\(\int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [276]

Optimal result

Integrand size = 44, antiderivative size = 203 \[ \int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2}-\frac {(6 c e f+4 c d g-5 b e g) (8 c d-3 b e+2 c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^3 e^2}+\frac {(2 c d-b e)^2 (6 c e f+4 c d g-5 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{7/2} e^2} \] Output:

-1/3*g*(e*x+d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/c/e^2-1/24*(-5*b*e 
*g+4*c*d*g+6*c*e*f)*(2*c*e*x-3*b*e+8*c*d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2) 
^(1/2)/c^3/e^2+1/8*(-b*e+2*c*d)^2*(-5*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^(7/2)/e^2
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {(-2 c d+b e)^2 \left (-\frac {\sqrt {c} (d+e x) (-b e+c (d-e x)) \left (15 b^2 e^2 g-2 b c e (9 e f+26 d g+5 e g x)+4 c^2 \left (10 d^2 g+6 d e (2 f+g x)+e^2 x (3 f+2 g x)\right )\right )}{(-2 c d+b e)^2}-3 (6 c e f+4 c d g-5 b e g) \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )\right )}{24 c^{7/2} e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

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

Output:

((-2*c*d + b*e)^2*(-((Sqrt[c]*(d + e*x)*(-(b*e) + c*(d - e*x))*(15*b^2*e^2 
*g - 2*b*c*e*(9*e*f + 26*d*g + 5*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))))/(-2*c*d + b*e)^2) - 3*(6*c*e*f + 4*c*d*g - 5* 
b*e*g)*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]*ArcTan[Sqrt[c*d - b*e - c*e*x 
]/(Sqrt[c]*Sqrt[d + e*x])]))/(24*c^(7/2)*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d 
 - e*x))])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1221, 1134, 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.

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

\(\Big \downarrow \) 1221

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

-1/3*(g*(d + e*x)^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c*e^2) + ( 
(6*c*e*f + 4*c*d*g - 5*b*e*g)*(-1/2*((d + e*x)*Sqrt[d*(c*d - b*e) - b*e^2* 
x - c*e^2*x^2])/(c*e) + (3*(2*c*d - b*e)*(-(Sqrt[d*(c*d - b*e) - b*e^2*x - 
 c*e^2*x^2]/(c*e)) + ((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*c^(3/2)*e)))/(4*c)))/(6*c*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 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 1134

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d + e*x)^ 
(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ 
c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*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 1221

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 
)/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   Int[(d + e*x)^m*(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] 
 && NeQ[m + 2*p + 2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(772\) vs. \(2(187)=374\).

Time = 2.02 (sec) , antiderivative size = 773, normalized size of antiderivative = 3.81

method	result	size
default	\(\frac {d^{2} f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}}+e \left (2 d g +e f \right ) \left (-\frac {x \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{2 c \,e^{2}}-\frac {3 b \left (-\frac {\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{c \,e^{2}}-\frac {b \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \sqrt {c \,e^{2}}}\right )}{4 c}+\frac {\left (-b d e +c \,d^{2}\right ) \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \,e^{2} \sqrt {c \,e^{2}}}\right )+d \left (d g +2 e f \right ) \left (-\frac {\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{c \,e^{2}}-\frac {b \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \sqrt {c \,e^{2}}}\right )+e^{2} g \left (-\frac {x^{2} \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{3 c \,e^{2}}-\frac {5 b \left (-\frac {x \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{2 c \,e^{2}}-\frac {3 b \left (-\frac {\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{c \,e^{2}}-\frac {b \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \sqrt {c \,e^{2}}}\right )}{4 c}+\frac {\left (-b d e +c \,d^{2}\right ) \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \,e^{2} \sqrt {c \,e^{2}}}\right )}{6 c}+\frac {2 \left (-b d e +c \,d^{2}\right ) \left (-\frac {\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{c \,e^{2}}-\frac {b \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \sqrt {c \,e^{2}}}\right )}{3 c \,e^{2}}\right )\)	\(773\)

