\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^5} \, dx\) [154]

Optimal result

Integrand size = 44, antiderivative size = 210 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {2 c g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{5 e^2 (2 c d-b e) (d+e x)^5}+\frac {2 c^{3/2} g \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{e^2} \] Output:

2*c*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)-2/3*g*(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/(-b*e+2*c*d)/(e*x+d)^5+2*c^(3/2)*g*arctan(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 = 0.73 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.13 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\frac {2 ((d+e x) (-b e+c (d-e x)))^{3/2} \left (\frac {-b^2 e^2 (3 e f+2 d g+5 e g x)-2 b c e \left (3 d^2 g+d e (-3 f+7 g x)+e^2 x (3 f+10 g x)\right )+c^2 \left (23 d^3 g-3 e^3 f x^2-3 d^2 e (f-18 g x)+d e^2 x (6 f+43 g x)\right )}{(2 c d-b e) (d+e x)^4 (-b e+c (d-e x))}-\frac {15 c^{3/2} g \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{3/2} (-b e+c (d-e x))^{3/2}}\right )}{15 e^2} \] Input:

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x 
)^5,x]
 

Output:

(2*((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((-(b^2*e^2*(3*e*f + 2*d*g + 5 
*e*g*x)) - 2*b*c*e*(3*d^2*g + d*e*(-3*f + 7*g*x) + e^2*x*(3*f + 10*g*x)) + 
 c^2*(23*d^3*g - 3*e^3*f*x^2 - 3*d^2*e*(f - 18*g*x) + d*e^2*x*(6*f + 43*g* 
x)))/((2*c*d - b*e)*(d + e*x)^4*(-(b*e) + c*(d - e*x))) - (15*c^(3/2)*g*Ar 
cTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/((d + e*x)^(3/2)*(- 
(b*e) + c*(d - e*x))^(3/2))))/(15*e^2)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1220, 1130, 1125, 27, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1125

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 217

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

Input:

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^5,x]
 

Output:

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(5*e^2*(2*c*d 
 - b*e)*(d + e*x)^5) + (g*((-2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2) 
)/(3*e*(d + e*x)^3) - c*((-2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(e 
*(d + e*x)) - (Sqrt[c]*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e 
) - b*e^2*x - c*e^2*x^2])])/e)))/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 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. \(562\) vs. \(2(194)=388\).

Time = 5.42 (sec) , antiderivative size = 563, normalized size of antiderivative = 2.68

method	result	size
default	\(\frac {g \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{3 \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{4}}-\frac {2 c \,e^{2} \left (-\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{3}}-\frac {4 c \,e^{2} \left (\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {6 c \,e^{2} \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (-b \,e^{2}+2 d e c \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 d e c \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 d e c \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 d e c}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2}\right )}{-b \,e^{2}+2 d e c}\right )}{-b \,e^{2}+2 d e c}\right )}{3 \left (-b \,e^{2}+2 d e c \right )}\right )}{e^{5}}+\frac {2 \left (d g -e f \right ) \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5 e^{6} \left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{5}}\)	\(563\)

Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^5,x,method=_RET 
URNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (194) = 388\).

Time = 5.39 (sec) , antiderivative size = 879, normalized size of antiderivative = 4.19 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^5,x, algo 
rithm="fricas")
 

Output:

[1/30*(15*((2*c^2*d*e^3 - b*c*e^4)*g*x^3 + 3*(2*c^2*d^2*e^2 - b*c*d*e^3)*g 
*x^2 + 3*(2*c^2*d^3*e - b*c*d^2*e^2)*g*x + (2*c^2*d^4 - b*c*d^3*e)*g)*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* 
sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((3*c^2*e^3*f - (43*c^2*d*e^2 - 
 20*b*c*e^3)*g)*x^2 + 3*(c^2*d^2*e - 2*b*c*d*e^2 + b^2*e^3)*f - (23*c^2*d^ 
3 - 6*b*c*d^2*e - 2*b^2*d*e^2)*g - (6*(c^2*d*e^2 - b*c*e^3)*f + (54*c^2*d^ 
2*e - 14*b*c*d*e^2 - 5*b^2*e^3)*g)*x))/(2*c*d^4*e^2 - b*d^3*e^3 + (2*c*d*e 
^5 - b*e^6)*x^3 + 3*(2*c*d^2*e^4 - b*d*e^5)*x^2 + 3*(2*c*d^3*e^3 - b*d^2*e 
^4)*x), -1/15*(15*((2*c^2*d*e^3 - b*c*e^4)*g*x^3 + 3*(2*c^2*d^2*e^2 - b*c* 
d*e^3)*g*x^2 + 3*(2*c^2*d^3*e - b*c*d^2*e^2)*g*x + (2*c^2*d^4 - b*c*d^3*e) 
*g)*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*sqrt(-c 
*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*((3*c^2*e^3*f - (43*c^2*d*e^2 - 20*b*c 
*e^3)*g)*x^2 + 3*(c^2*d^2*e - 2*b*c*d*e^2 + b^2*e^3)*f - (23*c^2*d^3 - 6*b 
*c*d^2*e - 2*b^2*d*e^2)*g - (6*(c^2*d*e^2 - b*c*e^3)*f + (54*c^2*d^2*e - 1 
4*b*c*d*e^2 - 5*b^2*e^3)*g)*x))/(2*c*d^4*e^2 - b*d^3*e^3 + (2*c*d*e^5 - b* 
e^6)*x^3 + 3*(2*c*d^2*e^4 - b*d*e^5)*x^2 + 3*(2*c*d^3*e^3 - b*d^2*e^4)*x)]
 

