\(\int \frac {(d+e x)^2 (f+g x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [275]

Optimal result

Integrand size = 46, antiderivative size = 348 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {g (16 c e f-2 c d g-7 b e g) (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 c^2 e^3}-\frac {g^2 (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^3}-\frac {\left (35 b^2 e^2 g^2-20 b c e g (4 e f+3 d g)+4 c^2 \left (12 e^2 f^2+16 d e f g+7 d^2 g^2\right )\right ) (8 c d-3 b e+2 c e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{192 c^4 e^3}+\frac {(2 c d-b e)^2 \left (35 b^2 e^2 g^2-20 b c e g (4 e f+3 d g)+4 c^2 \left (12 e^2 f^2+16 d e f g+7 d^2 g^2\right )\right ) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{64 c^{9/2} e^3} \] Output:

-1/24*g*(-7*b*e*g-2*c*d*g+16*c*e*f)*(e*x+d)^2*(d*(-b*e+c*d)-b*e^2*x-c*e^2* 
x^2)^(1/2)/c^2/e^3-1/4*g^2*(e*x+d)^3*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2 
)/c/e^3-1/192*(35*b^2*e^2*g^2-20*b*c*e*g*(3*d*g+4*e*f)+4*c^2*(7*d^2*g^2+16 
*d*e*f*g+12*e^2*f^2))*(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^4/e^3+1/64*(-b*e+2*c*d)^2*(35*b^2*e^2*g^2-20*b*c*e*g*(3*d*g+4*e 
*f)+4*c^2*(7*d^2*g^2+16*d*e*f*g+12*e^2*f^2))*arctan(c^(1/2)*(e*x+d)/(d*(-b 
*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(9/2)/e^3
 

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {-\sqrt {c} (d+e x) (-b e+c (d-e x)) \left (-105 b^3 e^3 g^2+10 b^2 c e^2 g (24 e f+46 d g+7 e g x)+8 c^3 \left (32 d^3 g^2+d^2 e g (80 f+21 g x)+16 d e^2 \left (3 f^2+3 f g x+g^2 x^2\right )+2 e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )\right )-4 b c^2 e \left (155 d^2 g^2+2 d e g (104 f+29 g x)+2 e^2 \left (18 f^2+20 f g x+7 g^2 x^2\right )\right )\right )-3 (-2 c d+b e)^2 \left (35 b^2 e^2 g^2-20 b c e g (4 e f+3 d g)+4 c^2 \left (12 e^2 f^2+16 d e f g+7 d^2 g^2\right )\right ) \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{192 c^{9/2} e^3 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

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

Output:

(-(Sqrt[c]*(d + e*x)*(-(b*e) + c*(d - e*x))*(-105*b^3*e^3*g^2 + 10*b^2*c*e 
^2*g*(24*e*f + 46*d*g + 7*e*g*x) + 8*c^3*(32*d^3*g^2 + d^2*e*g*(80*f + 21* 
g*x) + 16*d*e^2*(3*f^2 + 3*f*g*x + g^2*x^2) + 2*e^3*x*(6*f^2 + 8*f*g*x + 3 
*g^2*x^2)) - 4*b*c^2*e*(155*d^2*g^2 + 2*d*e*g*(104*f + 29*g*x) + 2*e^2*(18 
*f^2 + 20*f*g*x + 7*g^2*x^2)))) - 3*(-2*c*d + b*e)^2*(35*b^2*e^2*g^2 - 20* 
b*c*e*g*(4*e*f + 3*d*g) + 4*c^2*(12*e^2*f^2 + 16*d*e*f*g + 7*d^2*g^2))*Sqr 
t[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])])/(192*c^(9/2)*e^3*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1262, 27, 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.

