g+616*b^2*c^ 2*d*e^3*f-2150*b*c^3*d^3*e*g-1106*b*c^3*d^2*e^2*f+814*c^4*d^4*g+637*c^4*d^ 3*e*f)/(c*e*x+b*e-c*d)/c^5/e^2
Time = 0.09 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (5 \, c^{4} e^{4} g x^{4} + {\left (7 \, c^{4} e^{4} f + {\left (29 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} g\right )} x^{3} + {\left (7 \, {\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} f + {\left (93 \, c^{4} d^{2} e^{2} - 74 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} - 7 \, {\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} f - 2 \, {\left (407 \, c^{4} d^{4} - 1075 \, b c^{3} d^{3} e + 1028 \, b^{2} c^{2} d^{2} e^{2} - 424 \, b^{3} c d e^{3} + 64 \, b^{4} e^{4}\right )} g + {\left (7 \, {\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} f + {\left (407 \, c^{4} d^{3} e - 668 \, b c^{3} d^{2} e^{2} + 360 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{35 \, {\left (c^{6} e^{4} x^{2} + b c^{5} e^{4} x - c^{6} d^{2} e^{2} + b c^{5} d e^{3}\right )}} \] Input:
integrate((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="fricas")
Output:
2/35*(5*c^4*e^4*g*x^4 + (7*c^4*e^4*f + (29*c^4*d*e^3 - 8*b*c^3*e^4)*g)*x^3 + (7*(7*c^4*d*e^3 - 2*b*c^3*e^4)*f + (93*c^4*d^2*e^2 - 74*b*c^3*d*e^3 + 1 6*b^2*c^2*e^4)*g)*x^2 - 7*(91*c^4*d^3*e - 158*b*c^3*d^2*e^2 + 88*b^2*c^2*d *e^3 - 16*b^3*c*e^4)*f - 2*(407*c^4*d^4 - 1075*b*c^3*d^3*e + 1028*b^2*c^2* d^2*e^2 - 424*b^3*c*d*e^3 + 64*b^4*e^4)*g + (7*(43*c^4*d^2*e^2 - 36*b*c^3* d*e^3 + 8*b^2*c^2*e^4)*f + (407*c^4*d^3*e - 668*b*c^3*d^2*e^2 + 360*b^2*c^ 2*d*e^3 - 64*b^3*c*e^4)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*s qrt(e*x + d)/(c^6*e^4*x^2 + b*c^5*e^4*x - c^6*d^2*e^2 + b*c^5*d*e^3)
Timed out. \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x+d)**(9/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/ 2),x)
Output:
Timed out
Time = 0.09 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (c^{3} e^{3} x^{3} - 91 \, c^{3} d^{3} + 158 \, b c^{2} d^{2} e - 88 \, b^{2} c d e^{2} + 16 \, b^{3} e^{3} + {\left (7 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} + {\left (43 \, c^{3} d^{2} e - 36 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{5 \, \sqrt {-c e x + c d - b e} c^{4} e} - \frac {2 \, {\left (5 \, c^{4} e^{4} x^{4} - 814 \, c^{4} d^{4} + 2150 \, b c^{3} d^{3} e - 2056 \, b^{2} c^{2} d^{2} e^{2} + 848 \, b^{3} c d e^{3} - 128 \, b^{4} e^{4} + {\left (29 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} + {\left (93 \, c^{4} d^{2} e^{2} - 74 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (407 \, c^{4} d^{3} e - 668 \, b c^{3} d^{2} e^{2} + 360 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{35 \, \sqrt {-c e x + c d - b e} c^{5} e^{2}} \] Input:
integrate((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="maxima")
Output:
-2/5*(c^3*e^3*x^3 - 91*c^3*d^3 + 158*b*c^2*d^2*e - 88*b^2*c*d*e^2 + 16*b^3 *e^3 + (7*c^3*d*e^2 - 2*b*c^2*e^3)*x^2 + (43*c^3*d^2*e - 36*b*c^2*d*e^2 + 8*b^2*c*e^3)*x)*f/(sqrt(-c*e*x + c*d - b*e)*c^4*e) - 2/35*(5*c^4*e^4*x^4 - 814*c^4*d^4 + 2150*b*c^3*d^3*e - 2056*b^2*c^2*d^2*e^2 + 848*b^3*c*d*e^3 - 128*b^4*e^4 + (29*c^4*d*e^3 - 8*b*c^3*e^4)*x^3 + (93*c^4*d^2*e^2 - 74*b*c ^3*d*e^3 + 16*b^2*c^2*e^4)*x^2 + (407*c^4*d^3*e - 668*b*c^3*d^2*e^2 + 360* b^2*c^2*d*e^3 - 64*b^3*c*e^4)*x)*g/(sqrt(-c*e*x + c*d - b*e)*c^5*e^2)
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (313) = 626\).
