\(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx\) [146]

Optimal result

Integrand size = 44, antiderivative size = 439 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{13 e^2 (2 c d-b e) (d+e x)^8}-\frac {2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^7}+\frac {16 c (13 b e g-2 c (5 e f+8 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^6}-\frac {32 c^2 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3003 e^2 (2 c d-b e)^4 (d+e x)^5}+\frac {128 c^3 (13 b e g-2 c (5 e f+8 d g)) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{15015 e^2 (2 c d-b e)^5 (d+e x)^4}-\frac {256 c^4 (10 c e f+16 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{45045 e^2 (2 c d-b e)^6 (d+e x)^3} \] Output:

-2/13*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)/( 
e*x+d)^8-2/143*(-13*b*e*g+16*c*d*g+10*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x 
^2)^(3/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^7+16/1287*c*(13*b*e*g-2*c*(8*d*g+5*e* 
f))*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)^3/(e*x+d)^6-32 
/3003*c^2*(-13*b*e*g+16*c*d*g+10*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^( 
3/2)/e^2/(-b*e+2*c*d)^4/(e*x+d)^5+128/15015*c^3*(13*b*e*g-2*c*(8*d*g+5*e*f 
))*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(-b*e+2*c*d)^5/(e*x+d)^4-256 
/45045*c^4*(-13*b*e*g+16*c*d*g+10*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^ 
(3/2)/e^2/(-b*e+2*c*d)^6/(e*x+d)^3
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.06 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx=\frac {2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))} \left (-315 b^5 e^5 (11 e f+2 d g+13 e g x)-40 b^3 c^2 e^3 \left (736 d^3 g+2 e^3 x^2 (35 f+39 g x)+85 d^2 e (49 f+59 g x)+2 d e^2 x (385 f+446 g x)\right )+70 b^4 c e^4 \left (97 d^2 g+e^2 x (45 f+52 g x)+d e (540 f+644 g x)\right )+32 c^5 \left (911 d^6 g+40 e^6 f x^5+64 d e^5 x^4 (5 f+g x)+32 d^3 e^3 x^2 (85 f+59 g x)+4 d^2 e^4 x^3 (295 f+128 g x)+8 d^5 e (775 f+911 g x)+d^4 e^2 x (4555 f+4352 g x)\right )+48 b^2 c^3 e^2 \left (1337 d^4 g+2 e^4 x^3 (25 f+26 g x)+4 d e^3 x^2 (125 f+137 g x)+d^2 e^2 x (2425 f+2802 g x)+d^3 e (7750 f+9418 g x)\right )-16 b c^4 e \left (4378 d^5 g+8 e^5 x^4 (15 f+13 g x)+44 d^2 e^3 x^2 (105 f+109 g x)+8 d e^4 x^3 (135 f+128 g x)+4 d^3 e^2 x (3195 f+3616 g x)+d^4 e (26445 f+32291 g x)\right )\right )}{45045 e^2 (-2 c d+b e)^6 (d+e x)^7} \] Input:

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

Output:

(2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-315*b^5 
*e^5*(11*e*f + 2*d*g + 13*e*g*x) - 40*b^3*c^2*e^3*(736*d^3*g + 2*e^3*x^2*( 
35*f + 39*g*x) + 85*d^2*e*(49*f + 59*g*x) + 2*d*e^2*x*(385*f + 446*g*x)) + 
 70*b^4*c*e^4*(97*d^2*g + e^2*x*(45*f + 52*g*x) + d*e*(540*f + 644*g*x)) + 
 32*c^5*(911*d^6*g + 40*e^6*f*x^5 + 64*d*e^5*x^4*(5*f + g*x) + 32*d^3*e^3* 
x^2*(85*f + 59*g*x) + 4*d^2*e^4*x^3*(295*f + 128*g*x) + 8*d^5*e*(775*f + 9 
11*g*x) + d^4*e^2*x*(4555*f + 4352*g*x)) + 48*b^2*c^3*e^2*(1337*d^4*g + 2* 
e^4*x^3*(25*f + 26*g*x) + 4*d*e^3*x^2*(125*f + 137*g*x) + d^2*e^2*x*(2425* 
f + 2802*g*x) + d^3*e*(7750*f + 9418*g*x)) - 16*b*c^4*e*(4378*d^5*g + 8*e^ 
5*x^4*(15*f + 13*g*x) + 44*d^2*e^3*x^2*(105*f + 109*g*x) + 8*d*e^4*x^3*(13 
5*f + 128*g*x) + 4*d^3*e^2*x*(3195*f + 3616*g*x) + d^4*e*(26445*f + 32291* 
g*x))))/(45045*e^2*(-2*c*d + b*e)^6*(d + e*x)^7)
 

