\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}}{(d+e x)^6} \, dx\) [167]

Optimal result

Integrand size = 44, antiderivative size = 281 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {c^2 g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2}-\frac {2 c (c e f-5 c d g+2 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}+\frac {2 (c e f-3 c d g+b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (d+e x)^5}+\frac {c^{3/2} (2 c e f-12 c d g+5 b e g) \arctan \left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {c} (d+e x)}\right )}{e^2} \] Output:

c^2*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2-2*c*(2*b*e*g-5*c*d*g+c*e* 
f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)+2/3*(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)/e^2/(e*x+d)^3-2/5*(-d*g+e*f)* 
(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/e^2/(e*x+d)^5+c^(3/2)*(5*b*e*g-12*c 
*d*g+2*c*e*f)*arctan(1/c^(1/2)/(e*x+d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1 
/2))/e^2
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.88 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {-2 b^2 e^2 (3 e f+2 d g+5 e g x)-2 b c e \left (16 d^2 g-d e (f-39 g x)+e^2 x (11 f+35 g x)\right )+c^2 \left (141 d^3 g+e^3 x^2 (-46 f+15 g x)+3 d e^2 x (-16 f+77 g x)+d^2 e (-26 f+333 g x)\right )}{(d+e x)^5 (-c d+b e+c e x)^2}+\frac {15 c^{3/2} (5 b e g+2 c (e f-6 d g)) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{5/2} (-b e+c (d-e x))^{5/2}}\right )}{15 e^2} \] 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 
)^6,x]
 

Output:

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((-2*b^2*e^2*(3*e*f + 2*d*g + 5* 
e*g*x) - 2*b*c*e*(16*d^2*g - d*e*(f - 39*g*x) + e^2*x*(11*f + 35*g*x)) + c 
^2*(141*d^3*g + e^3*x^2*(-46*f + 15*g*x) + 3*d*e^2*x*(-16*f + 77*g*x) + d^ 
2*e*(-26*f + 333*g*x)))/((d + e*x)^5*(-(c*d) + b*e + c*e*x)^2) + (15*c^(3/ 
2)*(5*b*e*g + 2*c*(e*f - 6*d*g))*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*S 
qrt[d + e*x])])/((d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(5/2))))/(15*e^2)
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1220, 1130, 1125, 27, 1160, 1092, 217}

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.

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

\(\Big \downarrow \) 1220

\(\displaystyle -\frac {(5 b e g-12 c d g+2 c e f) \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}}{(d+e x)^5}dx}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle -\frac {(5 b e g-12 c d g+2 c e f) \left (-\frac {5}{3} c \int \frac {\left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}}{(d+e x)^3}dx-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}\)

\(\Big \downarrow \) 1125

\(\displaystyle -\frac {(5 b e g-12 c d g+2 c e f) \left (-\frac {5}{3} c \left (-\frac {\int \frac {c e^4 (3 c d-2 b e-c e x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{e^4}-\frac {2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(5 b e g-12 c d g+2 c e f) \left (-\frac {5}{3} c \left (-c \int \frac {3 c d-2 b e-c e x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}\)

\(\Big \downarrow \) 1160

\(\displaystyle -\frac {(5 b e g-12 c d g+2 c e f) \left (-\frac {5}{3} c \left (-c \left (\frac {3}{2} (2 c d-b e) \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {(5 b e g-12 c d g+2 c e f) \left (-\frac {5}{3} c \left (-c \left (3 (2 c d-b e) \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right )}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {\left (-\frac {5}{3} c \left (-c \left (\frac {3 (2 c d-b e) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt {c} e}+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )-\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e (d+e x)^4}\right ) (5 b e g-12 c d g+2 c e f)}{5 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{5 e^2 (d+e x)^6 (2 c d-b e)}\)

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)^6,x]
 

Output:

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(5*e^2*(2*c*d 
 - b*e)*(d + e*x)^6) - ((2*c*e*f - 12*c*d*g + 5*b*e*g)*((-2*(d*(c*d - b*e) 
 - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e*(d + e*x)^4) - (5*c*((-2*(2*c*d - b*e) 
*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e*(d + e*x)) - c*(Sqrt[d*(c*d 
 - b*e) - b*e^2*x - c*e^2*x^2]/e + (3*(2*c*d - b*e)*ArcTan[(e*(b + 2*c*x)) 
/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(2*Sqrt[c]*e))))/ 
3))/(5*e*(2*c*d - b*e))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1125

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[-2*e^(2*m + 3)*(Sqrt[a + b*x + c*x^2]/((-2*c*d + b*e)^(m + 
2)*(d + e*x))), x] - Simp[e^(2*m + 2)   Int[(1/Sqrt[a + b*x + c*x^2])*Expan 
dToSum[((-2*c*d + b*e)^(-m - 1) - ((-c)*d + b*e + c*e*x)^(-m - 1))/(d + e*x 
), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && ILtQ[m, 0] && EqQ[m + p, -3/2]
 

rule 1130

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + p + 1))), x] 
- Simp[c*(p/(e^2*(m + p + 1)))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p 
 - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] & 
& IntegerQ[2*p]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1423\) vs. \(2(263)=526\).

Time = 7.12 (sec) , antiderivative size = 1424, normalized size of antiderivative = 5.07

method	result	size
default	\(\text {Expression too large to display}\)	\(1424\)

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)^6,x,method=_RET 
URNVERBOSE)
 

Output:

g/e^6*(-2/3/(-b*e^2+2*c*d*e)/(x+d/e)^5*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)* 
(x+d/e))^(7/2)-4/3*c*e^2/(-b*e^2+2*c*d*e)*(-2/(-b*e^2+2*c*d*e)/(x+d/e)^4*( 
-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-6*c*e^2/(-b*e^2+2*c*d*e)* 
(2/(-b*e^2+2*c*d*e)/(x+d/e)^3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^ 
(7/2)+8*c*e^2/(-b*e^2+2*c*d*e)*(2/3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^2*(x+ 
d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+10/3*c*e^2/(-b*e^2+2*c*d*e)*(1/5*(- 
c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(5/2)+1/2*(-b*e^2+2*c*d*e)*(-1/8 
*(-2*c*e^2*(x+d/e)-b*e^2+2*d*e*c)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e) 
*(x+d/e))^(3/2)+3/16*(-b*e^2+2*c*d*e)^2/c/e^2*(-1/4*(-2*c*e^2*(x+d/e)-b*e^ 
2+2*d*e*c)/c/e^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)+1/8*(-b 
*e^2+2*c*d*e)^2/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+d/e-1/2*(-b*e^ 
2+2*c*d*e)/c/e^2)/(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(1/2)))))))) 
)-(d*g-e*f)/e^7*(-2/5/(-b*e^2+2*c*d*e)/(x+d/e)^6*(-c*e^2*(x+d/e)^2+(-b*e^2 
+2*c*d*e)*(x+d/e))^(7/2)-2/5*c*e^2/(-b*e^2+2*c*d*e)*(-2/3/(-b*e^2+2*c*d*e) 
/(x+d/e)^5*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)-4/3*c*e^2/(-b 
*e^2+2*c*d*e)*(-2/(-b*e^2+2*c*d*e)/(x+d/e)^4*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c 
*d*e)*(x+d/e))^(7/2)-6*c*e^2/(-b*e^2+2*c*d*e)*(2/(-b*e^2+2*c*d*e)/(x+d/e)^ 
3*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d/e))^(7/2)+8*c*e^2/(-b*e^2+2*c*d* 
e)*(2/3/(-b*e^2+2*c*d*e)/(x+d/e)^2*(-c*e^2*(x+d/e)^2+(-b*e^2+2*c*d*e)*(x+d 
/e))^(7/2)+10/3*c*e^2/(-b*e^2+2*c*d*e)*(1/5*(-c*e^2*(x+d/e)^2+(-b*e^2+2...
 

