\(\int \frac {(d+e x)^2}{(f+g x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [294]

Optimal result

Integrand size = 44, antiderivative size = 424 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 c d (d+e x)}{3 (c d f-a e g)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 c d \left (a e^2 (e f-8 d g)+c d^2 (e f+6 d g)-e \left (7 a e^2 g-c d (2 e f+5 d g)\right ) x\right )}{3 \left (c d^2-a e^2\right )^2 (c d f-a e g)^2 (f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {g \left (3 a^2 e^4 g^2+2 a c d e^2 g (8 e f-11 d g)-c^2 d^2 \left (4 e^2 f^2+8 d e f g-15 d^2 g^2\right )\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right )^2 (e f-d g) (c d f-a e g)^3 (f+g x)}-\frac {g^2 \left (a e^2 g-c d (6 e f-5 d g)\right ) \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{(e f-d g)^{3/2} (c d f-a e g)^{7/2}} \] Output:

-2/3*c*d*(e*x+d)/(-a*e*g+c*d*f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+ 
2/3*c*d*(a*e^2*(-8*d*g+e*f)+c*d^2*(6*d*g+e*f)-e*(7*a*e^2*g-c*d*(5*d*g+2*e* 
f))*x)/(-a*e^2+c*d^2)^2/(-a*e*g+c*d*f)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c* 
d*e*x^2)^(1/2)-1/3*g*(3*a^2*e^4*g^2+2*a*c*d*e^2*g*(-11*d*g+8*e*f)-c^2*d^2* 
(-15*d^2*g^2+8*d*e*f*g+4*e^2*f^2))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2) 
/(-a*e^2+c*d^2)^2/(-d*g+e*f)/(-a*e*g+c*d*f)^3/(g*x+f)-g^2*(a*e^2*g-c*d*(-5 
*d*g+6*e*f))*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 = 2.98 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\frac {(d+e x) \left (3 a^4 e^6 g^3-6 a^3 c d e^4 g^3 (d-e x)-c^4 d^4 \left (4 e^2 f^2 x (f+g x)+d^2 g \left (2 f^2-10 f g x-15 g^2 x^2\right )-2 d e f \left (f^2-3 f g x-4 g^2 x^2\right )\right )+3 a^2 c^2 d^2 e^2 g \left (d^2 g^2-2 d e g (3 f+5 g x)+e^2 \left (6 f^2+6 f g x+g^2 x^2\right )\right )+2 a c^3 d^3 e \left (d^2 g^2 (7 f+10 g x)+e^2 f \left (-3 f^2+5 f g x+8 g^2 x^2\right )-d e g \left (4 f^2+12 f g x+11 g^2 x^2\right )\right )\right )}{\left (c d^2-a e^2\right )^2 (-e f+d g) (c d f-a e g)^3 (a e+c d x) (f+g x)}-\frac {3 g^2 \left (a e^2 g+c d (-6 e f+5 d g)\right ) \sqrt {a e+c d x} \sqrt {d+e x} \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )}{(-e f+d g)^{3/2} (c d f-a e g)^{7/2}}}{3 \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(d + e*x)^2/((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2) 
^(5/2)),x]
 

Output:

(((d + e*x)*(3*a^4*e^6*g^3 - 6*a^3*c*d*e^4*g^3*(d - e*x) - c^4*d^4*(4*e^2* 
f^2*x*(f + g*x) + d^2*g*(2*f^2 - 10*f*g*x - 15*g^2*x^2) - 2*d*e*f*(f^2 - 3 
*f*g*x - 4*g^2*x^2)) + 3*a^2*c^2*d^2*e^2*g*(d^2*g^2 - 2*d*e*g*(3*f + 5*g*x 
) + e^2*(6*f^2 + 6*f*g*x + g^2*x^2)) + 2*a*c^3*d^3*e*(d^2*g^2*(7*f + 10*g* 
x) + e^2*f*(-3*f^2 + 5*f*g*x + 8*g^2*x^2) - d*e*g*(4*f^2 + 12*f*g*x + 11*g 
^2*x^2))))/((c*d^2 - a*e^2)^2*(-(e*f) + d*g)*(c*d*f - a*e*g)^3*(a*e + c*d* 
x)*(f + g*x)) - (3*g^2*(a*e^2*g + c*d*(-6*e*f + 5*d*g))*Sqrt[a*e + c*d*x]* 
Sqrt[d + e*x]*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)))/(3*S 
qrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 416, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1264, 27, 2177, 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.

