\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^2} \, dx\) [151]

Optimal result

Integrand size = 44, antiderivative size = 211 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {(6 c e f-4 c d g-b e g) (8 c d-5 b e-2 c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c e^2}-\frac {g (c d-b e-c e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}+\frac {(2 c d-b e)^2 (6 c e f-4 c d g-b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e^2} \] Output:

1/24*(-b*e*g-4*c*d*g+6*c*e*f)*(-2*c*e*x-5*b*e+8*c*d)*(d*(-b*e+c*d)-b*e^2*x 
-c*e^2*x^2)^(1/2)/c/e^2-1/3*g*(-c*e*x-b*e+c*d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e 
^2*x^2)^(1/2)/c/e^2+1/8*(-b*e+2*c*d)^2*(-b*e*g-4*c*d*g+6*c*e*f)*arctan(c^( 
1/2)*(e*x+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(3/2)/e^2
 

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.04 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {\sqrt {d+e x} \sqrt {c d-b e-c e x} \left (-\sqrt {c} \sqrt {d+e x} \sqrt {c d-b e-c e x} \left (3 b^2 e^2 g+2 b c e (15 e f-14 d g+7 e g x)+4 c^2 \left (10 d^2 g-6 d e (2 f+g x)+e^2 x (3 f+2 g x)\right )\right )+3 (-2 c d+b e)^2 (-6 c e f+4 c d g+b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )\right )}{24 c^{3/2} e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] 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 
)^2,x]
 

Output:

(Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]*(-(Sqrt[c]*Sqrt[d + e*x]*Sqrt[c*d - 
 b*e - c*e*x]*(3*b^2*e^2*g + 2*b*c*e*(15*e*f - 14*d*g + 7*e*g*x) + 4*c^2*( 
10*d^2*g - 6*d*e*(2*f + g*x) + e^2*x*(3*f + 2*g*x)))) + 3*(-2*c*d + b*e)^2 
*(-6*c*e*f + 4*c*d*g + b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt 
[d + e*x])]))/(24*c^(3/2)*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.24, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1220, 1131, 1087, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(-b e g-4 c d g+6 c e f) \left (\frac {1}{2} (2 c d-b e) \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right )}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2 (2 c d-b e)}\)

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (\frac {1}{2} (2 c d-b e) \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e}\right ) (-b e g-4 c d g+6 c e f)}{e (2 c d-b e)}+\frac {2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2 (2 c d-b e)}\)

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)^2,x]
 

Output:

(2*(e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - 
b*e)*(d + e*x)^2) + ((6*c*e*f - 4*c*d*g - b*e*g)*((d*(c*d - b*e) - b*e^2*x 
 - c*e^2*x^2)^(3/2)/(3*e) + ((2*c*d - b*e)*(((b + 2*c*x)*Sqrt[d*(c*d - b*e 
) - b*e^2*x - c*e^2*x^2])/(4*c) + ((2*c*d - b*e)^2*ArcTan[(e*(b + 2*c*x))/ 
(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(8*c^(3/2)*e)))/2) 
)/(e*(2*c*d - b*e))
 

Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1131

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 + 2*p + 1))), x 
] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1)))   Int[(d + e*x)^(m + 1)*(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] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne 
Q[m + 2*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. \(563\) vs. \(2(195)=390\).

Time = 2.86 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.67

method	result	size
default	\(\frac {g \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (-b \,e^{2}+2 d e c \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 d e c \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 d e c \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 d e c}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2}\right )}{e^{2}}-\frac {\left (d g -e f \right ) \left (\frac {2 \left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {6 c \,e^{2} \left (\frac {\left (-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (-b \,e^{2}+2 d e c \right ) \left (-\frac {\left (-2 c \,e^{2} \left (x +\frac {d}{e}\right )-b \,e^{2}+2 d e c \right ) \sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}{4 c \,e^{2}}+\frac {\left (-b \,e^{2}+2 d e c \right )^{2} \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {d}{e}-\frac {-b \,e^{2}+2 d e c}{2 c \,e^{2}}\right )}{\sqrt {-c \,e^{2} \left (x +\frac {d}{e}\right )^{2}+\left (-b \,e^{2}+2 d e c \right ) \left (x +\frac {d}{e}\right )}}\right )}{8 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{2}\right )}{-b \,e^{2}+2 d e c}\right )}{e^{3}}\)	\(564\)

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)^2,x,method=_RET 
URNVERBOSE)
 

Output:

g/e^2*(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)*ar 
ctan((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^3*(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)*arct 
an((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 = 0.22 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.69 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\left [\frac {3 \, {\left (6 \, {\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f - {\left (16 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (8 \, c^{3} e^{2} g x^{2} - 6 \, {\left (8 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} f + {\left (40 \, c^{3} d^{2} - 28 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + 2 \, {\left (6 \, c^{3} e^{2} f - {\left (12 \, c^{3} d e - 7 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{96 \, c^{2} e^{2}}, -\frac {3 \, {\left (6 \, {\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f - {\left (16 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, {\left (8 \, c^{3} e^{2} g x^{2} - 6 \, {\left (8 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} f + {\left (40 \, c^{3} d^{2} - 28 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} g + 2 \, {\left (6 \, c^{3} e^{2} f - {\left (12 \, c^{3} d e - 7 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, c^{2} e^{2}}\right ] \] 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)^2,x, algo 
rithm="fricas")
 