Time = 0.36 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.05 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (8 \, c^{4} d^{3} e f - 12 \, b c^{3} d^{2} e^{2} f + 6 \, b^{2} c^{2} d e^{3} f - b^{3} c e^{4} f + 8 \, c^{4} d^{4} g - 20 \, b c^{3} d^{3} e g + 18 \, b^{2} c^{2} d^{2} e^{2} g - 7 \, b^{3} c d e^{3} g + b^{4} e^{4} g\right )}}{\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{5} e} + \frac {420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{33} d^{2} e^{7} f - 420 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{32} d e^{8} f + 105 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{31} e^{9} f + 700 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{33} d^{3} e^{6} g - 1260 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{32} d^{2} e^{7} g + 735 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{31} d e^{8} g - 140 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{3} c^{30} e^{9} g - 70 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{32} d e^{7} f + 35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{31} e^{8} f - 210 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{32} d^{2} e^{6} g + 245 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{31} d e^{7} g - 70 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} c^{30} e^{8} g + 7 \, {\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^{31} e^{7} f + 49 \, {\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^{31} d e^{6} g - 28 \, {\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} b c^{30} e^{7} g + 5 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{30} e^{6} g}{c^{35} e^{7}}\right )}}{35 \, e} \] Input:
integrate((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x, algorithm="giac")
Output:
2/35*(35*(8*c^4*d^3*e*f - 12*b*c^3*d^2*e^2*f + 6*b^2*c^2*d*e^3*f - b^3*c*e ^4*f + 8*c^4*d^4*g - 20*b*c^3*d^3*e*g + 18*b^2*c^2*d^2*e^2*g - 7*b^3*c*d*e ^3*g + b^4*e^4*g)/(sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^5*e) + (420*sqrt(-(e *x + d)*c + 2*c*d - b*e)*c^33*d^2*e^7*f - 420*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^32*d*e^8*f + 105*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^31*e^9*f + 700*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^33*d^3*e^6*g - 1260*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^32*d^2*e^7*g + 735*sqrt(-(e*x + d)*c + 2*c*d - b*e )*b^2*c^31*d*e^8*g - 140*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^30*e^9*g - 70*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^32*d*e^7*f + 35*(-(e*x + d)*c + 2 *c*d - b*e)^(3/2)*b*c^31*e^8*f - 210*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^ 32*d^2*e^6*g + 245*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^31*d*e^7*g - 70* (-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^30*e^8*g + 7*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^31*e^7*f + 49*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^31*d*e^6*g - 28*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^30*e^7*g + 5*(( e*x + d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^30*e^6*g)/( c^35*e^7))/e
