\(\int \frac {(d+e x)^2}{(f+g x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [281]

Optimal result

Integrand size = 46, antiderivative size = 410 \[ \int \frac {(d+e x)^2}{(f+g x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 g (c e f+c d g-b e g) (f+g x)^3}+\frac {e (7 b e g-2 c (e f+6 d g)) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{12 g (c e f+c d g-b e g)^2 (f+g x)^2}-\frac {e^2 \left (3 b^2 e^2 g^2+4 b c e g (4 e f-7 d g)-4 c^2 \left (e^2 f^2+6 d e f g-10 d^2 g^2\right )\right ) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{24 g (e f-d g) (b e g-c (e f+d g))^3 (f+g x)}+\frac {e^3 (2 c d-b e)^2 (6 c e f-4 c d g-b e g) \arctan \left (\frac {\sqrt {c e f+c d g-b e g} (d+e x)}{\sqrt {e f-d g} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 (e f-d g)^{3/2} (c e f+c d g-b e g)^{7/2}} \] Output:

1/3*(-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)^3+1/12*e*(7*b*e*g-2*c*(6*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)^2-1/24*e^2*(3*b^2*e^2*g^2+4*b* 
c*e*g*(-7*d*g+4*e*f)-4*c^2*(-10*d^2*g^2+6*d*e*f*g+e^2*f^2))*(d*(-b*e+c*d)- 
b*e^2*x-c*e^2*x^2)^(1/2)/g/(-d*g+e*f)/(b*e*g-c*(d*g+e*f))^3/(g*x+f)+1/8*e^ 
3*(-b*e+2*c*d)^2*(-b*e*g-4*c*d*g+6*c*e*f)*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)^(3/2)/(-b*e*g+c*d*g+c*e*f)^(7/2)
 

Mathematica [A] (verified)

Time = 4.35 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^2}{(f+g x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {e^3 (-2 c d+b e)^2 \left (\frac {(d+e x) (-c d+b e+c e x) \left (b^2 e^2 g \left (-8 d^2 g^2+2 d e g (f-7 g x)+e^2 \left (3 f^2+8 f g x-3 g^2 x^2\right )\right )-2 b c e \left (-8 d^3 g^3-d^2 e g^2 (11 f+19 g x)-2 d e^2 g \left (-2 f^2+9 f g x+7 g^2 x^2\right )+e^3 f \left (9 f^2+25 f g x+8 g^2 x^2\right )\right )-4 c^2 \left (2 d^4 g^3+6 d^3 e g^2 (f+g x)-e^4 f^2 x (3 f+g x)-6 d e^3 f \left (2 f^2+3 f g x+g^2 x^2\right )+d^2 e^2 g \left (7 f^2+21 f g x+10 g^2 x^2\right )\right )\right )}{e^3 (-2 c d+b e)^2 (e f-d g) (c e f+c d g-b e g)^3 (f+g x)^3}-\frac {3 (6 c e f-4 c d g-b e g) \sqrt {d+e x} \sqrt {c d-b e-c e x} \arctan \left (\frac {\sqrt {e f-d g} \sqrt {c d-b e-c e x}}{\sqrt {c e f+c d g-b e g} \sqrt {d+e x}}\right )}{(e f-d g)^{3/2} (c e f+c d g-b e g)^{7/2}}\right )}{24 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

Integrate[(d + e*x)^2/((f + g*x)^4*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^ 
2]),x]
 

Output:

(e^3*(-2*c*d + b*e)^2*(((d + e*x)*(-(c*d) + b*e + c*e*x)*(b^2*e^2*g*(-8*d^ 
2*g^2 + 2*d*e*g*(f - 7*g*x) + e^2*(3*f^2 + 8*f*g*x - 3*g^2*x^2)) - 2*b*c*e 
*(-8*d^3*g^3 - d^2*e*g^2*(11*f + 19*g*x) - 2*d*e^2*g*(-2*f^2 + 9*f*g*x + 7 
*g^2*x^2) + e^3*f*(9*f^2 + 25*f*g*x + 8*g^2*x^2)) - 4*c^2*(2*d^4*g^3 + 6*d 
^3*e*g^2*(f + g*x) - e^4*f^2*x*(3*f + g*x) - 6*d*e^3*f*(2*f^2 + 3*f*g*x + 
g^2*x^2) + d^2*e^2*g*(7*f^2 + 21*f*g*x + 10*g^2*x^2))))/(e^3*(-2*c*d + b*e 
)^2*(e*f - d*g)*(c*e*f + c*d*g - b*e*g)^3*(f + g*x)^3) - (3*(6*c*e*f - 4*c 
*d*g - b*e*g)*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])])/( 
(e*f - d*g)^(3/2)*(c*e*f + c*d*g - b*e*g)^(7/2))))/(24*Sqrt[(d + e*x)*(-(b 
*e) + c*(d - e*x))])
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {1266, 27, 1237, 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.

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

\(\Big \downarrow \) 1266

\(\displaystyle \frac {\int \frac {e (e f-d g) \left (12 c g d^2+b e (e f-7 d g)+2 e (c e f+5 c d g-3 b e g) x\right )}{2 g (f+g x)^3 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{3 (e f-d g) (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 g (f+g x)^3 (-b e g+c d g+c e f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \int \frac {12 c g d^2+b e (e f-7 d g)+2 e (c e f+5 c d g-3 b e g) x}{(f+g x)^3 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{6 g (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 g (f+g x)^3 (-b e g+c d g+c e f)}\)

\(\Big \downarrow \) 1237

\(\displaystyle \frac {e \left (\frac {\int \frac {e (e f-d g) \left (40 c^2 g d^2+3 b^2 e^2 g+2 b c e (e f-14 d g)+2 c e (2 c e f+12 c d g-7 b e g) x\right )}{2 (f+g x)^2 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 (e f-d g) (-b e g+c d g+c e f)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+12 c d g+2 c e f)}{2 (f+g x)^2 (-b e g+c d g+c e f)}\right )}{6 g (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 g (f+g x)^3 (-b e g+c d g+c e f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \left (\frac {e \int \frac {40 c^2 g d^2+3 b^2 e^2 g+2 b c e (e f-14 d g)+2 c e (2 c e f+12 c d g-7 b e g) x}{(f+g x)^2 \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{4 (-b e g+c d g+c e f)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+12 c d g+2 c e f)}{2 (f+g x)^2 (-b e g+c d g+c e f)}\right )}{6 g (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 g (f+g x)^3 (-b e g+c d g+c e f)}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {e \left (\frac {e \left (\frac {3 e g (2 c d-b e)^2 (-b e g-4 c d g+6 c e f) \int \frac {1}{(f+g x) \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{2 (e f-d g) (-b e g+c d g+c e f)}+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \left (3 b^2 e^2 g^2+4 b c e g (4 e f-7 d g)-4 c^2 \left (-10 d^2 g^2+6 d e f g+e^2 f^2\right )\right )}{(f+g x) (e f-d g) (-b e g+c d g+c e f)}\right )}{4 (-b e g+c d g+c e f)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+12 c d g+2 c e f)}{2 (f+g x)^2 (-b e g+c d g+c e f)}\right )}{6 g (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 g (f+g x)^3 (-b e g+c d g+c e f)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {e \left (\frac {e \left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \left (3 b^2 e^2 g^2+4 b c e g (4 e f-7 d g)-4 c^2 \left (-10 d^2 g^2+6 d e f g+e^2 f^2\right )\right )}{(f+g x) (e f-d g) (-b e g+c d g+c e f)}-\frac {3 e g (2 c d-b e)^2 (-b e g-4 c d g+6 c e f) \int \frac {1}{-\frac {\left (2 c g d^2+b e (e f-2 d g)+e^2 (2 c f-b g) x\right )^2}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 (e f-d g) (c e f+c d g-b e g)}d\frac {2 c g d^2+b e (e f-2 d g)+e^2 (2 c f-b g) x}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}}{(e f-d g) (-b e g+c d g+c e f)}\right )}{4 (-b e g+c d g+c e f)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+12 c d g+2 c e f)}{2 (f+g x)^2 (-b e g+c d g+c e f)}\right )}{6 g (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 g (f+g x)^3 (-b e g+c d g+c e f)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {e \left (\frac {e \left (\frac {3 e g (2 c d-b e)^2 (-b e g-4 c d g+6 c e f) \arctan \left (\frac {e^2 x (2 c f-b g)+b e (e f-2 d g)+2 c d^2 g}{2 \sqrt {e f-d g} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \sqrt {-b e g+c d g+c e f}}\right )}{2 (e f-d g)^{3/2} (-b e g+c d g+c e f)^{3/2}}+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} \left (3 b^2 e^2 g^2+4 b c e g (4 e f-7 d g)-4 c^2 \left (-10 d^2 g^2+6 d e f g+e^2 f^2\right )\right )}{(f+g x) (e f-d g) (-b e g+c d g+c e f)}\right )}{4 (-b e g+c d g+c e f)}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+12 c d g+2 c e f)}{2 (f+g x)^2 (-b e g+c d g+c e f)}\right )}{6 g (-b e g+c d g+c e f)}+\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{3 g (f+g x)^3 (-b e g+c d g+c e f)}\)