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1216, 1218, 1129, 1129, 1129, 1129, 1123}

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.

\(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^8} \, dx\)

\(\Big \downarrow \) 1216

\(\displaystyle \int \frac {(f+g x) \left (\frac {c d^2-b d e}{d}-c e x\right )^8}{\left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{15/2}}dx\)

\(\Big \downarrow \) 1218

\(\displaystyle \frac {(-13 b e g+16 c d g+10 c e f) \int \frac {(c d-b e-c e x)^7}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{13/2}}dx}{13 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^8}{13 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(-13 b e g+16 c d g+10 c e f) \left (-\frac {8 \int \frac {(c d-b e-c e x)^8}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{13/2}}dx}{3 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^7}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^8}{13 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(-13 b e g+16 c d g+10 c e f) \left (-\frac {8 \left (-\frac {6 \int \frac {(c d-b e-c e x)^9}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{13/2}}dx}{5 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^8}{5 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{3 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^7}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^8}{13 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(-13 b e g+16 c d g+10 c e f) \left (-\frac {8 \left (-\frac {6 \left (-\frac {4 \int \frac {(c d-b e-c e x)^{10}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{13/2}}dx}{7 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^9}{7 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{5 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^8}{5 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{3 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^7}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^8}{13 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(-13 b e g+16 c d g+10 c e f) \left (-\frac {8 \left (-\frac {6 \left (-\frac {4 \left (-\frac {2 \int \frac {(c d-b e-c e x)^{11}}{\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{13/2}}dx}{9 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^{10}}{9 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{7 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^9}{7 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{5 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^8}{5 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{3 (2 c d-b e)}-\frac {2 (-b e+c d-c e x)^7}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^8}{13 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {\left (-\frac {2 (-b e+c d-c e x)^7}{3 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}-\frac {8 \left (-\frac {2 (-b e+c d-c e x)^8}{5 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}-\frac {6 \left (-\frac {2 (-b e+c d-c e x)^9}{7 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}-\frac {4 \left (\frac {4 (-b e+c d-c e x)^{11}}{99 e (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}-\frac {2 (-b e+c d-c e x)^{10}}{9 e (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{11/2}}\right )}{7 (2 c d-b e)}\right )}{5 (2 c d-b e)}\right )}{3 (2 c d-b e)}\right ) (-13 b e g+16 c d g+10 c e f)}{13 e (2 c d-b e)}-\frac {2 (e f-d g) (-b e+c d-c e x)^8}{13 e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{13/2}}\)

Input:

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^8,x]
 

Output:

(-2*(e*f - d*g)*(c*d - b*e - c*e*x)^8)/(13*e^2*(2*c*d - b*e)*(d*(c*d - b*e 
) - b*e^2*x - c*e^2*x^2)^(13/2)) + ((10*c*e*f + 16*c*d*g - 13*b*e*g)*((-2* 
(c*d - b*e - c*e*x)^7)/(3*e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2 
*x^2)^(11/2)) - (8*((-2*(c*d - b*e - c*e*x)^8)/(5*e*(2*c*d - b*e)*(d*(c*d 
- b*e) - b*e^2*x - c*e^2*x^2)^(11/2)) - (6*((-2*(c*d - b*e - c*e*x)^9)/(7* 
e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(11/2)) - (4*((-2*(c 
*d - b*e - c*e*x)^10)/(9*e*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2* 
x^2)^(11/2)) + (4*(c*d - b*e - c*e*x)^11)/(99*e*(2*c*d - b*e)^2*(d*(c*d - 
b*e) - b*e^2*x - c*e^2*x^2)^(11/2))))/(7*(2*c*d - b*e))))/(5*(2*c*d - b*e) 
)))/(3*(2*c*d - b*e))))/(13*e*(2*c*d - b*e))
 

Defintions of rubi rules used

rule 1123

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)/((p + 1)*(2*c*d - b 
*e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 
0] && EqQ[m + 2*p + 2, 0]
 

rule 1129

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]
 

rule 1216

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_.)*Sqrt[(a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2], x_Symbol] :> Int[((f + g*x)^n*(a + b*x + c*x^2)^(m + 
1/2))/(a/d + c*(x/e))^m, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c* 
d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && IntegerQ[n]
 

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 = 11.36 (sec) , antiderivative size = 782, normalized size of antiderivative = 1.78