Time = 11.35 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.18 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {2\,g\,x^4\,\sqrt {d+e\,x}}{7\,c^2}-\frac {\sqrt {d+e\,x}\,\left (256\,g\,b^4\,e^4-1696\,g\,b^3\,c\,d\,e^3-224\,f\,b^3\,c\,e^4+4112\,g\,b^2\,c^2\,d^2\,e^2+1232\,f\,b^2\,c^2\,d\,e^3-4300\,g\,b\,c^3\,d^3\,e-2212\,f\,b\,c^3\,d^2\,e^2+1628\,g\,c^4\,d^4+1274\,f\,c^4\,d^3\,e\right )}{35\,c^6\,e^4}+\frac {x^2\,\sqrt {d+e\,x}\,\left (32\,g\,b^2\,c^2\,e^4-148\,g\,b\,c^3\,d\,e^3-28\,f\,b\,c^3\,e^4+186\,g\,c^4\,d^2\,e^2+98\,f\,c^4\,d\,e^3\right )}{35\,c^6\,e^4}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (29\,c\,d\,g-8\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^3\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,g\,b^3\,c\,e^4+720\,g\,b^2\,c^2\,d\,e^3+112\,f\,b^2\,c^2\,e^4-1336\,g\,b\,c^3\,d^2\,e^2-504\,f\,b\,c^3\,d\,e^3+814\,g\,c^4\,d^3\,e+602\,f\,c^4\,d^2\,e^2\right )}{35\,c^6\,e^4}\right )}{x^2+\frac {b\,x}{c}+\frac {d\,\left (b\,e-c\,d\right )}{c\,e^2}} \] Input:
int(((f + g*x)*(d + e*x)^(9/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2 ),x)
Output:
((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*((2*g*x^4*(d + e*x)^(1/2))/(7 *c^2) - ((d + e*x)^(1/2)*(256*b^4*e^4*g + 1628*c^4*d^4*g - 224*b^3*c*e^4*f + 1274*c^4*d^3*e*f - 4300*b*c^3*d^3*e*g - 1696*b^3*c*d*e^3*g - 2212*b*c^3 *d^2*e^2*f + 1232*b^2*c^2*d*e^3*f + 4112*b^2*c^2*d^2*e^2*g))/(35*c^6*e^4) + (x^2*(d + e*x)^(1/2)*(32*b^2*c^2*e^4*g + 186*c^4*d^2*e^2*g - 28*b*c^3*e^ 4*f + 98*c^4*d*e^3*f - 148*b*c^3*d*e^3*g))/(35*c^6*e^4) + (2*x^3*(d + e*x) ^(1/2)*(29*c*d*g - 8*b*e*g + 7*c*e*f))/(35*c^3*e) + (x*(d + e*x)^(1/2)*(11 2*b^2*c^2*e^4*f + 602*c^4*d^2*e^2*f - 128*b^3*c*e^4*g + 814*c^4*d^3*e*g - 504*b*c^3*d*e^3*f - 1336*b*c^3*d^2*e^2*g + 720*b^2*c^2*d*e^3*g))/(35*c^6*e ^4)))/(x^2 + (b*x)/c + (d*(b*e - c*d))/(c*e^2))
Time = 0.21 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^{9/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {1628}{35} c^{4} d^{4} g -\frac {1696}{35} b^{3} c d \,e^{3} g +\frac {4112}{35} b^{2} c^{2} d^{2} e^{2} g +\frac {176}{5} b^{2} c^{2} d \,e^{3} f -\frac {16}{5} b^{2} c^{2} e^{4} f x -\frac {860}{7} b \,c^{3} d^{3} e g -\frac {14}{5} c^{4} d \,e^{3} f \,x^{2}+\frac {256}{35} b^{4} e^{4} g -\frac {2}{7} c^{4} e^{4} g \,x^{4}+\frac {128}{35} b^{3} c \,e^{4} g x -\frac {32}{35} b^{2} c^{2} e^{4} g \,x^{2}-\frac {316}{5} b \,c^{3} d^{2} e^{2} f +\frac {4}{5} b \,c^{3} e^{4} f \,x^{2}+\frac {16}{35} b \,c^{3} e^{4} g \,x^{3}-\frac {814}{35} c^{4} d^{3} e g x -\frac {186}{35} c^{4} d^{2} e^{2} g \,x^{2}-\frac {58}{35} c^{4} d \,e^{3} g \,x^{3}-\frac {32}{5} b^{3} c \,e^{4} f +\frac {182}{5} c^{4} d^{3} e f -\frac {2}{5} c^{4} e^{4} f \,x^{3}-\frac {86}{5} c^{4} d^{2} e^{2} f x -\frac {144}{7} b^{2} c^{2} d \,e^{3} g x +\frac {1336}{35} b \,c^{3} d^{2} e^{2} g x +\frac {72}{5} b \,c^{3} d \,e^{3} f x +\frac {148}{35} b \,c^{3} d \,e^{3} g \,x^{2}}{\sqrt {-c e x -b e +c d}\, c^{5} e^{2}} \] Input:
int((e*x+d)^(9/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
Output:
(2*(128*b**4*e**4*g - 848*b**3*c*d*e**3*g - 112*b**3*c*e**4*f + 64*b**3*c* e**4*g*x + 2056*b**2*c**2*d**2*e**2*g + 616*b**2*c**2*d*e**3*f - 360*b**2* c**2*d*e**3*g*x - 56*b**2*c**2*e**4*f*x - 16*b**2*c**2*e**4*g*x**2 - 2150* b*c**3*d**3*e*g - 1106*b*c**3*d**2*e**2*f + 668*b*c**3*d**2*e**2*g*x + 252 *b*c**3*d*e**3*f*x + 74*b*c**3*d*e**3*g*x**2 + 14*b*c**3*e**4*f*x**2 + 8*b *c**3*e**4*g*x**3 + 814*c**4*d**4*g + 637*c**4*d**3*e*f - 407*c**4*d**3*e* g*x - 301*c**4*d**2*e**2*f*x - 93*c**4*d**2*e**2*g*x**2 - 49*c**4*d*e**3*f *x**2 - 29*c**4*d*e**3*g*x**3 - 7*c**4*e**4*f*x**3 - 5*c**4*e**4*g*x**4))/ (35*sqrt( - b*e + c*d - c*e*x)*c**5*e**2)