\(\int \frac {(d+e x)^{11/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [241]

Optimal result

Integrand size = 46, antiderivative size = 339 \[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (2 c d-b e)^3 (c e f+c d g-b e g) (d+e x)^{3/2}}{3 c^5 e^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {2 (2 c d-b e)^2 (3 c e f+5 c d g-4 b e g) \sqrt {d+e x}}{c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {6 (2 c d-b e) (c e f+3 c d g-2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c^5 e^2 \sqrt {d+e x}}+\frac {2 (c e f+7 c d g-4 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c^5 e^2 (d+e x)^{3/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 c^5 e^2 (d+e x)^{5/2}} \] Output:

2/3*(-b*e+2*c*d)^3*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(3/2)/c^5/e^2/(d*(-b*e+c*d 
)-b*e^2*x-c*e^2*x^2)^(3/2)-2*(-b*e+2*c*d)^2*(-4*b*e*g+5*c*d*g+3*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)-6*(-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)^(1/2)/c^5/e^2/(e* 
x+d)^(1/2)+2/3*(-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)^( 
3/2)/c^5/e^2/(e*x+d)^(3/2)-2/5*g*(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)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (128 b^4 e^4 g-16 b^3 c e^3 (5 e f+47 d g-12 e g x)+24 b^2 c^2 e^2 \left (67 d^2 g+3 d e (5 f-13 g x)+e^2 x (-5 f+2 g x)\right )-2 b c^3 e \left (741 d^3 g+3 d^2 e (85 f-246 g x)+e^3 x^2 (15 f+4 g x)+3 d e^2 x (-70 f+31 g x)\right )+c^4 \left (498 d^4 g+9 d^3 e (25 f-83 g x)+e^4 x^3 (5 f+3 g x)+d e^3 x^2 (75 f+23 g x)+3 d^2 e^2 x (-115 f+61 g x)\right )\right )}{15 c^5 e^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

Integrate[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^ 
2)^(5/2),x]
 

Output:

(2*Sqrt[d + e*x]*(128*b^4*e^4*g - 16*b^3*c*e^3*(5*e*f + 47*d*g - 12*e*g*x) 
 + 24*b^2*c^2*e^2*(67*d^2*g + 3*d*e*(5*f - 13*g*x) + e^2*x*(-5*f + 2*g*x)) 
 - 2*b*c^3*e*(741*d^3*g + 3*d^2*e*(85*f - 246*g*x) + e^3*x^2*(15*f + 4*g*x 
) + 3*d*e^2*x*(-70*f + 31*g*x)) + c^4*(498*d^4*g + 9*d^3*e*(25*f - 83*g*x) 
 + e^4*x^3*(5*f + 3*g*x) + d*e^3*x^2*(75*f + 23*g*x) + 3*d^2*e^2*x*(-115*f 
 + 61*g*x))))/(15*c^5*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + 
c*(d - e*x))])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.04, 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.

\(\displaystyle \int \frac {(d+e x)^{11/2} (f+g x)}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1218

\(\displaystyle \frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-8 b e g+11 c d g+5 c e f) \int \frac {(d+e x)^{9/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-8 b e g+11 c d g+5 c e f) \left (\frac {6 (2 c d-b e) \int \frac {(d+e x)^{7/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{5 c}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-8 b e g+11 c d g+5 c e f) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \int \frac {(d+e x)^{5/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 c}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {(-8 b e g+11 c d g+5 c e f) \left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {2 (2 c d-b e) \int \frac {(d+e x)^{3/2}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}dx}{c}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 c}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c e (2 c d-b e)}\)

\(\Big \downarrow \) 1122

\(\displaystyle \frac {2 (d+e x)^{11/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {\left (\frac {6 (2 c d-b e) \left (\frac {4 (2 c d-b e) \left (\frac {4 \sqrt {d+e x} (2 c d-b e)}{c^2 e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac {2 (d+e x)^{3/2}}{c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{3 c}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{5 c}-\frac {2 (d+e x)^{7/2}}{5 c e \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-8 b e g+11 c d g+5 c e f)}{3 c e (2 c d-b e)}\)

Input:

Int[((d + e*x)^(11/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/ 
2),x]
 

Output:

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(11/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c* 
d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)) - ((5*c*e*f + 11*c*d*g - 8*b*e*g)*( 
(-2*(d + e*x)^(7/2))/(5*c*e*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + ( 
6*(2*c*d - b*e)*((-2*(d + e*x)^(5/2))/(3*c*e*Sqrt[d*(c*d - b*e) - b*e^2*x 
- c*e^2*x^2]) + (4*(2*c*d - b*e)*((4*(2*c*d - b*e)*Sqrt[d + e*x])/(c^2*e*S 
qrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) - (2*(d + e*x)^(3/2))/(c*e*Sqrt[ 
d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])))/(3*c)))/(5*c)))/(3*c*e*(2*c*d - b* 
e))
 

Defintions of rubi rules used

rule 1122

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]
 

rule 1128

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]
 

rule 1218

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]
 

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.06

method	result	size
default	\(-\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (3 g \,e^{4} x^{4} c^{4}-8 b \,c^{3} e^{4} g \,x^{3}+23 c^{4} d \,e^{3} g \,x^{3}+5 c^{4} e^{4} f \,x^{3}+48 b^{2} c^{2} e^{4} g \,x^{2}-186 b \,c^{3} d \,e^{3} g \,x^{2}-30 b \,c^{3} e^{4} f \,x^{2}+183 c^{4} d^{2} e^{2} g \,x^{2}+75 c^{4} d \,e^{3} f \,x^{2}+192 b^{3} c \,e^{4} g x -936 b^{2} c^{2} d \,e^{3} g x -120 b^{2} c^{2} e^{4} f x +1476 b \,c^{3} d^{2} e^{2} g x +420 b \,c^{3} d \,e^{3} f x -747 c^{4} d^{3} e g x -345 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -752 b^{3} c d \,e^{3} g -80 b^{3} c \,e^{4} f +1608 b^{2} c^{2} d^{2} e^{2} g +360 b^{2} c^{2} d \,e^{3} f -1482 b \,c^{3} d^{3} e g -510 b \,c^{3} d^{2} e^{2} f +498 c^{4} d^{4} g +225 d^{3} f \,c^{4} e \right )}{15 \sqrt {e x +d}\, \left (c e x +b e -c d \right )^{2} c^{5} e^{2}}\)	\(361\)
gosper	\(\frac {2 \left (c e x +b e -c d \right ) \left (3 g \,e^{4} x^{4} c^{4}-8 b \,c^{3} e^{4} g \,x^{3}+23 c^{4} d \,e^{3} g \,x^{3}+5 c^{4} e^{4} f \,x^{3}+48 b^{2} c^{2} e^{4} g \,x^{2}-186 b \,c^{3} d \,e^{3} g \,x^{2}-30 b \,c^{3} e^{4} f \,x^{2}+183 c^{4} d^{2} e^{2} g \,x^{2}+75 c^{4} d \,e^{3} f \,x^{2}+192 b^{3} c \,e^{4} g x -936 b^{2} c^{2} d \,e^{3} g x -120 b^{2} c^{2} e^{4} f x +1476 b \,c^{3} d^{2} e^{2} g x +420 b \,c^{3} d \,e^{3} f x -747 c^{4} d^{3} e g x -345 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -752 b^{3} c d \,e^{3} g -80 b^{3} c \,e^{4} f +1608 b^{2} c^{2} d^{2} e^{2} g +360 b^{2} c^{2} d \,e^{3} f -1482 b \,c^{3} d^{3} e g -510 b \,c^{3} d^{2} e^{2} f +498 c^{4} d^{4} g +225 d^{3} f \,c^{4} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{15 c^{5} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(367\)
orering	\(\frac {2 \left (c e x +b e -c d \right ) \left (3 g \,e^{4} x^{4} c^{4}-8 b \,c^{3} e^{4} g \,x^{3}+23 c^{4} d \,e^{3} g \,x^{3}+5 c^{4} e^{4} f \,x^{3}+48 b^{2} c^{2} e^{4} g \,x^{2}-186 b \,c^{3} d \,e^{3} g \,x^{2}-30 b \,c^{3} e^{4} f \,x^{2}+183 c^{4} d^{2} e^{2} g \,x^{2}+75 c^{4} d \,e^{3} f \,x^{2}+192 b^{3} c \,e^{4} g x -936 b^{2} c^{2} d \,e^{3} g x -120 b^{2} c^{2} e^{4} f x +1476 b \,c^{3} d^{2} e^{2} g x +420 b \,c^{3} d \,e^{3} f x -747 c^{4} d^{3} e g x -345 c^{4} d^{2} e^{2} f x +128 b^{4} e^{4} g -752 b^{3} c d \,e^{3} g -80 b^{3} c \,e^{4} f +1608 b^{2} c^{2} d^{2} e^{2} g +360 b^{2} c^{2} d \,e^{3} f -1482 b \,c^{3} d^{3} e g -510 b \,c^{3} d^{2} e^{2} f +498 c^{4} d^{4} g +225 d^{3} f \,c^{4} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{15 c^{5} e^{2} \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(367\)