Output:

[1/96*(3*(6*(4*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*f - (16*c^3*d^3 - 12 
*b*c^2*d^2*e + b^3*e^3)*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*(8*c^3*e^2*g*x^2 - 6*(8*c^3*d*e - 5*b*c^2*e 
^2)*f + (40*c^3*d^2 - 28*b*c^2*d*e + 3*b^2*c*e^2)*g + 2*(6*c^3*e^2*f - (12 
*c^3*d*e - 7*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/ 
(c^2*e^2), -1/48*(3*(6*(4*c^3*d^2*e - 4*b*c^2*d*e^2 + b^2*c*e^3)*f - (16*c 
^3*d^3 - 12*b*c^2*d^2*e + b^3*e^3)*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*(8*c^3*e^2*g*x^2 - 6*(8*c^3*d*e - 5*b*c^2*e^2)* 
f + (40*c^3*d^2 - 28*b*c^2*d*e + 3*b^2*c*e^2)*g + 2*(6*c^3*e^2*f - (12*c^3 
*d*e - 7*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^2 
*e^2)]
                                                                                    
                                                                                    
 

Sympy [F]

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] 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)^2,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. 1297 vs. \(2 (194) = 388\).

Time = 0.66 (sec) , antiderivative size = 1297, normalized size of antiderivative = 6.15 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Too large to display} \] 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)^2,x, algo 
rithm="giac")
 

Output:

-1/24*(3*(24*c^3*d^2*e*f*sgn(1/(e*x + d))*sgn(e) - 24*b*c^2*d*e^2*f*sgn(1/ 
(e*x + d))*sgn(e) + 6*b^2*c*e^3*f*sgn(1/(e*x + d))*sgn(e) - 16*c^3*d^3*g*s 
gn(1/(e*x + d))*sgn(e) + 12*b*c^2*d^2*e*g*sgn(1/(e*x + d))*sgn(e) - b^3*e^ 
3*g*sgn(1/(e*x + d))*sgn(e))*arctan(sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + 
 d))/sqrt(c))/(c^(3/2)*e^3) - (120*(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^2*e*f*sgn(1/(e*x + d))*s 
gn(e) + 72*c^5*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^2*e*f*sgn(1/(e 
*x + d))*sgn(e) + 192*c^4*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/2)*d^2 
*e*f*sgn(1/(e*x + d))*sgn(e) - 120*b*(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*e^2*f*sgn(1/(e*x + d)) 
*sgn(e) - 72*b*c^4*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d*e^2*f*sgn( 
1/(e*x + d))*sgn(e) - 192*b*c^3*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3/ 
2)*d*e^2*f*sgn(1/(e*x + d))*sgn(e) + 30*b^2*(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))*e^3*f*sgn(1/(e*x + 
d))*sgn(e) + 18*b^2*c^3*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*e^3*f*s 
gn(1/(e*x + d))*sgn(e) + 48*b^2*c^2*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d)) 
^(3/2)*e^3*f*sgn(1/(e*x + d))*sgn(e) - 144*(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*g*sgn(1/(e*x + 
 d))*sgn(e) - 48*c^5*sqrt(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))*d^3*g*sgn( 
1/(e*x + d))*sgn(e) - 128*c^4*(-c + 2*c*d/(e*x + d) - b*e/(e*x + d))^(3...
 

Mupad [F(-1)]

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

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 839, normalized size of antiderivative = 3.98 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx =\text {Too large to display} \] 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)^2,x)
 

Output:

(i*( - 3*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d) 
)*b**4*e**4*g + 6*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e 
 + 2*c*d))*b**3*c*d*e**3*g + 18*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)* 
i)/sqrt( - b*e + 2*c*d))*b**3*c*e**4*f + 36*sqrt(c)*asinh((sqrt( - b*e + c 
*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**2*d**2*e**2*g - 108*sqrt(c)*a 
sinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**2*d*e**3 
*f - 120*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d) 
)*b*c**3*d**3*e*g + 216*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
 - b*e + 2*c*d))*b*c**3*d**2*e**2*f + 96*sqrt(c)*asinh((sqrt( - b*e + c*d 
- c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**4*d**4*g - 144*sqrt(c)*asinh((sqrt( - 
 b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**4*d**3*e*f - 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* 
c*e**2*g + 28*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*c**2*d*e*g - 30*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*c**2*e**2*f - 14*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*c** 
2*e**2*g*x - 40*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( 
 - b*e + c*d - c*e*x)*c**3*d**2*g + 48*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)*c**3*d*e*f + 24*sqrt(d + e*x) 
*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**3...