\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^{10}} \, dx\) [171]

Optimal result

Integrand size = 44, antiderivative size = 285 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (2 c d-b e) (d+e x)^{10}}-\frac {2 (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{143 e^2 (2 c d-b e)^2 (d+e x)^9}-\frac {8 c (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{1287 e^2 (2 c d-b e)^3 (d+e x)^8}-\frac {16 c^2 (6 c e f+20 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9009 e^2 (2 c d-b e)^4 (d+e x)^7} \] Output:

-2/13*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)/( 
e*x+d)^10-2/143*(-13*b*e*g+20*c*d*g+6*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x 
^2)^(7/2)/e^2/(-b*e+2*c*d)^2/(e*x+d)^9-8/1287*c*(-13*b*e*g+20*c*d*g+6*c*e* 
f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/2)/e^2/(-b*e+2*c*d)^3/(e*x+d)^8-16/ 
9009*c^2*(-13*b*e*g+20*c*d*g+6*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(7/ 
2)/e^2/(-b*e+2*c*d)^4/(e*x+d)^7
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.88 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\frac {2 (-c d+b e+c e x)^3 \sqrt {(d+e x) (-b e+c (d-e x))} \left (-63 b^3 e^3 (11 e f+2 d g+13 e g x)+14 b^2 c e^2 \left (53 d^2 g+e^2 x (27 f+26 g x)+4 d e (81 f+94 g x)\right )+8 c^3 \left (97 d^4 g+6 e^4 f x^3+20 d e^3 x^2 (3 f+g x)+10 d^3 e (93 f+97 g x)+d^2 e^2 x (291 f+200 g x)\right )-4 b c^2 e \left (348 d^3 g+2 e^3 x^2 (21 f+13 g x)+2 d e^2 x (231 f+200 g x)+d^2 e (2499 f+2801 g x)\right )\right )}{9009 e^2 (-2 c d+b e)^4 (d+e x)^7} \] Input:

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x 
)^10,x]
 

Output:

(2*(-(c*d) + b*e + c*e*x)^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-63*b^ 
3*e^3*(11*e*f + 2*d*g + 13*e*g*x) + 14*b^2*c*e^2*(53*d^2*g + e^2*x*(27*f + 
 26*g*x) + 4*d*e*(81*f + 94*g*x)) + 8*c^3*(97*d^4*g + 6*e^4*f*x^3 + 20*d*e 
^3*x^2*(3*f + g*x) + 10*d^3*e*(93*f + 97*g*x) + d^2*e^2*x*(291*f + 200*g*x 
)) - 4*b*c^2*e*(348*d^3*g + 2*e^3*x^2*(21*f + 13*g*x) + 2*d*e^2*x*(231*f + 
 200*g*x) + d^2*e*(2499*f + 2801*g*x))))/(9009*e^2*(-2*c*d + b*e)^4*(d + e 
*x)^7)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1220, 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.

\(\displaystyle \int \frac {(f+g x) \left (-b d e-b e^2 x+c d^2-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(-13 b e g+20 c d g+6 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^9}dx}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (d+e x)^{10} (2 c d-b e)}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(-13 b e g+20 c d g+6 c e f) \left (\frac {4 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^8}dx}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e (d+e x)^9 (2 c d-b e)}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (d+e x)^{10} (2 c d-b e)}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(-13 b e g+20 c d g+6 c e f) \left (\frac {4 c \left (\frac {2 c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^7}dx}{9 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e (d+e x)^8 (2 c d-b e)}\right )}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e (d+e x)^9 (2 c d-b e)}\right )}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (d+e x)^{10} (2 c d-b e)}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {\left (\frac {4 c \left (-\frac {4 c \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{63 e (d+e x)^7 (2 c d-b e)^2}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 e (d+e x)^8 (2 c d-b e)}\right )}{11 (2 c d-b e)}-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{11 e (d+e x)^9 (2 c d-b e)}\right ) (-13 b e g+20 c d g+6 c e f)}{13 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{13 e^2 (d+e x)^{10} (2 c d-b e)}\)

Input:

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2))/(d + e*x)^10,x 
]
 

Output:

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(13*e^2*(2*c* 
d - b*e)*(d + e*x)^10) + ((6*c*e*f + 20*c*d*g - 13*b*e*g)*((-2*(d*(c*d - b 
*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(11*e*(2*c*d - b*e)*(d + e*x)^9) + (4*c* 
((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*e*(2*c*d - b*e)*(d + 
e*x)^8) - (4*c*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(63*e*(2*c*d - 
 b*e)^2*(d + e*x)^7)))/(11*(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 1220

