\(\int \frac {(d+e x)^2}{(f+g x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [288]

Optimal result

Integrand size = 44, antiderivative size = 342 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 c^2 d^2 (d+e x)}{(c d f-a e g)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (c d f-a e g)^2 (f+g x)^2}-\frac {g \left (a e^2 g+c d (6 e f-7 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (e f-d g) (c d f-a e g)^3 (f+g x)}-\frac {\left (a^2 e^4 g^2-2 a c d e^2 g (4 e f-3 d g)-c^2 d^2 \left (8 e^2 f^2-24 d e f g+15 d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 (e f-d g)^{3/2} (c d f-a e g)^{7/2}} \] Output:

-2*c^2*d^2*(e*x+d)/(-a*e*g+c*d*f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2 
)-1/2*g*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^2/(g*x+f)^2 
-1/4*g*(a*e^2*g+c*d*(-7*d*g+6*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2 
)/(-d*g+e*f)/(-a*e*g+c*d*f)^3/(g*x+f)-1/4*(a^2*e^4*g^2-2*a*c*d*e^2*g*(-3*d 
*g+4*e*f)-c^2*d^2*(15*d^2*g^2-24*d*e*f*g+8*e^2*f^2))*arctanh((-a*e*g+c*d*f 
)^(1/2)*(e*x+d)/(-d*g+e*f)^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/ 
(-d*g+e*f)^(3/2)/(-a*e*g+c*d*f)^(7/2)
 

Mathematica [A] (verified)

Time = 3.36 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\frac {(a e+c d x) (d+e x)^2 \left (a^2 e^2 g^2 (-e f+2 d g+e g x)+a c d e g \left (-d g (9 f+5 g x)+e \left (8 f^2+5 f g x+g^2 x^2\right )\right )+c^2 d^2 \left (2 e f \left (4 f^2+12 f g x+7 g^2 x^2\right )-d g \left (8 f^2+25 f g x+15 g^2 x^2\right )\right )\right )}{(e f-d g) (-c d f+a e g)^3 (f+g x)^2}+\frac {\left (-a^2 e^4 g^2+2 a c d e^2 g (4 e f-3 d g)+c^2 d^2 \left (8 e^2 f^2-24 d e f g+15 d^2 g^2\right )\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )}{(-e f+d g)^{3/2} (c d f-a e g)^{7/2}}}{4 ((a e+c d x) (d+e x))^{3/2}} \] Input:

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

Output:

(((a*e + c*d*x)*(d + e*x)^2*(a^2*e^2*g^2*(-(e*f) + 2*d*g + e*g*x) + a*c*d* 
e*g*(-(d*g*(9*f + 5*g*x)) + e*(8*f^2 + 5*f*g*x + g^2*x^2)) + c^2*d^2*(2*e* 
f*(4*f^2 + 12*f*g*x + 7*g^2*x^2) - d*g*(8*f^2 + 25*f*g*x + 15*g^2*x^2))))/ 
((e*f - d*g)*(-(c*d*f) + a*e*g)^3*(f + g*x)^2) + ((-(a^2*e^4*g^2) + 2*a*c* 
d*e^2*g*(4*e*f - 3*d*g) + c^2*d^2*(8*e^2*f^2 - 24*d*e*f*g + 15*d^2*g^2))*( 
a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*ArcTan[(Sqrt[c*d*f - a*e*g]*Sqrt[d + e* 
x])/(Sqrt[-(e*f) + d*g]*Sqrt[a*e + c*d*x])])/((-(e*f) + d*g)^(3/2)*(c*d*f 
- a*e*g)^(7/2)))/(4*((a*e + c*d*x)*(d + e*x))^(3/2))
 

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.25, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1264, 27, 2181, 27, 1228, 1154, 219}

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.