Fricas [A] (verification not implemented)

Time = 7.31 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.26 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] 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)^6,x, algo 
rithm="fricas")
 

Output:

[1/60*(15*(2*c^2*d^3*e*f + (2*c^2*e^4*f - (12*c^2*d*e^3 - 5*b*c*e^4)*g)*x^ 
3 + 3*(2*c^2*d*e^3*f - (12*c^2*d^2*e^2 - 5*b*c*d*e^3)*g)*x^2 - (12*c^2*d^4 
 - 5*b*c*d^3*e)*g + 3*(2*c^2*d^2*e^2*f - (12*c^2*d^3*e - 5*b*c*d^2*e^2)*g) 
*x)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2 
*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(- 
c)) + 4*(15*c^2*e^3*g*x^3 - (46*c^2*e^3*f - 7*(33*c^2*d*e^2 - 10*b*c*e^3)* 
g)*x^2 - 2*(13*c^2*d^2*e - b*c*d*e^2 + 3*b^2*e^3)*f + (141*c^2*d^3 - 32*b* 
c*d^2*e - 4*b^2*d*e^2)*g - (2*(24*c^2*d*e^2 + 11*b*c*e^3)*f - (333*c^2*d^2 
*e - 78*b*c*d*e^2 - 10*b^2*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - 
b*d*e))/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2), 1/30*(15*(2*c^2*d 
^3*e*f + (2*c^2*e^4*f - (12*c^2*d*e^3 - 5*b*c*e^4)*g)*x^3 + 3*(2*c^2*d*e^3 
*f - (12*c^2*d^2*e^2 - 5*b*c*d*e^3)*g)*x^2 - (12*c^2*d^4 - 5*b*c*d^3*e)*g 
+ 3*(2*c^2*d^2*e^2*f - (12*c^2*d^3*e - 5*b*c*d^2*e^2)*g)*x)*sqrt(c)*arctan 
(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c 
^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) + 2*(15*c^2*e^3*g*x^3 - (46*c 
^2*e^3*f - 7*(33*c^2*d*e^2 - 10*b*c*e^3)*g)*x^2 - 2*(13*c^2*d^2*e - b*c*d* 
e^2 + 3*b^2*e^3)*f + (141*c^2*d^3 - 32*b*c*d^2*e - 4*b^2*d*e^2)*g - (2*(24 
*c^2*d*e^2 + 11*b*c*e^3)*f - (333*c^2*d^2*e - 78*b*c*d*e^2 - 10*b^2*e^3)*g 
)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(e^5*x^3 + 3*d*e^4*x^2 + 
3*d^2*e^3*x + d^3*e^2)]
 

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {5}{2}} \left (f + g x\right )}{\left (d + e x\right )^{6}}\, dx \] 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)**6,x 
)
 

Output:

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(f + g*x)/(d + e*x)**6, x 
)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\text {Exception raised: ValueError} \] 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)^6,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 818 vs. \(2 (263) = 526\).

Time = 72.48 (sec) , antiderivative size = 818, normalized size of antiderivative = 2.91 \[ \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)^6} \, dx=\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} c^{2} g}{e^{2}} - \frac {{\left (2 \, \sqrt {-c} c^{2} e f - 12 \, \sqrt {-c} c^{2} d g + 5 \, b \sqrt {-c} c e g\right )} \log \left ({\left | -b \sqrt {-c} c^{3} d^{6} e^{2} - 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} c^{4} d^{6} {\left | e \right |} - 6 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b c^{3} d^{5} e {\left | e \right |} + 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} \sqrt {-c} c^{3} d^{5} e + 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b \sqrt {-c} c^{2} d^{4} e^{2} + 30 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} c^{3} d^{4} {\left | e \right |} + 20 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} b c^{2} d^{3} e {\left | e \right |} - 40 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{4} \sqrt {-c} c^{2} d^{3} e - 15 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{4} b \sqrt {-c} c d^{2} e^{2} - 30 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{5} c^{2} d^{2} {\left | e \right |} - 6 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{5} b c d e {\left | e \right |} + 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{6} \sqrt {-c} c d e + {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{6} b \sqrt {-c} e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{7} c {\left | e \right |} \right |}\right )}{14 \, e {\left | e \right |}} \] 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)^6,x, algo 
rithm="giac")
 