Input:

int((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 585, normalized size of antiderivative = 2.88 \[ \int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\left [\frac {3 \, {\left (6 \, {\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (16 \, c^{3} d^{3} - 36 \, b c^{2} d^{2} e + 24 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} + 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) - 4 \, {\left (8 \, c^{3} e^{2} g x^{2} + 6 \, {\left (8 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f + {\left (40 \, c^{3} d^{2} - 52 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + 2 \, {\left (6 \, c^{3} e^{2} f + {\left (12 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{96 \, c^{4} e^{2}}, -\frac {3 \, {\left (6 \, {\left (4 \, c^{3} d^{2} e - 4 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + {\left (16 \, c^{3} d^{3} - 36 \, b c^{2} d^{2} e + 24 \, b^{2} c d e^{2} - 5 \, b^{3} e^{3}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, {\left (8 \, c^{3} e^{2} g x^{2} + 6 \, {\left (8 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} f + {\left (40 \, c^{3} d^{2} - 52 \, b c^{2} d e + 15 \, b^{2} c e^{2}\right )} g + 2 \, {\left (6 \, c^{3} e^{2} f + {\left (12 \, c^{3} d e - 5 \, b c^{2} e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{48 \, c^{4} e^{2}}\right ] \] Input:

integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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 - 36 
*b*c^2*d^2*e + 24*b^2*c*d*e^2 - 5*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 - 3*b*c^2*e^2)*f + (40*c^3*d^2 - 52*b*c^2*d*e + 15*b^2*c*e^2)*g + 
2*(6*c^3*e^2*f + (12*c^3*d*e - 5*b*c^2*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2* 
x + c*d^2 - b*d*e))/(c^4*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 - 36*b*c^2*d^2*e + 24*b^2*c*d*e^2 - 5*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 - 3*b*c^2*e^2)*f + (40*c^3*d^2 - 52*b*c^2*d*e + 15 
*b^2*c*e^2)*g + 2*(6*c^3*e^2*f + (12*c^3*d*e - 5*b*c^2*e^2)*g)*x)*sqrt(-c* 
e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^4*e^2)]
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (194) = 388\).

Time = 1.37 (sec) , antiderivative size = 660, normalized size of antiderivative = 3.25 \[ \int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\begin {cases} \left (- \frac {b \left (- \frac {3 b \left (- \frac {5 b e^{2} g}{6 c} + 2 d e g + e^{2} f\right )}{4 c} + d^{2} g + 2 d e f + \frac {g \left (- 2 b d e + 2 c d^{2}\right )}{3 c}\right )}{2 c} + d^{2} f + \frac {\left (- b d e + c d^{2}\right ) \left (- \frac {5 b e^{2} g}{6 c} + 2 d e g + e^{2} f\right )}{2 c e^{2}}\right ) \left (\begin {cases} \frac {\log {\left (- b e^{2} - 2 c e^{2} x + 2 \sqrt {- c e^{2}} \sqrt {- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}} \right )}}{\sqrt {- c e^{2}}} & \text {for}\: \frac {b^{2} e^{2}}{4 c} - b d e + c d^{2} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {- c e^{2} \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \left (- \frac {g x^{2}}{3 c} - \frac {x \left (- \frac {5 b e^{2} g}{6 c} + 2 d e g + e^{2} f\right )}{2 c e^{2}} - \frac {- \frac {3 b \left (- \frac {5 b e^{2} g}{6 c} + 2 d e g + e^{2} f\right )}{4 c} + d^{2} g + 2 d e f + \frac {g \left (- 2 b d e + 2 c d^{2}\right )}{3 c}}{c e^{2}}\right ) \sqrt {- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}} & \text {for}\: c e^{2} \neq 0 \\- \frac {2 \left (- \frac {g \left (- b d e - b e^{2} x + c d^{2}\right )^{\frac {7}{2}}}{7 b^{3} e^{4}} - \frac {\left (- b d e - b e^{2} x + c d^{2}\right )^{\frac {5}{2}} \left (b d e g - b e^{2} f - 3 c d^{2} g\right )}{5 b^{3} e^{4}} - \frac {\left (- b d e - b e^{2} x + c d^{2}\right )^{\frac {3}{2}} \left (- 2 b c d^{3} e g + 2 b c d^{2} e^{2} f + 3 c^{2} d^{4} g\right )}{3 b^{3} e^{4}} - \frac {\sqrt {- b d e - b e^{2} x + c d^{2}} \left (b c^{2} d^{5} e g - b c^{2} d^{4} e^{2} f - c^{3} d^{6} g\right )}{b^{3} e^{4}}\right )}{b e^{2}} & \text {for}\: b e^{2} \neq 0 \\\frac {d^{2} f x + \frac {e^{2} g x^{4}}{4} + \frac {x^{3} \cdot \left (2 d e g + e^{2} f\right )}{3} + \frac {x^{2} \left (d^{2} g + 2 d e f\right )}{2}}{\sqrt {- b d e + c d^{2}}} & \text {otherwise} \end {cases} \] Input:

integrate((e*x+d)**2*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x 
)
 

Output:

Piecewise(((-b*(-3*b*(-5*b*e**2*g/(6*c) + 2*d*e*g + e**2*f)/(4*c) + d**2*g 
 + 2*d*e*f + g*(-2*b*d*e + 2*c*d**2)/(3*c))/(2*c) + d**2*f + (-b*d*e + c*d 
**2)*(-5*b*e**2*g/(6*c) + 2*d*e*g + e**2*f)/(2*c*e**2))*Piecewise((log(-b* 
e**2 - 2*c*e**2*x + 2*sqrt(-c*e**2)*sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e* 
*2*x**2))/sqrt(-c*e**2), Ne(b**2*e**2/(4*c) - b*d*e + c*d**2, 0)), ((b/(2* 
c) + x)*log(b/(2*c) + x)/sqrt(-c*e**2*(b/(2*c) + x)**2), True)) + (-g*x**2 
/(3*c) - x*(-5*b*e**2*g/(6*c) + 2*d*e*g + e**2*f)/(2*c*e**2) - (-3*b*(-5*b 
*e**2*g/(6*c) + 2*d*e*g + e**2*f)/(4*c) + d**2*g + 2*d*e*f + g*(-2*b*d*e + 
 2*c*d**2)/(3*c))/(c*e**2))*sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2) 
, Ne(c*e**2, 0)), (-2*(-g*(-b*d*e - b*e**2*x + c*d**2)**(7/2)/(7*b**3*e**4 
) - (-b*d*e - b*e**2*x + c*d**2)**(5/2)*(b*d*e*g - b*e**2*f - 3*c*d**2*g)/ 
(5*b**3*e**4) - (-b*d*e - b*e**2*x + c*d**2)**(3/2)*(-2*b*c*d**3*e*g + 2*b 
*c*d**2*e**2*f + 3*c**2*d**4*g)/(3*b**3*e**4) - sqrt(-b*d*e - b*e**2*x + c 
*d**2)*(b*c**2*d**5*e*g - b*c**2*d**4*e**2*f - c**3*d**6*g)/(b**3*e**4))/( 
b*e**2), Ne(b*e**2, 0)), ((d**2*f*x + e**2*g*x**4/4 + x**3*(2*d*e*g + e**2 
*f)/3 + x**2*(d**2*g + 2*d*e*f)/2)/sqrt(-b*d*e + c*d**2), True))
 

Maxima [F(-2)]

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

integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),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(e*(b*e-2*c*d)>0)', see `assume?` 
 for more
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.39 \[ \int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {1}{24} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (\frac {4 \, g x}{c} + \frac {6 \, c^{2} e^{3} f + 12 \, c^{2} d e^{2} g - 5 \, b c e^{3} g}{c^{3} e^{3}}\right )} x + \frac {48 \, c^{2} d e^{2} f - 18 \, b c e^{3} f + 40 \, c^{2} d^{2} e g - 52 \, b c d e^{2} g + 15 \, b^{2} e^{3} g}{c^{3} e^{3}}\right )} - \frac {{\left (24 \, c^{3} d^{2} e f - 24 \, b c^{2} d e^{2} f + 6 \, b^{2} c e^{3} f + 16 \, c^{3} d^{3} g - 36 \, b c^{2} d^{2} e g + 24 \, b^{2} c d e^{2} g - 5 \, b^{3} e^{3} g\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{16 \, \sqrt {-c} c^{3} e {\left | e \right |}} \] Input:

integrate((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo 
rithm="giac")
 

Output:

-1/24*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*g*x/c + (6*c^2*e^3* 
f + 12*c^2*d*e^2*g - 5*b*c*e^3*g)/(c^3*e^3))*x + (48*c^2*d*e^2*f - 18*b*c* 
e^3*f + 40*c^2*d^2*e*g - 52*b*c*d*e^2*g + 15*b^2*e^3*g)/(c^3*e^3)) - 1/16* 
(24*c^3*d^2*e*f - 24*b*c^2*d*e^2*f + 6*b^2*c*e^3*f + 16*c^3*d^3*g - 36*b*c 
^2*d^2*e*g + 24*b^2*c*d*e^2*g - 5*b^3*e^3*g)*log(abs(-b*e^2 + 2*(sqrt(-c*e 
^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*abs(e)))/(sqr 
t(-c)*c^3*e*abs(e))
 

Mupad [F(-1)]

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

int(((f + g*x)*(d + e*x)^2)/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2),x)
 

Output:

int(((f + g*x)*(d + e*x)^2)/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2), x 
)
 

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 839, normalized size of antiderivative = 4.13 \[ \int \frac {(d+e x)^2 (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^2*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
 

Output:

(i*( - 15*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d 
))*b**4*e**4*g + 102*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 - 252*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)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**2*d* 
e**3*f + 264*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)/s 
qrt( - 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((sqr 
t( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**4*d**3*e*f - 15*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 + 52*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*c**2*d*e*g + 18*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 + 10*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 
)*sqrt( - 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)...