method	result	size
gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (1664 b \,c^{4} e^{6} g \,x^{5}-2048 c^{5} d \,e^{5} g \,x^{5}-1280 c^{5} e^{6} f \,x^{5}-2496 b^{2} c^{3} e^{6} g \,x^{4}+16384 b \,c^{4} d \,e^{5} g \,x^{4}+1920 b \,c^{4} e^{6} f \,x^{4}-16384 c^{5} d^{2} e^{4} g \,x^{4}-10240 c^{5} d \,e^{5} f \,x^{4}+3120 b^{3} c^{2} e^{6} g \,x^{3}-26304 b^{2} c^{3} d \,e^{5} g \,x^{3}-2400 b^{2} c^{3} e^{6} f \,x^{3}+76736 b \,c^{4} d^{2} e^{4} g \,x^{3}+17280 b \,c^{4} d \,e^{5} f \,x^{3}-60416 c^{5} d^{3} e^{3} g \,x^{3}-37760 c^{5} d^{2} e^{4} f \,x^{3}-3640 b^{4} c \,e^{6} g \,x^{2}+35680 b^{3} c^{2} d \,e^{5} g \,x^{2}+2800 b^{3} c^{2} e^{6} f \,x^{2}-134496 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-24000 b^{2} c^{3} d \,e^{5} f \,x^{2}+231424 b \,c^{4} d^{3} e^{3} g \,x^{2}+73920 b \,c^{4} d^{2} e^{4} f \,x^{2}-139264 c^{5} d^{4} e^{2} g \,x^{2}-87040 c^{5} d^{3} e^{3} f \,x^{2}+4095 b^{5} e^{6} g x -45080 b^{4} c d \,e^{5} g x -3150 b^{4} c \,e^{6} f x +200600 b^{3} c^{2} d^{2} e^{4} g x +30800 b^{3} c^{2} d \,e^{5} f x -452064 b^{2} c^{3} d^{3} e^{3} g x -116400 b^{2} c^{3} d^{2} e^{4} f x +516656 b \,c^{4} d^{4} e^{2} g x +204480 b \,c^{4} d^{3} e^{3} f x -233216 c^{5} d^{5} e g x -145760 c^{5} d^{4} e^{2} f x +630 b^{5} d \,e^{5} g +3465 b^{5} e^{6} f -6790 b^{4} c \,d^{2} e^{4} g -37800 b^{4} c d \,e^{5} f +29440 b^{3} c^{2} d^{3} e^{3} g +166600 b^{3} c^{2} d^{2} e^{4} f -64176 b^{2} c^{3} d^{4} e^{2} g -372000 b^{2} c^{3} d^{3} e^{3} f +70048 b \,c^{4} d^{5} e g +423120 b \,c^{4} d^{4} e^{2} f -29152 c^{5} d^{6} g -198400 c^{5} d^{5} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{45045 \left (e x +d \right )^{7} e^{2} \left (b^{6} e^{6}-12 b^{5} c d \,e^{5}+60 b^{4} c^{2} d^{2} e^{4}-160 b^{3} c^{3} d^{3} e^{3}+240 b^{2} c^{4} d^{4} e^{2}-192 b \,c^{5} d^{5} e +64 c^{6} d^{6}\right )}\)	\(782\)
orering	\(-\frac {2 \left (c e x +b e -c d \right ) \left (1664 b \,c^{4} e^{6} g \,x^{5}-2048 c^{5} d \,e^{5} g \,x^{5}-1280 c^{5} e^{6} f \,x^{5}-2496 b^{2} c^{3} e^{6} g \,x^{4}+16384 b \,c^{4} d \,e^{5} g \,x^{4}+1920 b \,c^{4} e^{6} f \,x^{4}-16384 c^{5} d^{2} e^{4} g \,x^{4}-10240 c^{5} d \,e^{5} f \,x^{4}+3120 b^{3} c^{2} e^{6} g \,x^{3}-26304 b^{2} c^{3} d \,e^{5} g \,x^{3}-2400 b^{2} c^{3} e^{6} f \,x^{3}+76736 b \,c^{4} d^{2} e^{4} g \,x^{3}+17280 b \,c^{4} d \,e^{5} f \,x^{3}-60416 c^{5} d^{3} e^{3} g \,x^{3}-37760 c^{5} d^{2} e^{4} f \,x^{3}-3640 b^{4} c \,e^{6} g \,x^{2}+35680 b^{3} c^{2} d \,e^{5} g \,x^{2}+2800 b^{3} c^{2} e^{6} f \,x^{2}-134496 b^{2} c^{3} d^{2} e^{4} g \,x^{2}-24000 b^{2} c^{3} d \,e^{5} f \,x^{2}+231424 b \,c^{4} d^{3} e^{3} g \,x^{2}+73920 b \,c^{4} d^{2} e^{4} f \,x^{2}-139264 c^{5} d^{4} e^{2} g \,x^{2}-87040 c^{5} d^{3} e^{3} f \,x^{2}+4095 b^{5} e^{6} g x -45080 b^{4} c d \,e^{5} g x -3150 b^{4} c \,e^{6} f x +200600 b^{3} c^{2} d^{2} e^{4} g x +30800 b^{3} c^{2} d \,e^{5} f x -452064 b^{2} c^{3} d^{3} e^{3} g x -116400 b^{2} c^{3} d^{2} e^{4} f x +516656 b \,c^{4} d^{4} e^{2} g x +204480 b \,c^{4} d^{3} e^{3} f x -233216 c^{5} d^{5} e g x -145760 c^{5} d^{4} e^{2} f x +630 b^{5} d \,e^{5} g +3465 b^{5} e^{6} f -6790 b^{4} c \,d^{2} e^{4} g -37800 b^{4} c d \,e^{5} f +29440 b^{3} c^{2} d^{3} e^{3} g +166600 b^{3} c^{2} d^{2} e^{4} f -64176 b^{2} c^{3} d^{4} e^{2} g -372000 b^{2} c^{3} d^{3} e^{3} f +70048 b \,c^{4} d^{5} e g +423120 b \,c^{4} d^{4} e^{2} f -29152 c^{5} d^{6} g -198400 c^{5} d^{5} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{45045 \left (e x +d \right )^{7} e^{2} \left (b^{6} e^{6}-12 b^{5} c d \,e^{5}+60 b^{4} c^{2} d^{2} e^{4}-160 b^{3} c^{3} d^{3} e^{3}+240 b^{2} c^{4} d^{4} e^{2}-192 b \,c^{5} d^{5} e +64 c^{6} d^{6}\right )}\)	\(782\)
default	\(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{11 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {8 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {2 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{5}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (-b \,e^{2}+2 d e c \right )}\right )}{3 \left (-b \,e^{2}+2 d e c \right )}\right )}{11 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{8}}-\frac {\left (d g -e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{13 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{8}}+\frac {10 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{11 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{7}}+\frac {8 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {2 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{5}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (-b \,e^{2}+2 d e c \right )}\right )}{3 \left (-b \,e^{2}+2 d e c \right )}\right )}{11 \left (-b \,e^{2}+2 d e c \right )}\right )}{13 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{9}}\)	\(870\)
trager	\(\text {Expression too large to display}\)	\(1010\)

Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x,method=_RET 
URNVERBOSE)
 

Output:

-2/45045*(c*e*x+b*e-c*d)*(1664*b*c^4*e^6*g*x^5-2048*c^5*d*e^5*g*x^5-1280*c 
^5*e^6*f*x^5-2496*b^2*c^3*e^6*g*x^4+16384*b*c^4*d*e^5*g*x^4+1920*b*c^4*e^6 
*f*x^4-16384*c^5*d^2*e^4*g*x^4-10240*c^5*d*e^5*f*x^4+3120*b^3*c^2*e^6*g*x^ 
3-26304*b^2*c^3*d*e^5*g*x^3-2400*b^2*c^3*e^6*f*x^3+76736*b*c^4*d^2*e^4*g*x 
^3+17280*b*c^4*d*e^5*f*x^3-60416*c^5*d^3*e^3*g*x^3-37760*c^5*d^2*e^4*f*x^3 
-3640*b^4*c*e^6*g*x^2+35680*b^3*c^2*d*e^5*g*x^2+2800*b^3*c^2*e^6*f*x^2-134 
496*b^2*c^3*d^2*e^4*g*x^2-24000*b^2*c^3*d*e^5*f*x^2+231424*b*c^4*d^3*e^3*g 
*x^2+73920*b*c^4*d^2*e^4*f*x^2-139264*c^5*d^4*e^2*g*x^2-87040*c^5*d^3*e^3* 
f*x^2+4095*b^5*e^6*g*x-45080*b^4*c*d*e^5*g*x-3150*b^4*c*e^6*f*x+200600*b^3 
*c^2*d^2*e^4*g*x+30800*b^3*c^2*d*e^5*f*x-452064*b^2*c^3*d^3*e^3*g*x-116400 
*b^2*c^3*d^2*e^4*f*x+516656*b*c^4*d^4*e^2*g*x+204480*b*c^4*d^3*e^3*f*x-233 
216*c^5*d^5*e*g*x-145760*c^5*d^4*e^2*f*x+630*b^5*d*e^5*g+3465*b^5*e^6*f-67 
90*b^4*c*d^2*e^4*g-37800*b^4*c*d*e^5*f+29440*b^3*c^2*d^3*e^3*g+166600*b^3* 
c^2*d^2*e^4*f-64176*b^2*c^3*d^4*e^2*g-372000*b^2*c^3*d^3*e^3*f+70048*b*c^4 
*d^5*e*g+423120*b*c^4*d^4*e^2*f-29152*c^5*d^6*g-198400*c^5*d^5*e*f)*(-c*e^ 
2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^7/e^2/(b^6*e^6-12*b^5*c*d*e^5+60* 
b^4*c^2*d^2*e^4-160*b^3*c^3*d^3*e^3+240*b^2*c^4*d^4*e^2-192*b*c^5*d^5*e+64 
*c^6*d^6)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x, algo 
rithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{8}}\, dx \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**8,x 
)
 

Output:

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**8, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x, algo 
rithm="maxima")
 

Output:

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
 

Giac [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x, algo 
rithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 37.47 (sec) , antiderivative size = 19572, normalized size of antiderivative = 44.58 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx=\text {Too large to display} \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^8,x)
 