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 
]
 

Maple [A] (verified)

Time = 26.77 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.34

method	result	size
gosper	\(-\frac {2 \left (c e x +b e -c d \right ) \left (104 b \,c^{2} e^{4} g \,x^{3}-160 c^{3} d \,e^{3} g \,x^{3}-48 c^{3} e^{4} f \,x^{3}-364 b^{2} c \,e^{4} g \,x^{2}+1600 b \,c^{2} d \,e^{3} g \,x^{2}+168 b \,c^{2} e^{4} f \,x^{2}-1600 c^{3} d^{2} e^{2} g \,x^{2}-480 c^{3} d \,e^{3} f \,x^{2}+819 b^{3} e^{4} g x -5264 b^{2} c d \,e^{3} g x -378 b^{2} c \,e^{4} f x +11204 b \,c^{2} d^{2} e^{2} g x +1848 b \,c^{2} d \,e^{3} f x -7760 c^{3} d^{3} e g x -2328 c^{3} d^{2} e^{2} f x +126 b^{3} d \,e^{3} g +693 b^{3} e^{4} f -742 b^{2} c \,d^{2} e^{2} g -4536 b^{2} c d \,e^{3} f +1392 b \,c^{2} d^{3} e g +9996 b \,c^{2} d^{2} e^{2} f -776 c^{3} d^{4} g -7440 c^{3} d^{3} e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{9009 \left (e x +d \right )^{9} e^{2} \left (b^{4} e^{4}-8 d \,e^{3} b^{3} c +24 d^{2} e^{2} b^{2} c^{2}-32 d^{3} e b \,c^{3}+16 d^{4} c^{4}\right )}\)	\(382\)
orering	\(-\frac {2 \left (c e x +b e -c d \right ) \left (104 b \,c^{2} e^{4} g \,x^{3}-160 c^{3} d \,e^{3} g \,x^{3}-48 c^{3} e^{4} f \,x^{3}-364 b^{2} c \,e^{4} g \,x^{2}+1600 b \,c^{2} d \,e^{3} g \,x^{2}+168 b \,c^{2} e^{4} f \,x^{2}-1600 c^{3} d^{2} e^{2} g \,x^{2}-480 c^{3} d \,e^{3} f \,x^{2}+819 b^{3} e^{4} g x -5264 b^{2} c d \,e^{3} g x -378 b^{2} c \,e^{4} f x +11204 b \,c^{2} d^{2} e^{2} g x +1848 b \,c^{2} d \,e^{3} f x -7760 c^{3} d^{3} e g x -2328 c^{3} d^{2} e^{2} f x +126 b^{3} d \,e^{3} g +693 b^{3} e^{4} f -742 b^{2} c \,d^{2} e^{2} g -4536 b^{2} c d \,e^{3} f +1392 b \,c^{2} d^{3} e g +9996 b \,c^{2} d^{2} e^{2} f -776 c^{3} d^{4} g -7440 c^{3} d^{3} e f \right ) \left (-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}\right )^{\frac {5}{2}}}{9009 \left (e x +d \right )^{9} e^{2} \left (b^{4} e^{4}-8 d \,e^{3} b^{3} c +24 d^{2} e^{2} b^{2} c^{2}-32 d^{3} e b \,c^{3}+16 d^{4} c^{4}\right )}\)	\(382\)
default	\(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{9}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{10}}-\frac {\left (d g -e f \right ) \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{13 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{10}}+\frac {6 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{11 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{9}}+\frac {4 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{9 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{8}}-\frac {4 c \,e^{2} \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{63 \left (-b \,e^{2}+2 d e c \right )^{2} \left (x +\frac {d}{e}\right )^{7}}\right )}{11 \left (-b \,e^{2}+2 d e c \right )}\right )}{13 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{11}}\)	\(534\)