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

\(\Big \downarrow \) 1262

\(\displaystyle -\frac {\int -\frac {e^2 (d+e x)^2 \left (8 c e^2 f^2+6 c d^2 g^2-7 b d e g^2+e g (16 c e f-2 c d g-7 b e g) x\right )}{2 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{4 c e^4}-\frac {g^2 (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(d+e x)^2 \left (8 c e^2 f^2+6 c d^2 g^2-7 b d e g^2+e g (16 c e f-2 c d g-7 b e g) x\right )}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c e^2}-\frac {g^2 (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^3}\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {\frac {\left (35 b^2 e^2 g^2-20 b c e g (3 d g+4 e f)+4 c^2 \left (7 d^2 g^2+16 d e f g+12 e^2 f^2\right )\right ) \int \frac {(d+e x)^2}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{6 c}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g-2 c d g+16 c e f)}{3 c e}}{8 c e^2}-\frac {g^2 (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^3}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {\frac {\left (35 b^2 e^2 g^2-20 b c e g (3 d g+4 e f)+4 c^2 \left (7 d^2 g^2+16 d e f g+12 e^2 f^2\right )\right ) \left (\frac {3 (2 c d-b e) \int \frac {d+e x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{4 c}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c e}\right )}{6 c}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g-2 c d g+16 c e f)}{3 c e}}{8 c e^2}-\frac {g^2 (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^3}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\left (35 b^2 e^2 g^2-20 b c e g (3 d g+4 e f)+4 c^2 \left (7 d^2 g^2+16 d e f g+12 e^2 f^2\right )\right ) \left (\frac {3 (2 c d-b e) \left (\frac {(2 c d-b e) \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 c}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e}\right )}{4 c}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c e}\right )}{6 c}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g-2 c d g+16 c e f)}{3 c e}}{8 c e^2}-\frac {g^2 (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^3}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\left (35 b^2 e^2 g^2-20 b c e g (3 d g+4 e f)+4 c^2 \left (7 d^2 g^2+16 d e f g+12 e^2 f^2\right )\right ) \left (\frac {3 (2 c d-b e) \left (\frac {(2 c d-b e) \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{c}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e}\right )}{4 c}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c e}\right )}{6 c}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g-2 c d g+16 c e f)}{3 c e}}{8 c e^2}-\frac {g^2 (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^3}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\left (\frac {3 (2 c d-b e) \left (\frac {(2 c d-b e) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 c^{3/2} e}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{c e}\right )}{4 c}-\frac {(d+e x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 c e}\right ) \left (35 b^2 e^2 g^2-20 b c e g (3 d g+4 e f)+4 c^2 \left (7 d^2 g^2+16 d e f g+12 e^2 f^2\right )\right )}{6 c}-\frac {g (d+e x)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g-2 c d g+16 c e f)}{3 c e}}{8 c e^2}-\frac {g^2 (d+e x)^3 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c e^3}\)

Input:

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

Output:

-1/4*(g^2*(d + e*x)^3*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(c*e^3) + 
 (-1/3*(g*(16*c*e*f - 2*c*d*g - 7*b*e*g)*(d + e*x)^2*Sqrt[d*(c*d - b*e) - 
b*e^2*x - c*e^2*x^2])/(c*e) + ((35*b^2*e^2*g^2 - 20*b*c*e*g*(4*e*f + 3*d*g 
) + 4*c^2*(12*e^2*f^2 + 16*d*e*f*g + 7*d^2*g^2))*(-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))/(8*c*e^2)
 

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 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]
 

rule 1262

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n + e*g^n*( 
m + p + n)*(d + e*x)^(n - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x], x] /; 
 FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 
IGtQ[n, 0] && NeQ[m + n + 2*p + 1, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1457\) vs. \(2(328)=656\).

Time = 2.76 (sec) , antiderivative size = 1458, normalized size of antiderivative = 4.19

method	result	size
default	\(\text {Expression too large to display}\)	\(1458\)

Input:

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

Output:

d^2*f^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*e*g*(d*g+e*f)*(-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)*arc 
tan((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* 
d*f*(d*g+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)))+(d^2*g^2+4*d*e*f*g+e^2*f^2)*(-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)))+e^2*g^2*(-1/ 
4*x^3/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-7/8*b/c*(-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^...
 

Fricas [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.86 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^2*(g*x+f)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, al 
gorithm="fricas")
 

Output:

[-1/768*(3*(48*(4*c^4*d^2*e^2 - 4*b*c^3*d*e^3 + b^2*c^2*e^4)*f^2 + 16*(16* 
c^4*d^3*e - 36*b*c^3*d^2*e^2 + 24*b^2*c^2*d*e^3 - 5*b^3*c*e^4)*f*g + (112* 
c^4*d^4 - 352*b*c^3*d^3*e + 408*b^2*c^2*d^2*e^2 - 200*b^3*c*d*e^3 + 35*b^4 
*e^4)*g^2)*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*(48*c^4*e^3*g^2*x^3 + 48*(8*c^4*d*e^2 - 3*b*c^3*e^3)*f^2 + 
16*(40*c^4*d^2*e - 52*b*c^3*d*e^2 + 15*b^2*c^2*e^3)*f*g + (256*c^4*d^3 - 6 
20*b*c^3*d^2*e + 460*b^2*c^2*d*e^2 - 105*b^3*c*e^3)*g^2 + 8*(16*c^4*e^3*f* 
g + (16*c^4*d*e^2 - 7*b*c^3*e^3)*g^2)*x^2 + 2*(48*c^4*e^3*f^2 + 16*(12*c^4 
*d*e^2 - 5*b*c^3*e^3)*f*g + (84*c^4*d^2*e - 116*b*c^3*d*e^2 + 35*b^2*c^2*e 
^3)*g^2)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^5*e^3), -1/384* 
(3*(48*(4*c^4*d^2*e^2 - 4*b*c^3*d*e^3 + b^2*c^2*e^4)*f^2 + 16*(16*c^4*d^3* 
e - 36*b*c^3*d^2*e^2 + 24*b^2*c^2*d*e^3 - 5*b^3*c*e^4)*f*g + (112*c^4*d^4 
- 352*b*c^3*d^3*e + 408*b^2*c^2*d^2*e^2 - 200*b^3*c*d*e^3 + 35*b^4*e^4)*g^ 
2)*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*(48*c^4* 
e^3*g^2*x^3 + 48*(8*c^4*d*e^2 - 3*b*c^3*e^3)*f^2 + 16*(40*c^4*d^2*e - 52*b 
*c^3*d*e^2 + 15*b^2*c^2*e^3)*f*g + (256*c^4*d^3 - 620*b*c^3*d^2*e + 460*b^ 
2*c^2*d*e^2 - 105*b^3*c*e^3)*g^2 + 8*(16*c^4*e^3*f*g + (16*c^4*d*e^2 - 7*b 
*c^3*e^3)*g^2)*x^2 + 2*(48*c^4*e^3*f^2 + 16*(12*c^4*d*e^2 - 5*b*c^3*e^3...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1282 vs. \(2 (343) = 686\).