Input:

int((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method 
=_RETURNVERBOSE)
 

Output:

-2/15/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(3*c^4*e^4*g*x^4-8*b* 
c^3*e^4*g*x^3+23*c^4*d*e^3*g*x^3+5*c^4*e^4*f*x^3+48*b^2*c^2*e^4*g*x^2-186* 
b*c^3*d*e^3*g*x^2-30*b*c^3*e^4*f*x^2+183*c^4*d^2*e^2*g*x^2+75*c^4*d*e^3*f* 
x^2+192*b^3*c*e^4*g*x-936*b^2*c^2*d*e^3*g*x-120*b^2*c^2*e^4*f*x+1476*b*c^3 
*d^2*e^2*g*x+420*b*c^3*d*e^3*f*x-747*c^4*d^3*e*g*x-345*c^4*d^2*e^2*f*x+128 
*b^4*e^4*g-752*b^3*c*d*e^3*g-80*b^3*c*e^4*f+1608*b^2*c^2*d^2*e^2*g+360*b^2 
*c^2*d*e^3*f-1482*b*c^3*d^3*e*g-510*b*c^3*d^2*e^2*f+498*c^4*d^4*g+225*c^4* 
d^3*e*f)/(c*e*x+b*e-c*d)^2/c^5/e^2
 

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (3 \, c^{4} e^{4} g x^{4} + {\left (5 \, c^{4} e^{4} f + {\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} g\right )} x^{3} + 3 \, {\left (5 \, {\left (5 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} f + {\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} g\right )} x^{2} + 5 \, {\left (45 \, c^{4} d^{3} e - 102 \, b c^{3} d^{2} e^{2} + 72 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} f + 2 \, {\left (249 \, c^{4} d^{4} - 741 \, b c^{3} d^{3} e + 804 \, b^{2} c^{2} d^{2} e^{2} - 376 \, b^{3} c d e^{3} + 64 \, b^{4} e^{4}\right )} g - 3 \, {\left (5 \, {\left (23 \, c^{4} d^{2} e^{2} - 28 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} f + {\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{15 \, {\left (c^{7} e^{5} x^{3} + c^{7} d^{3} e^{2} - 2 \, b c^{6} d^{2} e^{3} + b^{2} c^{5} d e^{4} - {\left (c^{7} d e^{4} - 2 \, b c^{6} e^{5}\right )} x^{2} - {\left (c^{7} d^{2} e^{3} - b^{2} c^{5} e^{5}\right )} x\right )}} \] Input:

integrate((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, 
 algorithm="fricas")
 

Output:

-2/15*(3*c^4*e^4*g*x^4 + (5*c^4*e^4*f + (23*c^4*d*e^3 - 8*b*c^3*e^4)*g)*x^ 
3 + 3*(5*(5*c^4*d*e^3 - 2*b*c^3*e^4)*f + (61*c^4*d^2*e^2 - 62*b*c^3*d*e^3 
+ 16*b^2*c^2*e^4)*g)*x^2 + 5*(45*c^4*d^3*e - 102*b*c^3*d^2*e^2 + 72*b^2*c^ 
2*d*e^3 - 16*b^3*c*e^4)*f + 2*(249*c^4*d^4 - 741*b*c^3*d^3*e + 804*b^2*c^2 
*d^2*e^2 - 376*b^3*c*d*e^3 + 64*b^4*e^4)*g - 3*(5*(23*c^4*d^2*e^2 - 28*b*c 
^3*d*e^3 + 8*b^2*c^2*e^4)*f + (249*c^4*d^3*e - 492*b*c^3*d^2*e^2 + 312*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 
)*sqrt(e*x + d)/(c^7*e^5*x^3 + c^7*d^3*e^2 - 2*b*c^6*d^2*e^3 + b^2*c^5*d*e 
^4 - (c^7*d*e^4 - 2*b*c^6*e^5)*x^2 - (c^7*d^2*e^3 - b^2*c^5*e^5)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(11/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5 
/2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (c^{3} e^{3} x^{3} + 45 \, c^{3} d^{3} - 102 \, b c^{2} d^{2} e + 72 \, b^{2} c d e^{2} - 16 \, b^{3} e^{3} + 3 \, {\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} x^{2} - 3 \, {\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} x\right )} f}{3 \, {\left (c^{5} e^{2} x - c^{5} d e + b c^{4} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (3 \, c^{4} e^{4} x^{4} + 498 \, c^{4} d^{4} - 1482 \, b c^{3} d^{3} e + 1608 \, b^{2} c^{2} d^{2} e^{2} - 752 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + {\left (23 \, c^{4} d e^{3} - 8 \, b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (61 \, c^{4} d^{2} e^{2} - 62 \, b c^{3} d e^{3} + 16 \, b^{2} c^{2} e^{4}\right )} x^{2} - 3 \, {\left (249 \, c^{4} d^{3} e - 492 \, b c^{3} d^{2} e^{2} + 312 \, b^{2} c^{2} d e^{3} - 64 \, b^{3} c e^{4}\right )} x\right )} g}{15 \, {\left (c^{6} e^{3} x - c^{6} d e^{2} + b c^{5} e^{3}\right )} \sqrt {-c e x + c d - b e}} \] Input:

integrate((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, 
 algorithm="maxima")
 

Output:

2/3*(c^3*e^3*x^3 + 45*c^3*d^3 - 102*b*c^2*d^2*e + 72*b^2*c*d*e^2 - 16*b^3* 
e^3 + 3*(5*c^3*d*e^2 - 2*b*c^2*e^3)*x^2 - 3*(23*c^3*d^2*e - 28*b*c^2*d*e^2 
 + 8*b^2*c*e^3)*x)*f/((c^5*e^2*x - c^5*d*e + b*c^4*e^2)*sqrt(-c*e*x + c*d 
- b*e)) + 2/15*(3*c^4*e^4*x^4 + 498*c^4*d^4 - 1482*b*c^3*d^3*e + 1608*b^2* 
c^2*d^2*e^2 - 752*b^3*c*d*e^3 + 128*b^4*e^4 + (23*c^4*d*e^3 - 8*b*c^3*e^4) 
*x^3 + 3*(61*c^4*d^2*e^2 - 62*b*c^3*d*e^3 + 16*b^2*c^2*e^4)*x^2 - 3*(249*c 
^4*d^3*e - 492*b*c^3*d^2*e^2 + 312*b^2*c^2*d*e^3 - 64*b^3*c*e^4)*x)*g/((c^ 
6*e^3*x - c^6*d*e^2 + b*c^5*e^3)*sqrt(-c*e*x + c*d - b*e))
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (\frac {5 \, {\left (8 \, c^{4} d^{3} e f - 12 \, b c^{3} d^{2} e^{2} f + 6 \, b^{2} c^{2} d e^{3} f - b^{3} c e^{4} f + 8 \, c^{4} d^{4} g - 20 \, b c^{3} d^{3} e g + 18 \, b^{2} c^{2} d^{2} e^{2} g - 7 \, b^{3} c d e^{3} g + b^{4} e^{4} g + 36 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{3} d^{2} e f - 36 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b c^{2} d e^{2} f + 9 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b^{2} c e^{3} f + 60 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} c^{3} d^{3} g - 108 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b c^{2} d^{2} e g + 63 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b^{2} c d e^{2} g - 12 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} b^{3} e^{3} g\right )}}{{\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{5} e} + \frac {90 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{22} d e^{5} f - 45 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{21} e^{6} f + 270 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{22} d^{2} e^{4} g - 315 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{21} d e^{5} g + 90 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{20} e^{6} g - 5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{21} e^{5} f - 35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{21} d e^{4} g + 20 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{20} e^{5} g + 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{20} e^{4} g}{c^{25} e^{5}}\right )}}{15 \, e} \] Input:

integrate((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, 
 algorithm="giac")
 