Output:

(((d*((8*c^2*(14*b*e*g - 27*c*d*g + c*e*f))/(143*(9*b*e^2 - 18*c*d*e)*(b*e 
 - 2*c*d)^2) - (8*c^3*d*g)/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2)))/e 
- (8*c*(b*e - c*d)*(13*b*e*g - 26*c*d*g + c*e*f))/(143*e*(9*b*e^2 - 18*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)^5 - (((2*f*(b*e - c*d))/(13*b*e^2 - 26*c*d*e) - (d*((2*b*e*g - 2*c*d*g 
+ 2*c*e*f)/(13*b*e^2 - 26*c*d*e) - (2*c*d*g)/(13*b*e^2 - 26*c*d*e)))/e)*(c 
*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^7 - (((d*((4*c^2*(15* 
b*e*g - 28*c*d*g + 2*c*e*f))/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2) - 
(8*c^3*d*g)/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2)))/e - (144*c^3*d^2* 
g - 48*c^3*d*e*f + 28*b*c^2*e^2*f + 52*b^2*c*e^2*g - 176*b*c^2*d*e*g)/(143 
*e*(9*b*e^2 - 18*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)^5 - (((d*((4*c^2*e*f - 8*c^2*d*g + 6*b*c*e*g)/(13*( 
11*b*e^2 - 22*c*d*e)*(b*e - 2*c*d)) - (4*c^2*d*g)/(13*(11*b*e^2 - 22*c*d*e 
)*(b*e - 2*c*d))))/e - (2*b*(b*e*g - 2*c*d*g + c*e*f))/(13*(11*b*e^2 - 22* 
c*d*e)*(b*e - 2*c*d)))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e 
*x)^6 - (((d*((16*c^4*e*f - 64*c^4*d*g + 40*b*c^3*e*g)/(1287*(7*b*e^2 - 14 
*c*d*e)*(b*e - 2*c*d)^3) - (16*c^4*d*g)/(1287*(7*b*e^2 - 14*c*d*e)*(b*e - 
2*c*d)^3)))/e - (8*b*c^2*(2*b*e*g - 4*c*d*g + c*e*f))/(1287*(7*b*e^2 - 14* 
c*d*e)*(b*e - 2*c*d)^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + 
 e*x)^4 + (((d*((16*c^4*e*f - 240*c^4*d*g + 128*b*c^3*e*g)/(1287*(7*b*e...
 

Reduce [B] (verification not implemented)

Time = 8.32 (sec) , antiderivative size = 4939, normalized size of antiderivative = 11.25 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^8} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^8,x)
 

Output:

(2*i*( - 630*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - 
b*e + c*d - c*e*x)*b**6*d*e**6*g - 3465*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq 
rt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**6*e**7*f - 4095*sqrt(d + 
e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b** 
6*e**7*g*x + 7420*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqr 
t( - b*e + c*d - c*e*x)*b**5*c*d**2*e**5*g + 41265*sqrt(d + e*x)*sqrt(b*e 
- 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**5*c*d*e**6*f + 
 48545*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + 
c*d - c*e*x)*b**5*c*d*e**6*g*x - 315*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( 
 - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**5*c*e**7*f*x - 455*sqrt(d + 
e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b** 
5*c*e**7*g*x**2 - 36230*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c* 
d)*sqrt( - b*e + c*d - c*e*x)*b**4*c**2*d**3*e**4*g - 204400*sqrt(d + e*x) 
*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*c* 
*2*d**2*e**5*f - 238890*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c* 
d)*sqrt( - b*e + c*d - c*e*x)*b**4*c**2*d**2*e**5*g*x + 3850*sqrt(d + e*x) 
*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*c* 
*2*d*e**6*f*x + 5760*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)* 
sqrt( - b*e + c*d - c*e*x)*b**4*c**2*d*e**6*g*x**2 + 350*sqrt(d + e*x)*sqr 
t(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*c**...