Time = 1.29 (sec) , antiderivative size = 1282, normalized size of antiderivative = 3.68 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**2*(g*x+f)**2/(-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*(-7*b*e**2*g**2/(8*c) + 2*d*e*g**2 + 2*e**2*f*g 
)/(6*c) + d**2*g**2 + 4*d*e*f*g + e**2*f**2 + g**2*(-3*b*d*e + 3*c*d**2)/( 
4*c))/(4*c) + 2*d**2*f*g + 2*d*e*f**2 + (-2*b*d*e + 2*c*d**2)*(-7*b*e**2*g 
**2/(8*c) + 2*d*e*g**2 + 2*e**2*f*g)/(3*c*e**2))/(2*c) + d**2*f**2 + (-b*d 
*e + c*d**2)*(-5*b*(-7*b*e**2*g**2/(8*c) + 2*d*e*g**2 + 2*e**2*f*g)/(6*c) 
+ d**2*g**2 + 4*d*e*f*g + e**2*f**2 + g**2*(-3*b*d*e + 3*c*d**2)/(4*c))/(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)) + sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)*(-g**2* 
x**3/(4*c) - x**2*(-7*b*e**2*g**2/(8*c) + 2*d*e*g**2 + 2*e**2*f*g)/(3*c*e* 
*2) - x*(-5*b*(-7*b*e**2*g**2/(8*c) + 2*d*e*g**2 + 2*e**2*f*g)/(6*c) + d** 
2*g**2 + 4*d*e*f*g + e**2*f**2 + g**2*(-3*b*d*e + 3*c*d**2)/(4*c))/(2*c*e* 
*2) - (-3*b*(-5*b*(-7*b*e**2*g**2/(8*c) + 2*d*e*g**2 + 2*e**2*f*g)/(6*c) + 
 d**2*g**2 + 4*d*e*f*g + e**2*f**2 + g**2*(-3*b*d*e + 3*c*d**2)/(4*c))/(4* 
c) + 2*d**2*f*g + 2*d*e*f**2 + (-2*b*d*e + 2*c*d**2)*(-7*b*e**2*g**2/(8*c) 
 + 2*d*e*g**2 + 2*e**2*f*g)/(3*c*e**2))/(c*e**2)), Ne(c*e**2, 0)), (-2*(g* 
*2*(-b*d*e - b*e**2*x + c*d**2)**(9/2)/(9*b**4*e**6) + (-b*d*e - b*e**2*x 
+ c*d**2)**(7/2)*(2*b*d*e*g**2 - 2*b*e**2*f*g - 4*c*d**2*g**2)/(7*b**4*e** 
6) + (-b*d*e - b*e**2*x + c*d**2)**(5/2)*(b**2*d**2*e**2*g**2 - 2*b**2*...
 

Maxima [F(-2)]