trager	\(-\frac {2 \left (104 b \,c^{5} e^{7} g \,x^{6}-160 c^{6} d \,e^{6} g \,x^{6}-48 c^{6} e^{7} f \,x^{6}-52 b^{2} c^{4} e^{7} g \,x^{5}+808 b \,c^{5} d \,e^{6} g \,x^{5}+24 b \,c^{5} e^{7} f \,x^{5}-1120 c^{6} d^{2} e^{5} g \,x^{5}-336 c^{6} d \,e^{6} f \,x^{5}+39 b^{3} c^{3} e^{7} g \,x^{4}-476 b^{2} c^{4} d \,e^{6} g \,x^{4}-18 b^{2} c^{4} e^{7} f \,x^{4}+2876 b \,c^{5} d^{2} e^{5} g \,x^{4}+192 b \,c^{5} d \,e^{6} f \,x^{4}-3440 c^{6} d^{3} e^{4} g \,x^{4}-1032 c^{6} d^{2} e^{5} f \,x^{4}+1469 b^{4} c^{2} e^{7} g \,x^{3}-11611 b^{3} c^{3} d \,e^{6} g \,x^{3}+15 b^{3} c^{3} e^{7} f \,x^{3}+33962 b^{2} c^{4} d^{2} e^{5} g \,x^{3}-162 b^{2} c^{4} d \,e^{6} f \,x^{3}-41684 b \,c^{5} d^{3} e^{4} g \,x^{3}+708 b \,c^{5} d^{2} e^{5} f \,x^{3}+17864 c^{6} d^{4} e^{3} g \,x^{3}-1848 c^{6} d^{3} e^{4} f \,x^{3}+2093 b^{5} c \,e^{7} g \,x^{2}-17636 b^{4} c^{2} d \,e^{6} g \,x^{2}+1113 b^{4} c^{2} e^{7} f \,x^{2}+57557 b^{3} c^{3} d^{2} e^{5} g \,x^{2}-8859 b^{3} c^{3} d \,e^{6} f \,x^{2}-89930 b^{2} c^{4} d^{3} e^{4} g \,x^{2}+26334 b^{2} c^{4} d^{2} e^{5} f \,x^{2}+67268 b \,c^{5} d^{4} e^{3} g \,x^{2}-34404 b \,c^{5} d^{3} e^{4} f \,x^{2}-19352 c^{6} d^{5} e^{2} g \,x^{2}+15816 c^{6} d^{4} e^{3} f \,x^{2}+819 b^{6} e^{7} g x -7343 b^{5} c d \,e^{6} g x +1701 b^{5} c \,e^{7} f x +26471 b^{4} c^{2} d^{2} e^{5} g x -14784 b^{4} c^{2} d \,e^{6} f x -48977 b^{3} c^{3} d^{3} e^{4} g x +50277 b^{3} c^{3} d^{2} e^{5} f x +49250 b^{2} c^{4} d^{4} e^{3} g x -82998 b^{2} c^{4} d^{3} e^{4} f x -25652 b \,c^{5} d^{5} e^{2} g x +65796 b \,c^{5} d^{4} e^{3} f x +5432 c^{6} d^{6} e g x -19992 c^{6} d^{5} e^{2} f x +126 b^{6} d \,e^{6} g +693 b^{6} e^{7} f -1120 b^{5} c \,d^{2} e^{5} g -6615 b^{5} c d \,e^{6} f +3996 b^{4} c^{2} d^{3} e^{4} g +25683 b^{4} c^{2} d^{2} e^{5} f -7304 b^{3} c^{3} d^{4} e^{3} g -51729 b^{3} c^{3} d^{3} e^{4} f +7246 b^{2} c^{4} d^{5} e^{2} g +56844 b^{2} c^{4} d^{4} e^{3} f -3720 b \,c^{5} d^{6} e g -32316 b \,c^{5} d^{5} e^{2} f +776 c^{6} d^{7} g +7440 c^{6} d^{6} e f \right ) \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{9009 \left (b^{4} e^{4}-8 d \,e^{3} b^{3} c +24 d^{2} e^{2} b^{2} c^{2}-32 d^{3} e b \,c^{3}+16 d^{4} c^{4}\right ) e^{2} \left (e x +d \right )^{7}}\)	\(982\)

Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^10,x,method=_RE 
TURNVERBOSE)
 

Output:

-2/9009*(c*e*x+b*e-c*d)*(104*b*c^2*e^4*g*x^3-160*c^3*d*e^3*g*x^3-48*c^3*e^ 
4*f*x^3-364*b^2*c*e^4*g*x^2+1600*b*c^2*d*e^3*g*x^2+168*b*c^2*e^4*f*x^2-160 
0*c^3*d^2*e^2*g*x^2-480*c^3*d*e^3*f*x^2+819*b^3*e^4*g*x-5264*b^2*c*d*e^3*g 
*x-378*b^2*c*e^4*f*x+11204*b*c^2*d^2*e^2*g*x+1848*b*c^2*d*e^3*f*x-7760*c^3 
*d^3*e*g*x-2328*c^3*d^2*e^2*f*x+126*b^3*d*e^3*g+693*b^3*e^4*f-742*b^2*c*d^ 
2*e^2*g-4536*b^2*c*d*e^3*f+1392*b*c^2*d^3*e*g+9996*b*c^2*d^2*e^2*f-776*c^3 
*d^4*g-7440*c^3*d^3*e*f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^9/ 
e^2/(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)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^10,x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2)/(e*x+d)**10, 
x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^10,x, alg 
orithm="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. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^10,x, alg 
orithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 67.59 (sec) , antiderivative size = 51074, normalized size of antiderivative = 179.21 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx=\text {Too large to display} \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^10,x 
)
 

Output:

(((d*((2016*b^2*c^7*e^3*f - 17664*c^9*d^3*g + 3040*b^3*c^6*e^3*g + 5376*c^ 
9*d^2*e*f - 6528*b*c^8*d*e^2*f + 29184*b*c^8*d^2*e*g - 16224*b^2*c^7*d*e^2 
*g)/(135135*e*(b*e - 2*c*d)^7) - (d*((32*c^7*(21*b^2*e^2*g + 56*c^2*d^2*g 
+ 8*b*c*e^2*f - 12*c^2*d*e*f - 68*b*c*d*e*g))/(45045*(b*e - 2*c*d)^7) - (d 
*((128*c^8*e*(6*b*e*g - 9*c*d*g + c*e*f))/(135135*(b*e - 2*c*d)^7) - (128* 
c^9*d*e*g)/(135135*(b*e - 2*c*d)^7)))/e))/e))/e - (16*b*c^5*(69*b^3*e^3*g 
- 552*c^3*d^3*g + 52*b^2*c*e^3*f + 168*c^3*d^2*e*f - 186*b*c^2*d*e^2*f + 8 
28*b*c^2*d^2*e*g - 414*b^2*c*d*e^2*g))/(135135*e*(b*e - 2*c*d)^7))*(c*d^2 
- c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x) - (((d*((d*((d*((4*c^4*e^2 
*(19*b*e*g - 32*c*d*g + 2*c*e*f))/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^ 
2) - (8*c^5*d*e^2*g)/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2)))/e - (4*c 
^3*e*(51*b^2*e^2*g + 134*c^2*d^2*g + 19*b*c*e^2*f - 32*c^2*d*e*f - 166*b*c 
*d*e*g))/(143*(9*b*e^2 - 18*c*d*e)*(b*e - 2*c*d)^2)))/e + (204*b^2*c^3*e^4 
*f + 236*b^3*c^2*e^4*g + 536*c^5*d^2*e^2*f - 1216*c^5*d^3*e*g - 664*b*c^4* 
d*e^3*f + 2092*b*c^4*d^2*e^2*g - 1212*b^2*c^3*d*e^3*g)/(143*e*(9*b*e^2 - 1 
8*c*d*e)*(b*e - 2*c*d)^2)))/e - (960*c^5*d^4*g + 156*b^3*c^2*e^4*f + 100*b 
^4*c*e^4*g - 576*c^5*d^3*e*f - 2240*b*c^4*d^3*e*g + 1132*b*c^4*d^2*e^2*f - 
 732*b^2*c^3*d*e^3*f - 720*b^3*c^2*d*e^3*g + 1920*b^2*c^3*d^2*e^2*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*((3776*b^2*c^7*e^3*f - 44544*c^9*d^3*g +...
 

Reduce [B] (verification not implemented)

Time = 9.06 (sec) , antiderivative size = 4677, normalized size of antiderivative = 16.41 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^{10}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)/(e*x+d)^10,x)
 

Output:

(2*i*( - 126*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 - 693*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqr 
t( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**6*e**7*f - 819*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 + 1120*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**2*e**5*g + 6615*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 + 73 
43*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 - 1701*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 - 2093*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 - 3996*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 - 25683*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**2*d 
**2*e**5*f - 26471*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sq 
rt( - b*e + c*d - c*e*x)*b**4*c**2*d**2*e**5*g*x + 14784*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**2*d 
*e**6*f*x + 17636*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**4*c**2*d*e**6*g*x**2 - 1113*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*...