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

\(\Big \downarrow \) 1264

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2181

\(\displaystyle \frac {-\frac {\int \frac {\left (c d^2-a e^2\right )^2 (e f-d g) \left (a^2 g^2 e^3-a c d g (e f+7 d g) e-c^2 d^2 f (4 e f-11 d g)-2 c d g \left (3 a e^2 g-c d (e f+2 d g)\right ) x\right )}{2 (c d f-a e g)^2 (f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (e f-d g) (c d f-a e g)}-\frac {g \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (f+g x)^2 (c d f-a e g)^2}}{\left (c d^2-a e^2\right )^2}-\frac {2 c^2 d^2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\left (c d^2-a e^2\right )^2 \int \frac {a^2 g^2 e^3-a c d g (e f+7 d g) e-c^2 d^2 f (4 e f-11 d g)-2 c d g \left (3 a e^2 g-c d (e f+2 d g)\right ) x}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (c d f-a e g)^3}-\frac {g \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (f+g x)^2 (c d f-a e g)^2}}{\left (c d^2-a e^2\right )^2}-\frac {2 c^2 d^2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {-\frac {\left (c d^2-a e^2\right )^2 \left (\frac {\left (a^2 e^4 g^2-2 a c d e^2 g (4 e f-3 d g)-c^2 d^2 \left (15 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (e f-d g)}+\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (a e^2 g+c d (6 e f-7 d g)\right )}{(f+g x) (e f-d g)}\right )}{4 (c d f-a e g)^3}-\frac {g \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (f+g x)^2 (c d f-a e g)^2}}{\left (c d^2-a e^2\right )^2}-\frac {2 c^2 d^2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {\left (c d^2-a e^2\right )^2 \left (\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (a e^2 g+c d (6 e f-7 d g)\right )}{(f+g x) (e f-d g)}-\frac {\left (a^2 e^4 g^2-2 a c d e^2 g (4 e f-3 d g)-c^2 d^2 \left (15 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{4 (e f-d g) (c d f-a e g)-\frac {\left (c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\left (-\frac {c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{e f-d g}\right )}{4 (c d f-a e g)^3}-\frac {g \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (f+g x)^2 (c d f-a e g)^2}}{\left (c d^2-a e^2\right )^2}-\frac {2 c^2 d^2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\left (c d^2-a e^2\right )^2 \left (\frac {\left (a^2 e^4 g^2-2 a c d e^2 g (4 e f-3 d g)-c^2 d^2 \left (15 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) \text {arctanh}\left (\frac {-x \left (a e^2 g-c d (2 e f-d g)\right )+a e (e f-2 d g)+c d^2 f}{2 \sqrt {e f-d g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt {c d f-a e g}}\right )}{2 (e f-d g)^{3/2} \sqrt {c d f-a e g}}+\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (a e^2 g+c d (6 e f-7 d g)\right )}{(f+g x) (e f-d g)}\right )}{4 (c d f-a e g)^3}-\frac {g \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (f+g x)^2 (c d f-a e g)^2}}{\left (c d^2-a e^2\right )^2}-\frac {2 c^2 d^2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}\)

Input:

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

Output:

(-2*c^2*d^2*(d + e*x))/((c*d*f - a*e*g)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2]) + (-1/2*((c*d^2 - a*e^2)^2*g*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + 
 c*d*e*x^2])/((c*d*f - a*e*g)^2*(f + g*x)^2) - ((c*d^2 - a*e^2)^2*((g*(a*e 
^2*g + c*d*(6*e*f - 7*d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/( 
(e*f - d*g)*(f + g*x)) + ((a^2*e^4*g^2 - 2*a*c*d*e^2*g*(4*e*f - 3*d*g) - c 
^2*d^2*(8*e^2*f^2 - 24*d*e*f*g + 15*d^2*g^2))*ArcTanh[(c*d^2*f + a*e*(e*f 
- 2*d*g) - (a*e^2*g - c*d*(2*e*f - d*g))*x)/(2*Sqrt[e*f - d*g]*Sqrt[c*d*f 
- a*e*g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*(e*f - d*g)^(3/ 
2)*Sqrt[c*d*f - a*e*g])))/(4*(c*d*f - a*e*g)^3))/(c*d^2 - a*e^2)^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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[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 1264

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) 
^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x 
)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ 
(d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 
2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S 
imp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E 
xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) 
)/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] 
 && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, 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)*Qx + 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, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3620\) vs. \(2(320)=640\).

Time = 2.45 (sec) , antiderivative size = 3621, normalized size of antiderivative = 10.59

method	result	size
default	\(\text {Expression too large to display}\)	\(3621\)

Input:

int((e*x+d)^2/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_ 
RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2145 vs. \(2 (320) = 640\).

Time = 126.40 (sec) , antiderivative size = 4347, normalized size of antiderivative = 12.71 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, a 
lgorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate((e*x+d)**2/(g*x+f)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2 
),x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x+d)^2/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, a 
lgorithm="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(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x+d)^2/(g*x+f)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, a 
lgorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{-1,[1,1,11]%%%},[2,3,7,0]%%%}+%%%{%%%{4,[2,3,9]%%%},[2 
,3,6,0]%%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 14.21 (sec) , antiderivative size = 14503, normalized size of antiderivative = 42.41 \[ \int \frac {(d+e x)^2}{(f+g x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sq 
rt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*s 
qrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sq 
rt(c)*sqrt(d + e*x))*a**3*e**6*f**2*g**3 - 6*sqrt(d*g - e*f)*sqrt(a*e + c* 
d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sq 
rt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d 
**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**3*e**6*f*g* 
*4*x - 3*sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g) 
*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f 
)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d) 
*sqrt(c)*sqrt(d + e*x))*a**3*e**6*g**5*x**2 - 23*sqrt(d*g - e*f)*sqrt(a*e 
+ c*d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt( 
2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + 
 c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**2*c*d** 
2*e**4*f**2*g**3 - 46*sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*sqrt(a*e*g - c*d*f 
)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*s 
qrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sq 
rt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**2*c*d**2*e**4*f*g**4*x - 23*sqrt(d 
*g - e*f)*sqrt(a*e + c*d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a 
*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a*e*g...