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

integrate((e*x+d)^2*(g*x+f)^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, al 
gorithm="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 = 0.41 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.51 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {1}{192} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, {\left (\frac {6 \, g^{2} x}{c} + \frac {16 \, c^{3} e^{5} f g + 16 \, c^{3} d e^{4} g^{2} - 7 \, b c^{2} e^{5} g^{2}}{c^{4} e^{5}}\right )} x + \frac {48 \, c^{3} e^{5} f^{2} + 192 \, c^{3} d e^{4} f g - 80 \, b c^{2} e^{5} f g + 84 \, c^{3} d^{2} e^{3} g^{2} - 116 \, b c^{2} d e^{4} g^{2} + 35 \, b^{2} c e^{5} g^{2}}{c^{4} e^{5}}\right )} x + \frac {384 \, c^{3} d e^{4} f^{2} - 144 \, b c^{2} e^{5} f^{2} + 640 \, c^{3} d^{2} e^{3} f g - 832 \, b c^{2} d e^{4} f g + 240 \, b^{2} c e^{5} f g + 256 \, c^{3} d^{3} e^{2} g^{2} - 620 \, b c^{2} d^{2} e^{3} g^{2} + 460 \, b^{2} c d e^{4} g^{2} - 105 \, b^{3} e^{5} g^{2}}{c^{4} e^{5}}\right )} - \frac {{\left (192 \, c^{4} d^{2} e^{2} f^{2} - 192 \, b c^{3} d e^{3} f^{2} + 48 \, b^{2} c^{2} e^{4} f^{2} + 256 \, c^{4} d^{3} e f g - 576 \, b c^{3} d^{2} e^{2} f g + 384 \, b^{2} c^{2} d e^{3} f g - 80 \, b^{3} c e^{4} f g + 112 \, c^{4} d^{4} g^{2} - 352 \, b c^{3} d^{3} e g^{2} + 408 \, b^{2} c^{2} d^{2} e^{2} g^{2} - 200 \, b^{3} c d e^{3} g^{2} + 35 \, b^{4} e^{4} g^{2}\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{128 \, \sqrt {-c} c^{4} e^{2} {\left | e \right |}} \] Input:

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

Output:

-1/192*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(6*g^2*x/c + (16*c 
^3*e^5*f*g + 16*c^3*d*e^4*g^2 - 7*b*c^2*e^5*g^2)/(c^4*e^5))*x + (48*c^3*e^ 
5*f^2 + 192*c^3*d*e^4*f*g - 80*b*c^2*e^5*f*g + 84*c^3*d^2*e^3*g^2 - 116*b* 
c^2*d*e^4*g^2 + 35*b^2*c*e^5*g^2)/(c^4*e^5))*x + (384*c^3*d*e^4*f^2 - 144* 
b*c^2*e^5*f^2 + 640*c^3*d^2*e^3*f*g - 832*b*c^2*d*e^4*f*g + 240*b^2*c*e^5* 
f*g + 256*c^3*d^3*e^2*g^2 - 620*b*c^2*d^2*e^3*g^2 + 460*b^2*c*d*e^4*g^2 - 
105*b^3*e^5*g^2)/(c^4*e^5)) - 1/128*(192*c^4*d^2*e^2*f^2 - 192*b*c^3*d*e^3 
*f^2 + 48*b^2*c^2*e^4*f^2 + 256*c^4*d^3*e*f*g - 576*b*c^3*d^2*e^2*f*g + 38 
4*b^2*c^2*d*e^3*f*g - 80*b^3*c*e^4*f*g + 112*c^4*d^4*g^2 - 352*b*c^3*d^3*e 
*g^2 + 408*b^2*c^2*d^2*e^2*g^2 - 200*b^3*c*d*e^3*g^2 + 35*b^4*e^4*g^2)*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)))/(sqrt(-c)*c^4*e^2*abs(e))
 

Mupad [F(-1)]

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

int(((f + g*x)^2*(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)^2*(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 = 16.02 (sec) , antiderivative size = 1678, normalized size of antiderivative = 4.82 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(105*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d)) 
*b**5*e**5*g**2 - 810*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - 
 b*e + 2*c*d))*b**4*c*d*e**4*g**2 - 240*sqrt(c)*asinh((sqrt( - b*e + c*d - 
 c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c*e**5*f*g + 2424*sqrt(c)*asinh((sqr 
t( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**2*d**2*e**3*g**2 
+ 1632*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))* 
b**3*c**2*d*e**4*f*g + 144*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sq 
rt( - b*e + 2*c*d))*b**3*c**2*e**5*f**2 - 3504*sqrt(c)*asinh((sqrt( - b*e 
+ c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**3*d**3*e**2*g**2 - 4032*sq 
rt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**3 
*d**2*e**3*f*g - 864*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - 
b*e + 2*c*d))*b**2*c**3*d*e**4*f**2 + 2448*sqrt(c)*asinh((sqrt( - b*e + c* 
d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**4*d**4*e*g**2 + 4224*sqrt(c)*asin 
h((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**4*d**3*e**2*f* 
g + 1728*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d) 
)*b*c**4*d**2*e**3*f**2 - 672*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i) 
/sqrt( - b*e + 2*c*d))*c**5*d**5*g**2 - 1536*sqrt(c)*asinh((sqrt( - b*e + 
c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**5*d**4*e*f*g - 1152*sqrt(c)*asinh 
((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**5*d**3*e**2*f**2 
+ 105*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e ...