\(\int \frac {(d+e x)^2}{(f+g x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [280]

Optimal result

Integrand size = 46, antiderivative size = 265 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 g (c e f+c d g-b e g) (f+g x)^2}+\frac {e (5 b e g-2 c (e f+4 d g)) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 g (c e f+c d g-b e g)^2 (f+g x)}+\frac {3 e^2 (2 c d-b e)^2 \arctan \left (\frac {\sqrt {c e f+c d g-b e g} (d+e x)}{\sqrt {e f-d g} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 \sqrt {e f-d g} (c e f+c d g-b e g)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.27 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {e^2 (-2 c d+b e)^2 \left (\frac {(d+e x) (-c d+b e+c e x) \left (-b e (3 e f+2 d g+5 e g x)+2 c \left (d^2 g+e^2 f x+4 d e (f+g x)\right )\right )}{e^2 (-2 c d+b e)^2 (c e f+c d g-b e g)^2 (f+g x)^2}-\frac {3 \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {e f-d g} \sqrt {c d-b e-c e x}}{\sqrt {c e f+c d g-b e g} \sqrt {d+e x}}\right )}{\sqrt {e f-d g} (c e f+c d g-b e g)^{5/2}}\right )}{4 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1266, 27, 1228, 1154, 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.

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

\(\Big \downarrow \) 1266

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   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[Simplify[m + 2*p + 3], 0]
 

rule 1266

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^ 
n, d + e*x, x], R = PolynomialRemainder[(f + g*x)^n, d + e*x, x]}, Simp[(e* 
R*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a* 
e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1 
)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R 
*(m + 1) - b*e*R*(m + p + 2) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g, p}, x] && IGtQ[n, 1] && ILtQ[m, -1] && NeQ[c*d^2 - b* 
d*e + a*e^2, 0] && (NeQ[m + n, 0] || EqQ[p, -2^(-1)])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1560\) vs. \(2(245)=490\).

Time = 2.57 (sec) , antiderivative size = 1561, normalized size of antiderivative = 5.89

method	result	size
default	\(\text {Expression too large to display}\)	\(1561\)

Input:

int((e*x+d)^2/(g*x+f)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_R 
ETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1122 vs. \(2 (245) = 490\).

Time = 4.48 (sec) , antiderivative size = 2302, normalized size of antiderivative = 8.69 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2/(g*x+f)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, al 
gorithm="fricas")
 

Output:

[-1/16*(3*sqrt(-c*e^2*f^2 + b*e^2*f*g + (c*d^2 - b*d*e)*g^2)*((4*c^2*d^2*e 
^2 - 4*b*c*d*e^3 + b^2*e^4)*g^2*x^2 + 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2 
*e^4)*f*g*x + (4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*f^2)*log(-((4*c^2*d^ 
2*e^2 - 4*b*c*d*e^3 - b^2*e^4)*f^2 - 8*(b*c*d^2*e^2 - b^2*d*e^3)*f*g - 8*( 
c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*g^2 - (8*c^2*e^4*f^2 - 8*b*c*e^4*f*g 
- (4*c^2*d^2*e^2 - 4*b*c*d*e^3 - b^2*e^4)*g^2)*x^2 + 4*sqrt(-c*e^2*f^2 + b 
*e^2*f*g + (c*d^2 - b*d*e)*g^2)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e) 
*(b*e^2*f + 2*(c*d^2 - b*d*e)*g + (2*c*e^2*f - b*e^2*g)*x) - 2*(4*b*c*e^4* 
f^2 + (4*c^2*d^2*e^2 - 4*b*c*d*e^3 - 3*b^2*e^4)*f*g - 4*(b*c*d^2*e^2 - b^2 
*d*e^3)*g^2)*x)/(g^2*x^2 + 2*f*g*x + f^2)) + 4*sqrt(-c*e^2*x^2 - b*e^2*x + 
 c*d^2 - b*d*e)*((8*c^2*d*e^3 - 3*b*c*e^4)*f^3 + (2*c^2*d^2*e^2 - 10*b*c*d 
*e^3 + 3*b^2*e^4)*f^2*g - (8*c^2*d^3*e - 9*b*c*d^2*e^2 + b^2*d*e^3)*f*g^2 
- 2*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*g^3 + (2*c^2*e^4*f^3 + (8*c^2*d* 
e^3 - 7*b*c*e^4)*f^2*g - (2*c^2*d^2*e^2 + 6*b*c*d*e^3 - 5*b^2*e^4)*f*g^2 - 
 (8*c^2*d^3*e - 13*b*c*d^2*e^2 + 5*b^2*d*e^3)*g^3)*x))/(c^3*e^4*f^6 + (2*c 
^3*d*e^3 - 3*b*c^2*e^4)*f^5*g - 3*(b*c^2*d*e^3 - b^2*c*e^4)*f^4*g^2 - (2*c 
^3*d^3*e - 3*b*c^2*d^2*e^2 + b^3*e^4)*f^3*g^3 - (c^3*d^4 - 3*b*c^2*d^3*e + 
 3*b^2*c*d^2*e^2 - b^3*d*e^3)*f^2*g^4 + (c^3*e^4*f^4*g^2 + (2*c^3*d*e^3 - 
3*b*c^2*e^4)*f^3*g^3 - 3*(b*c^2*d*e^3 - b^2*c*e^4)*f^2*g^4 - (2*c^3*d^3*e 
- 3*b*c^2*d^2*e^2 + b^3*e^4)*f*g^5 - (c^3*d^4 - 3*b*c^2*d^3*e + 3*b^2*c...
 