Output:

sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*c^2*g/e^2 - 1/14*(2*sqrt(-c)*c^ 
2*e*f - 12*sqrt(-c)*c^2*d*g + 5*b*sqrt(-c)*c*e*g)*log(abs(-b*sqrt(-c)*c^3* 
d^6*e^2 - 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))* 
c^4*d^6*abs(e) - 6*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b 
*d*e))*b*c^3*d^5*e*abs(e) + 12*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x 
 + c*d^2 - b*d*e))^2*sqrt(-c)*c^3*d^5*e + 15*(sqrt(-c*e^2)*x - sqrt(-c*e^2 
*x^2 - b*e^2*x + c*d^2 - b*d*e))^2*b*sqrt(-c)*c^2*d^4*e^2 + 30*(sqrt(-c*e^ 
2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*c^3*d^4*abs(e) + 20*( 
sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*b*c^2*d^3*e 
*abs(e) - 40*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)) 
^4*sqrt(-c)*c^2*d^3*e - 15*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c 
*d^2 - b*d*e))^4*b*sqrt(-c)*c*d^2*e^2 - 30*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x 
^2 - b*e^2*x + c*d^2 - b*d*e))^5*c^2*d^2*abs(e) - 6*(sqrt(-c*e^2)*x - sqrt 
(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^5*b*c*d*e*abs(e) + 12*(sqrt(-c*e^2 
)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^6*sqrt(-c)*c*d*e + (sqrt 
(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^6*b*sqrt(-c)*e^2 
+ 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^7*c*abs( 
e)))/(e*abs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^6} \,d x \] 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)^6,x)
                                                                                    
                                                                                    
 

Output:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2))/(d + e*x)^6, x 
)
 

Reduce [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 2008, normalized size of antiderivative = 7.15 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}}{(d+e x)^6} \, dx =\text {Too large to display} \] 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)^6,x)
 

Output:

(i*( - 300*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c* 
d))*b**2*c*d**3*e**2*g - 900*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/ 
sqrt( - b*e + 2*c*d))*b**2*c*d**2*e**3*g*x - 900*sqrt(c)*asinh((sqrt( - b* 
e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c*d*e**4*g*x**2 - 300*sqrt( 
c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c*e**5* 
g*x**3 + 1320*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2 
*c*d))*b*c**2*d**4*e*g - 120*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/ 
sqrt( - b*e + 2*c*d))*b*c**2*d**3*e**2*f + 3960*sqrt(c)*asinh((sqrt( - b*e 
 + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d**3*e**2*g*x - 360*sqrt(c 
)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d**2*e 
**3*f*x + 3960*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 
2*c*d))*b*c**2*d**2*e**3*g*x**2 - 360*sqrt(c)*asinh((sqrt( - b*e + c*d - c 
*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d*e**4*f*x**2 + 1320*sqrt(c)*asinh(( 
sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d*e**4*g*x**3 - 
 120*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b* 
c**2*e**5*f*x**3 - 1440*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
 - b*e + 2*c*d))*c**3*d**5*g + 240*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e* 
x)*i)/sqrt( - b*e + 2*c*d))*c**3*d**4*e*f - 4320*sqrt(c)*asinh((sqrt( - b* 
e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**3*d**4*e*g*x + 720*sqrt(c)*as 
inh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**3*d**3*e**2...