\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^4} \, dx\) [153]

Optimal result

Integrand size = 44, antiderivative size = 220 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {c g \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2}+\frac {2 (c e f-3 c d g+b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^3}-\frac {\sqrt {c} (2 c e f-8 c d g+3 b e g) \arctan \left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {c} (d+e x)}\right )}{e^2} \] Output:

-c*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2+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)/e^2/(e*x+d)-2/3*(-d*g+e*f)*(d*(-b*e+c 
*d)-b*e^2*x-c*e^2*x^2)^(3/2)/e^2/(e*x+d)^3-c^(1/2)*(3*b*e*g-8*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 = 0.62 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.85 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx=\frac {((d+e x) (-b e+c (d-e x)))^{3/2} \left (\frac {c \left (-19 d^2 g+d e (4 f-26 g x)+e^2 x (8 f-3 g x)\right )+2 b e (2 d g+e (f+3 g x))}{(d+e x)^3 (-b e+c (d-e x))}-\frac {3 \sqrt {c} (2 c e f-8 c d g+3 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{3/2} (-b e+c (d-e x))^{3/2}}\right )}{3 e^2} \] 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 
)^4,x]
 

Output:

(((d + e*x)*(-(b*e) + c*(d - e*x)))^(3/2)*((c*(-19*d^2*g + d*e*(4*f - 26*g 
*x) + e^2*x*(8*f - 3*g*x)) + 2*b*e*(2*d*g + e*(f + 3*g*x)))/((d + e*x)^3*( 
-(b*e) + c*(d - e*x))) - (3*Sqrt[c]*(2*c*e*f - 8*c*d*g + 3*b*e*g)*ArcTan[S 
qrt[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))))/(3*e^2)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.18, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1220, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1125

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

\(\displaystyle -\frac {(3 b e g-8 c d g+2 c e f) \left (-c \left (\frac {3}{2} (2 c d-b e) \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^4 (2 c d-b e)}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {(3 b e g-8 c d g+2 c e f) \left (-c \left (3 (2 c d-b e) \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e}\right )-\frac {2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)}\right )}{3 e (2 c d-b e)}-\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{3 e^2 (d+e x)^4 (2 c d-b e)}\)

\(\Big \downarrow \) 217

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

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)^4,x]
 

Output:

(-2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(3*e^2*(2*c*d 
 - b*e)*(d + e*x)^4) - ((2*c*e*f - 8*c*d*g + 3*b*e*g)*((-2*(2*c*d - b*e)*S 
qrt[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 
*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 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. \(899\) vs. \(2(206)=412\).

Time = 4.45 (sec) , antiderivative size = 900, normalized size of antiderivative = 4.09

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}}}{\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 )}{e^{4}}-\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 {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}}\)	\(900\)

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)^4,x,method=_RET 
URNVERBOSE)
 

Output:

g/e^4*(-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))))))-(d*g-e*f)/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)))))))
 

Fricas [A] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.70 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx=\left [\frac {3 \, {\left (2 \, c d^{2} e f + {\left (2 \, c e^{3} f - {\left (8 \, c d e^{2} - 3 \, b e^{3}\right )} g\right )} x^{2} - {\left (8 \, c d^{3} - 3 \, b d^{2} e\right )} g + 2 \, {\left (2 \, c d e^{2} f - {\left (8 \, c d^{2} e - 3 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (3 \, c e^{2} g x^{2} - 2 \, {\left (2 \, c d e + b e^{2}\right )} f + {\left (19 \, c d^{2} - 4 \, b d e\right )} g - 2 \, {\left (4 \, c e^{2} f - {\left (13 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{12 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, -\frac {3 \, {\left (2 \, c d^{2} e f + {\left (2 \, c e^{3} f - {\left (8 \, c d e^{2} - 3 \, b e^{3}\right )} g\right )} x^{2} - {\left (8 \, c d^{3} - 3 \, b d^{2} e\right )} g + 2 \, {\left (2 \, c d e^{2} f - {\left (8 \, c d^{2} e - 3 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, {\left (3 \, c e^{2} g x^{2} - 2 \, {\left (2 \, c d e + b e^{2}\right )} f + {\left (19 \, c d^{2} - 4 \, b d e\right )} g - 2 \, {\left (4 \, c e^{2} f - {\left (13 \, c d e - 3 \, b e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{6 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \] 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)^4,x, algo 
rithm="fricas")
 

Output:

[1/12*(3*(2*c*d^2*e*f + (2*c*e^3*f - (8*c*d*e^2 - 3*b*e^3)*g)*x^2 - (8*c*d 
^3 - 3*b*d^2*e)*g + 2*(2*c*d*e^2*f - (8*c*d^2*e - 3*b*d*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*sq 
rt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(3* 
c*e^2*g*x^2 - 2*(2*c*d*e + b*e^2)*f + (19*c*d^2 - 4*b*d*e)*g - 2*(4*c*e^2* 
f - (13*c*d*e - 3*b*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)) 
/(e^4*x^2 + 2*d*e^3*x + d^2*e^2), -1/6*(3*(2*c*d^2*e*f + (2*c*e^3*f - (8*c 
*d*e^2 - 3*b*e^3)*g)*x^2 - (8*c*d^3 - 3*b*d^2*e)*g + 2*(2*c*d*e^2*f - (8*c 
*d^2*e - 3*b*d*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*(3*c*e^2*g*x^2 - 2*(2*c*d*e + b*e^2)*f + (19*c*d^2 - 4*b*d* 
e)*g - 2*(4*c*e^2*f - (13*c*d*e - 3*b*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x 
 + c*d^2 - b*d*e))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)]
 

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{4}}\, dx \] 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)**4,x 
)
 

Output:

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**4, x 
)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] 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)^4,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. 582 vs. \(2 (206) = 412\).

Time = 1.60 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.65 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx=-\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} c g}{e^{2}} + \frac {{\left (2 \, \sqrt {-c} c e f - 8 \, \sqrt {-c} c d g + 3 \, b \sqrt {-c} e g\right )} \log \left ({\left | b \sqrt {-c} c^{2} d^{4} e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} c^{3} d^{4} {\left | e \right |} + 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} b c^{2} d^{3} e {\left | e \right |} - 8 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} \sqrt {-c} c^{2} d^{3} e - 6 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{2} b \sqrt {-c} c d^{2} e^{2} - 12 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} c^{2} d^{2} {\left | e \right |} - 4 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{3} b c d e {\left | e \right |} + 8 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{4} \sqrt {-c} c d e + {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{4} b \sqrt {-c} e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )}^{5} c {\left | e \right |} \right |}\right )}{10 \, e {\left | e \right |}} \] 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)^4,x, algo 
rithm="giac")
 

Output:

-sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*c*g/e^2 + 1/10*(2*sqrt(-c)*c*e 
*f - 8*sqrt(-c)*c*d*g + 3*b*sqrt(-c)*e*g)*log(abs(b*sqrt(-c)*c^2*d^4*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^3*d^4*a 
bs(e) + 4*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*b* 
c^2*d^3*e*abs(e) - 8*(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^2*d^3*e - 6*(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*d^2*e^2 - 12*(sqrt(-c*e^2)*x - sqrt(- 
c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*c^2*d^2*abs(e) - 4*(sqrt(-c*e^2)*x 
 - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^3*b*c*d*e*abs(e) + 8*(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*d*e + 
 (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 
)*e^2 + 2*(sqrt(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))^5* 
c*abs(e)))/(e*abs(e))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^4} \,d x \] 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)^4,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)^4, x 
)
 

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 1184, normalized size of antiderivative = 5.38 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^4} \, dx =\text {Too large to display} \] 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)^4,x)
 

Output:

(i*(18*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))* 
b**2*d**2*e**2*g + 36*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - 
 b*e + 2*c*d))*b**2*d*e**3*g*x + 18*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e 
*x)*i)/sqrt( - b*e + 2*c*d))*b**2*e**4*g*x**2 - 84*sqrt(c)*asinh((sqrt( - 
b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c*d**3*e*g + 12*sqrt(c)*asin 
h((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c*d**2*e**2*f - 1 
68*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c* 
d**2*e**2*g*x + 24*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b* 
e + 2*c*d))*b*c*d*e**3*f*x - 84*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)* 
i)/sqrt( - b*e + 2*c*d))*b*c*d*e**3*g*x**2 + 12*sqrt(c)*asinh((sqrt( - b*e 
 + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c*e**4*f*x**2 + 96*sqrt(c)*asin 
h((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**2*d**4*g - 24*sq 
rt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**2*d**3 
*e*f + 192*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c* 
d))*c**2*d**3*e*g*x - 48*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt 
( - b*e + 2*c*d))*c**2*d**2*e**2*f*x + 96*sqrt(c)*asinh((sqrt( - b*e + c*d 
 - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**2*d**2*e**2*g*x**2 - 24*sqrt(c)*asin 
h((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**2*d*e**3*f*x**2 
+ 8*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*d*e*g + 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c...