Sympy [F]

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

integrate((e*x+d)**2/(g*x+f)**3/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2 
),x)
 

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^2/(g*x+f)^3/(-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((d*g-e*f)>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4728 vs. \(2 (245) = 490\).

Time = 11.78 (sec) , antiderivative size = 4728, normalized size of antiderivative = 17.84 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2/(g*x+f)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, al 
gorithm="giac")
 

Output:

1/4*(48*c^4*d^4*e^2*f^2*g^2*arctan((sqrt(-c*e^2)*f - sqrt(c*d^2 - b*d*e)*g 
)/sqrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g^2 + b*d*e*g^2)) - 96*b*c^3*d^3*e^3* 
f^2*g^2*arctan((sqrt(-c*e^2)*f - sqrt(c*d^2 - b*d*e)*g)/sqrt(c*e^2*f^2 - b 
*e^2*f*g - c*d^2*g^2 + b*d*e*g^2)) + 48*b^2*c^2*d^2*e^4*f^2*g^2*arctan((sq 
rt(-c*e^2)*f - sqrt(c*d^2 - b*d*e)*g)/sqrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g 
^2 + b*d*e*g^2)) - 3*b^4*e^6*f^2*g^2*arctan((sqrt(-c*e^2)*f - sqrt(c*d^2 - 
 b*d*e)*g)/sqrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g^2 + b*d*e*g^2)) - 48*sqrt( 
c*d^2 - b*d*e)*sqrt(-c*e^2)*b*c^2*d^2*e^2*f^2*g^2*arctan((sqrt(-c*e^2)*f - 
 sqrt(c*d^2 - b*d*e)*g)/sqrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g^2 + b*d*e*g^2 
)) + 48*sqrt(c*d^2 - b*d*e)*sqrt(-c*e^2)*b^2*c*d*e^3*f^2*g^2*arctan((sqrt( 
-c*e^2)*f - sqrt(c*d^2 - b*d*e)*g)/sqrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g^2 
+ b*d*e*g^2)) - 12*sqrt(c*d^2 - b*d*e)*sqrt(-c*e^2)*b^3*e^4*f^2*g^2*arctan 
((sqrt(-c*e^2)*f - sqrt(c*d^2 - b*d*e)*g)/sqrt(c*e^2*f^2 - b*e^2*f*g - c*d 
^2*g^2 + b*d*e*g^2)) + 8*sqrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g^2 + b*d*e*g^ 
2)*sqrt(-c*e^2)*c^3*d^2*e^2*f^3 - 8*sqrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g^2 
 + b*d*e*g^2)*sqrt(-c*e^2)*b*c^2*d*e^3*f^3 - 2*sqrt(c*e^2*f^2 - b*e^2*f*g 
- c*d^2*g^2 + b*d*e*g^2)*sqrt(-c*e^2)*b^2*c*e^4*f^3 + 8*sqrt(c*e^2*f^2 - b 
*e^2*f*g - c*d^2*g^2 + b*d*e*g^2)*sqrt(c*d^2 - b*d*e)*b*c^2*e^4*f^3 + 32*s 
qrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g^2 + b*d*e*g^2)*sqrt(-c*e^2)*c^3*d^3*e* 
f^2*g - 52*sqrt(c*e^2*f^2 - b*e^2*f*g - c*d^2*g^2 + b*d*e*g^2)*sqrt(-c*...
 

Mupad [F(-1)]

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

int((d + e*x)^2/((f + g*x)^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)), 
x)
 

Output:

int((d + e*x)^2/((f + g*x)^3*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)), 
 x)
 

Reduce [B] (verification not implemented)

Time = 3.91 (sec) , antiderivative size = 15967, normalized size of antiderivative = 60.25 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^2/(g*x+f)^3/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
 

Output:

(i*(4*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**3*d**2*e**2*g**5 + 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**3*d*e**3*f*g**4 + 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* 
*3*d*e**3*g**5*x - 6*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**3*e**4*f**2*g**3 - 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**3*e**4*f*g**4 
*x - 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**2*c*d**3*e*g**5 - 26*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*d**2*e**2*f*g**4 - 26*sq 
rt(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*d**2*e**2*g**5*x + 16*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*d*e**3*f**2*g**3 - 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 
**2*c*d*e**3*f*g**4*x + 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**2*c*e**4*f**3*g**2 + 34*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 
**4*f**2*g**3*x + 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*s 
qrt( - b*e + c*d - c*e*x)*b*c**2*d**4*g**5 + 32*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**3*e*f*g...