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{5}}\, dx \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**5,x 
)
 

Output:

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**5, x 
)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^5,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. 1592 vs. \(2 (194) = 388\).

Time = 0.44 (sec) , antiderivative size = 1592, normalized size of antiderivative = 7.58 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^5,x, algo 
rithm="giac")
 

Output:

-2/15*(15*c^(3/2)*g*arctan(sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))/sqrt 
(c))*sgn(1/(e*x + d))*sgn(e)/e^3 - (30*c^(5/2)*d*g*arctan(sqrt(-c)/sqrt(c) 
) - 15*b*c^(3/2)*e*g*arctan(sqrt(-c)/sqrt(c)) + 3*sqrt(-c)*c^2*e*f - 43*sq 
rt(-c)*c^2*d*g + 20*b*sqrt(-c)*c*e*g)*sgn(1/(e*x + d))*sgn(e)/(2*c*d*e^3 - 
 b*e^4) + (48*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*c^4*sqrt(-c + 2*c*d/ 
(e*x + d) - b*e/(e*x + d))*d^4*e^13*f*sgn(1/(e*x + d))*sgn(e) - 96*b*(c - 
2*c*d/(e*x + d) + b*e/(e*x + d))^2*c^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e* 
x + d))*d^3*e^14*f*sgn(1/(e*x + d))*sgn(e) + 72*b^2*(c - 2*c*d/(e*x + d) + 
 b*e/(e*x + d))^2*c^2*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^2*e^15* 
f*sgn(1/(e*x + d))*sgn(e) - 24*b^3*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2 
*c*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d*e^16*f*sgn(1/(e*x + d))*sg 
n(e) + 3*b^4*(c - 2*c*d/(e*x + d) + b*e/(e*x + d))^2*sqrt(-c + 2*c*d/(e*x 
+ d) - b*e/(e*x + d))*e^17*f*sgn(1/(e*x + d))*sgn(e) - 48*(c - 2*c*d/(e*x 
+ d) + b*e/(e*x + d))^2*c^4*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^5 
*e^12*g*sgn(1/(e*x + d))*sgn(e) - 480*c^6*sqrt(-c + 2*c*d/(e*x + d) - b*e/ 
(e*x + d))*d^5*e^12*g*sgn(1/(e*x + d))*sgn(e) + 160*c^5*(-c + 2*c*d/(e*x + 
 d) - b*e/(e*x + d))^(3/2)*d^5*e^12*g*sgn(1/(e*x + d))*sgn(e) + 96*b*(c - 
2*c*d/(e*x + d) + b*e/(e*x + d))^2*c^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e* 
x + d))*d^4*e^13*g*sgn(1/(e*x + d))*sgn(e) + 1200*b*c^5*sqrt(-c + 2*c*d/(e 
*x + d) - b*e/(e*x + d))*d^4*e^13*g*sgn(1/(e*x + d))*sgn(e) - 400*b*c^4...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^5} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^5,x)
 

Output:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(d + e*x)^5, x 
)
 

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 1667, normalized size of antiderivative = 7.94 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^5,x)
 

Output:

(2*i*(15*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 + 45*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqr 
t( - b*e + 2*c*d))*b**2*c*d**2*e**3*g*x + 45*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 + 15*sqrt(c)*as 
inh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c*e**5*g*x** 
3 - 60*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 - 180*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 - 180*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 - 60*sqrt(c)*as 
inh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d*e**4*g*x 
**3 + 60*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d) 
)*c**3*d**5*g + 180*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 + 180*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e* 
x)*i)/sqrt( - b*e + 2*c*d))*c**3*d**3*e**2*g*x**2 + 60*sqrt(c)*asinh((sqrt 
( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**3*d**2*e**3*g*x**3 + 2* 
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**2*d*e**2*g + 3*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**2*e**3*f + 5*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**2*e**3*g*x + 6*sqrt 
(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*...