Input:

Int[(d + e*x)^2/((f + g*x)^4*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])/(3*g*(c*e*f + c*d* 
g - b*e*g)*(f + g*x)^3) + (e*(-1/2*((2*c*e*f + 12*c*d*g - 7*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)^2) 
+ (e*(((3*b^2*e^2*g^2 + 4*b*c*e*g*(4*e*f - 7*d*g) - 4*c^2*(e^2*f^2 + 6*d*e 
*f*g - 10*d^2*g^2))*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/((e*f - d*g 
)*(c*e*f + c*d*g - b*e*g)*(f + g*x)) + (3*e*(2*c*d - b*e)^2*g*(6*c*e*f - 4 
*c*d*g - b*e*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*(e*f - d*g)^(3/2)*(c*e*f + c*d*g - b*e*g)^(3/2)))) 
/(4*(c*e*f + c*d*g - b*e*g))))/(6*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 1237

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)/((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*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(2825\) vs. \(2(386)=772\).

Time = 3.23 (sec) , antiderivative size = 2826, normalized size of antiderivative = 6.89

method	result	size
default	\(\text {Expression too large to display}\)	\(2826\)

Input:

int((e*x+d)^2/(g*x+f)^4/(-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^4*(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)))+(d^2*g^2-2*d*e*f*g+e^2*f^2)/g^6*(1/3/(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)^3*(-(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)-5/6*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)*(1/2/(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*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2501 vs. \(2 (386) = 772\).

Time = 53.14 (sec) , antiderivative size = 5060, normalized size of antiderivative = 12.34 \[ \int \frac {(d+e x)^2}{(f+g x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2/(g*x+f)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, al 
gorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate((e*x+d)**2/(g*x+f)**4/(-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)**4), 
 x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^2/(g*x+f)^4/(-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 [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)^2/(g*x+f)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, al 
gorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int((d + e*x)^2/((f + g*x)^4*(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)^4*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)), 
 x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 32.37 (sec) , antiderivative size = 38018, normalized size of antiderivative = 92.73 \[ \int \frac {(d+e x)^2}{(f+g x)^4 \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^2/(g*x+f)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
 

Output:

(i*(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**4*d**3*e**3*g**8 - 20*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**4*d**2*e**4*f*g**7 + 28*sqr 
t(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e* 
x)*b**4*d**2*e**4*g**8*x - 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**4*d*e**5*f**2*g**6 - 44*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**4*d 
*e**5*f*g**7*x + 6*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**4*d*e**5*g**8*x**2 + 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**4*e**6*f**3*g* 
*5 + 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**4*e**6*f**2*g**6*x - 6*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sq 
rt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*b**4*e**6*f*g**7*x**2 - 48*s 
qrt(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*c*d**4*e**2*g**8 - 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)*b**3*c*d**3*e**3*f*g**7 - 104*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*c*d**3*e**3*g**8*x + 122*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*c*d**2*e**4*f**2*g**6 - 36*sqrt(d 
 + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*...