Output:

-2/15*(5*(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 + 36*((e*x + d)*c - 2*c*d + b*e)*c^3*d^2*e*f - 36*((e*x + 
 d)*c - 2*c*d + b*e)*b*c^2*d*e^2*f + 9*((e*x + d)*c - 2*c*d + b*e)*b^2*c*e 
^3*f + 60*((e*x + d)*c - 2*c*d + b*e)*c^3*d^3*g - 108*((e*x + d)*c - 2*c*d 
 + b*e)*b*c^2*d^2*e*g + 63*((e*x + d)*c - 2*c*d + b*e)*b^2*c*d*e^2*g - 12* 
((e*x + d)*c - 2*c*d + b*e)*b^3*e^3*g)/(((e*x + d)*c - 2*c*d + b*e)*sqrt(- 
(e*x + d)*c + 2*c*d - b*e)*c^5*e) + (90*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c 
^22*d*e^5*f - 45*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^21*e^6*f + 270*sqrt( 
-(e*x + d)*c + 2*c*d - b*e)*c^22*d^2*e^4*g - 315*sqrt(-(e*x + d)*c + 2*c*d 
 - b*e)*b*c^21*d*e^5*g + 90*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^20*e^6* 
g - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^21*e^5*f - 35*(-(e*x + d)*c + 2 
*c*d - b*e)^(3/2)*c^21*d*e^4*g + 20*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c 
^20*e^5*g + 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b* 
e)*c^20*e^4*g)/(c^25*e^5))/e
 

Mupad [B] (verification not implemented)

Time = 12.35 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (256\,g\,b^4\,e^4-1504\,g\,b^3\,c\,d\,e^3-160\,f\,b^3\,c\,e^4+3216\,g\,b^2\,c^2\,d^2\,e^2+720\,f\,b^2\,c^2\,d\,e^3-2964\,g\,b\,c^3\,d^3\,e-1020\,f\,b\,c^3\,d^2\,e^2+996\,g\,c^4\,d^4+450\,f\,c^4\,d^3\,e\right )}{15\,c^7\,e^5}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (16\,g\,b^2\,e^2-62\,g\,b\,c\,d\,e-10\,f\,b\,c\,e^2+61\,g\,c^2\,d^2+25\,f\,c^2\,d\,e\right )}{5\,c^5\,e^3}+\frac {2\,x^3\,\sqrt {d+e\,x}\,\left (23\,c\,d\,g-8\,b\,e\,g+5\,c\,e\,f\right )}{15\,c^4\,e^2}+\frac {2\,g\,x^4\,\sqrt {d+e\,x}}{5\,c^3\,e}-\frac {x\,\sqrt {d+e\,x}\,\left (-384\,g\,b^3\,c\,e^4+1872\,g\,b^2\,c^2\,d\,e^3+240\,f\,b^2\,c^2\,e^4-2952\,g\,b\,c^3\,d^2\,e^2-840\,f\,b\,c^3\,d\,e^3+1494\,g\,c^4\,d^3\,e+690\,f\,c^4\,d^2\,e^2\right )}{15\,c^7\,e^5}\right )}{x^3+\frac {x\,\left (15\,b^2\,c^5\,e^5-15\,c^7\,d^2\,e^3\right )}{15\,c^7\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \] Input:

int(((f + g*x)*(d + e*x)^(11/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/ 
2),x)
 

Output:

-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*(((d + e*x)^(1/2)*(256*b^4*e 
^4*g + 996*c^4*d^4*g - 160*b^3*c*e^4*f + 450*c^4*d^3*e*f - 2964*b*c^3*d^3* 
e*g - 1504*b^3*c*d*e^3*g - 1020*b*c^3*d^2*e^2*f + 720*b^2*c^2*d*e^3*f + 32 
16*b^2*c^2*d^2*e^2*g))/(15*c^7*e^5) + (2*x^2*(d + e*x)^(1/2)*(16*b^2*e^2*g 
 + 61*c^2*d^2*g - 10*b*c*e^2*f + 25*c^2*d*e*f - 62*b*c*d*e*g))/(5*c^5*e^3) 
 + (2*x^3*(d + e*x)^(1/2)*(23*c*d*g - 8*b*e*g + 5*c*e*f))/(15*c^4*e^2) + ( 
2*g*x^4*(d + e*x)^(1/2))/(5*c^3*e) - (x*(d + e*x)^(1/2)*(240*b^2*c^2*e^4*f 
 + 690*c^4*d^2*e^2*f - 384*b^3*c*e^4*g + 1494*c^4*d^3*e*g - 840*b*c^3*d*e^ 
3*f - 2952*b*c^3*d^2*e^2*g + 1872*b^2*c^2*d*e^3*g))/(15*c^7*e^5)))/(x^3 + 
(x*(15*b^2*c^5*e^5 - 15*c^7*d^2*e^3))/(15*c^7*e^5) + (d*(b*e - c*d)^2)/(c^ 
2*e^3) + (x^2*(2*b*e - c*d))/(c*e))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^{11/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {\frac {332}{5} c^{4} d^{4} g -\frac {1504}{15} b^{3} c d \,e^{3} g +\frac {1072}{5} b^{2} c^{2} d^{2} e^{2} g +48 b^{2} c^{2} d \,e^{3} f -16 b^{2} c^{2} e^{4} f x -\frac {988}{5} b \,c^{3} d^{3} e g +10 c^{4} d \,e^{3} f \,x^{2}+\frac {256}{15} b^{4} e^{4} g +\frac {2}{5} c^{4} e^{4} g \,x^{4}+\frac {128}{5} b^{3} c \,e^{4} g x +\frac {32}{5} b^{2} c^{2} e^{4} g \,x^{2}-68 b \,c^{3} d^{2} e^{2} f -4 b \,c^{3} e^{4} f \,x^{2}-\frac {16}{15} b \,c^{3} e^{4} g \,x^{3}-\frac {498}{5} c^{4} d^{3} e g x +\frac {122}{5} c^{4} d^{2} e^{2} g \,x^{2}+\frac {46}{15} c^{4} d \,e^{3} g \,x^{3}-\frac {32}{3} b^{3} c \,e^{4} f +30 c^{4} d^{3} e f +\frac {2}{3} c^{4} e^{4} f \,x^{3}-46 c^{4} d^{2} e^{2} f x -\frac {624}{5} b^{2} c^{2} d \,e^{3} g x +\frac {984}{5} b \,c^{3} d^{2} e^{2} g x +56 b \,c^{3} d \,e^{3} f x -\frac {124}{5} b \,c^{3} d \,e^{3} g \,x^{2}}{\sqrt {-c e x -b e +c d}\, c^{5} e^{2} \left (c e x +b e -c d \right )} \] Input:

int((e*x+d)^(11/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
 

Output:

(2*(128*b**4*e**4*g - 752*b**3*c*d*e**3*g - 80*b**3*c*e**4*f + 192*b**3*c* 
e**4*g*x + 1608*b**2*c**2*d**2*e**2*g + 360*b**2*c**2*d*e**3*f - 936*b**2* 
c**2*d*e**3*g*x - 120*b**2*c**2*e**4*f*x + 48*b**2*c**2*e**4*g*x**2 - 1482 
*b*c**3*d**3*e*g - 510*b*c**3*d**2*e**2*f + 1476*b*c**3*d**2*e**2*g*x + 42 
0*b*c**3*d*e**3*f*x - 186*b*c**3*d*e**3*g*x**2 - 30*b*c**3*e**4*f*x**2 - 8 
*b*c**3*e**4*g*x**3 + 498*c**4*d**4*g + 225*c**4*d**3*e*f - 747*c**4*d**3* 
e*g*x - 345*c**4*d**2*e**2*f*x + 183*c**4*d**2*e**2*g*x**2 + 75*c**4*d*e** 
3*f*x**2 + 23*c**4*d*e**3*g*x**3 + 5*c**4*e**4*f*x**3 + 3*c**4*e**4*g*x**4 
))/(15*sqrt( - b*e + c*d - c*e*x)*c**5*e**2*(b*e - c*d + c*e*x))