\(\displaystyle \int \frac {(d+e x)^2}{(f+g x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1264

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2177

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {3 g^2 \left (c d^2-a e^2\right )^2 \left (-\frac {\left (a e^2 g-c d (6 e f-5 d g)\right ) \int \frac {1}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (e f-d g)}-\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(f+g x) (e f-d g)}\right )}{(c d f-a e g)^3}-\frac {2 c d \left (a^2 e^4 g+2 c d e x \left (4 a e^2 g-c d (3 d g+e f)\right )-a c d e^2 (e f-7 d g)-c^2 d^3 (6 d g+e f)\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}}{3 \left (c d^2-a e^2\right )^2}-\frac {2 c d (d+e x)}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {3 g^2 \left (c d^2-a e^2\right )^2 \left (\frac {\left (a e^2 g-c d (6 e f-5 d g)\right ) \int \frac {1}{4 (e f-d g) (c d f-a e g)-\frac {\left (c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\left (-\frac {c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{e f-d g}-\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(f+g x) (e f-d g)}\right )}{(c d f-a e g)^3}-\frac {2 c d \left (a^2 e^4 g+2 c d e x \left (4 a e^2 g-c d (3 d g+e f)\right )-a c d e^2 (e f-7 d g)-c^2 d^3 (6 d g+e f)\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}}{3 \left (c d^2-a e^2\right )^2}-\frac {2 c d (d+e x)}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {3 g^2 \left (c d^2-a e^2\right )^2 \left (-\frac {\left (a e^2 g-c d (6 e f-5 d g)\right ) \text {arctanh}\left (\frac {-x \left (a e^2 g-c d (2 e f-d g)\right )+a e (e f-2 d g)+c d^2 f}{2 \sqrt {e f-d g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt {c d f-a e g}}\right )}{2 (e f-d g)^{3/2} \sqrt {c d f-a e g}}-\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(f+g x) (e f-d g)}\right )}{(c d f-a e g)^3}-\frac {2 c d \left (a^2 e^4 g+2 c d e x \left (4 a e^2 g-c d (3 d g+e f)\right )-a c d e^2 (e f-7 d g)-c^2 d^3 (6 d g+e f)\right )}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^3}}{3 \left (c d^2-a e^2\right )^2}-\frac {2 c d (d+e x)}{3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)^2}\)

Input:

Int[(d + e*x)^2/((f + g*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2) 
),x]
 

Output:

(-2*c*d*(d + e*x))/(3*(c*d*f - a*e*g)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e 
*x^2)^(3/2)) + ((-2*c*d*(a^2*e^4*g - a*c*d*e^2*(e*f - 7*d*g) - c^2*d^3*(e* 
f + 6*d*g) + 2*c*d*e*(4*a*e^2*g - c*d*(e*f + 3*d*g))*x))/((c*d*f - a*e*g)^ 
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3*(c*d^2 - a*e^2)^2*g^2* 
(-((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*e^2*g - c*d*(6*e*f - 5*d*g))*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])))/(c*d*f - a*e*g)^3)/(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 2177

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3246\) vs. \(2(402)=804\).

Time = 2.03 (sec) , antiderivative size = 3247, normalized size of antiderivative = 7.66

method	result	size
default	\(\text {Expression too large to display}\)	\(3247\)

Input:

int((e*x+d)^2/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2),x,method=_ 
RETURNVERBOSE)
 

Output:

e^2/g^2*(2/3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d* 
e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+16/3*d*e*c/(4*a*c*d^2*e^2-(a*e^2+c*d^2) 
^2)^2*(2*c*d*e*x+a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))+1/g 
^4*(d^2*g^2-2*d*e*f*g+e^2*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)^(3/2)-5/2*(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)*(1/3/(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)^(3/2)-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)*(2/3*(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)^(3/2)+16/3*d*e*c/(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)^2*(2*d*e*c*(x+f/g)+(a*e^2*g+ 
c*d^2*g-2*c*d*e*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))+1/(a*d*e*g^2-a*e^ 
2*f*g-c*d^2*f*g+c*d*e*f^2)*g^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*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/(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3368 vs. \(2 (402) = 804\).

Time = 68.33 (sec) , antiderivative size = 6793, normalized size of antiderivative = 16.02 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^2/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, a 
lgorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**2/(g*x+f)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2 
),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^2/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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(((a*e^2)/g>0)', see `assume?` fo 
r more det
 

Giac [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)^2/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, a 
lgorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{{\left (f+g\,x\right )}^2\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \] Input:

int((d + e*x)^2/((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2) 
),x)
 

Output:

int((d + e*x)^2/((f + g*x)^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2) 
), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\left (e x +d \right )^{2}}{\left (g x +f \right )^{2} {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}d x \] Input:

int((e*x+d)^2/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
 

Output:

int((e*x+d)^2/(g*x+f)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)