\(\int \frac {(d+e x)^2}{(f+g x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [287]

Optimal result

Integrand size = 44, antiderivative size = 210 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 c d (d+e x)}{(c d f-a e g)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {g \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(c d f-a e g)^2 (f+g x)}+\frac {\left (a e^2 g+c d (2 e f-3 d g)\right ) \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {e f-d g} (c d f-a e g)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\frac {(a e+c d x) (d+e x)^2 (a e g+c d (2 f+3 g x))}{(c d f-a e g)^2 (f+g x)}-\frac {\left (a e^2 g+c d (2 e f-3 d g)\right ) (a e+c d x)^{3/2} (d+e x)^{3/2} \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )}{\sqrt {-e f+d g} (c d f-a e g)^{5/2}}}{((a e+c d x) (d+e x))^{3/2}} \] 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) 
^(3/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1264, 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.

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

\(\Big \downarrow \) 1264

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

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)^(3/2) 
),x]
 

Output:

(-2*c*d*(d + e*x))/((c*d*f - a*e*g)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d 
*e*x^2]) + (-((g*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(f + g*x)) + 
 ((a*e^2*g + c*d*(2*e*f - 3*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]*Sqr 
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sqrt[e*f - d*g]*Sqrt[c*d*f 
- a*e*g]))/(c*d*f - a*e*g)^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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1794\) vs. \(2(196)=392\).

Time = 2.17 (sec) , antiderivative size = 1795, normalized size of antiderivative = 8.55

method	result	size
default	\(\text {Expression too large to display}\)	\(1795\)

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)^(3/2),x,method=_ 
RETURNVERBOSE)
 

Output:

2*e^2/g^2*(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)^(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)^(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*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)))-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*(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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (196) = 392\).

Time = 9.97 (sec) , antiderivative size = 1923, normalized size of antiderivative = 9.16 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] 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)^(3/2),x, a 
lgorithm="fricas")
 

Output:

[1/4*((2*a*c*d*e^2*f^2 - (3*a*c*d^2*e - a^2*e^3)*f*g + (2*c^2*d^2*e*f*g - 
(3*c^2*d^3 - a*c*d*e^2)*g^2)*x^2 + (2*c^2*d^2*e*f^2 - 3*(c^2*d^3 - a*c*d*e 
^2)*f*g - (3*a*c*d^2*e - a^2*e^3)*g^2)*x)*sqrt(c*d*e*f^2 + a*d*e*g^2 - (c* 
d^2 + a*e^2)*f*g)*log((8*a^2*d^2*e^2*g^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2* 
e^4)*f^2 - 8*(a*c*d^3*e + a^2*d*e^3)*f*g + (8*c^2*d^2*e^2*f^2 - 8*(c^2*d^3 
*e + a*c*d*e^3)*f*g + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*g^2)*x^2 - 4*sqr 
t(c*d*e*f^2 + a*d*e*g^2 - (c*d^2 + a*e^2)*f*g)*sqrt(c*d*e*x^2 + a*d*e + (c 
*d^2 + a*e^2)*x)*(2*a*d*e*g - (c*d^2 + a*e^2)*f - (2*c*d*e*f - (c*d^2 + a* 
e^2)*g)*x) + 2*(4*(c^2*d^3*e + a*c*d*e^3)*f^2 - (3*c^2*d^4 + 10*a*c*d^2*e^ 
2 + 3*a^2*e^4)*f*g + 4*(a*c*d^3*e + a^2*d*e^3)*g^2)*x)/(g^2*x^2 + 2*f*g*x 
+ f^2)) - 4*(2*c^2*d^2*e*f^3 + a^2*d*e^2*g^3 - (2*c^2*d^3 + a*c*d*e^2)*f^2 
*g + (a*c*d^2*e - a^2*e^3)*f*g^2 + 3*(c^2*d^2*e*f^2*g + a*c*d^2*e*g^3 - (c 
^2*d^3 + a*c*d*e^2)*f*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)) 
/(a*c^3*d^3*e^2*f^5 + a^4*d*e^4*f*g^4 - (a*c^3*d^4*e + 3*a^2*c^2*d^2*e^3)* 
f^4*g + 3*(a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*f^3*g^2 - (3*a^3*c*d^2*e^3 + a^4 
*e^5)*f^2*g^3 + (c^4*d^4*e*f^4*g + a^3*c*d^2*e^3*g^5 - (c^4*d^5 + 3*a*c^3* 
d^3*e^2)*f^3*g^2 + 3*(a*c^3*d^4*e + a^2*c^2*d^2*e^3)*f^2*g^3 - (3*a^2*c^2* 
d^3*e^2 + a^3*c*d*e^4)*f*g^4)*x^2 + (c^4*d^4*e*f^5 + 2*a*c^3*d^4*e*f^3*g^2 
 + 2*a^3*c*d*e^4*f^2*g^3 + a^4*d*e^4*g^5 - (c^4*d^5 + 2*a*c^3*d^3*e^2)*f^4 
*g - (2*a^3*c*d^2*e^3 + a^4*e^5)*f*g^4)*x), 1/2*((2*a*c*d*e^2*f^2 - (3*...
 

Sympy [F(-1)]

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] 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)^(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(((a*e^2)/g>0)', see `assume?` fo 
r more det
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2888 vs. \(2 (196) = 392\).

Time = 8.49 (sec) , antiderivative size = 2888, normalized size of antiderivative = 13.75 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Too large to display} \] 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)^(3/2),x, a 
lgorithm="giac")
 

Output:

1/6*((2*c*d*e*f*g^3*log(abs(-8*sqrt(c*d*e)*c^3*d^3*e^2*f^3*abs(g) + 12*sqr 
t(c*d*e)*c^3*d^4*e*f^2*g*abs(g) + 12*sqrt(c*d*e)*a*c^2*d^2*e^3*f^2*g*abs(g 
) - 4*sqrt(c*d*e)*c^3*d^5*f*g^2*abs(g) - 16*sqrt(c*d*e)*a*c^2*d^3*e^2*f*g^ 
2*abs(g) - 4*sqrt(c*d*e)*a^2*c*d*e^4*f*g^2*abs(g) + 4*sqrt(c*d*e)*a*c^2*d^ 
4*e*g^3*abs(g) + 4*sqrt(c*d*e)*a^2*c*d^2*e^3*g^3*abs(g) + 8*sqrt(c*d*e*f^2 
 - c*d^2*f*g - a*e^2*f*g + a*d*e*g^2)*c^3*d^3*e^2*f^2*g - 8*sqrt(c*d*e*f^2 
 - c*d^2*f*g - a*e^2*f*g + a*d*e*g^2)*c^3*d^4*e*f*g^2 - 8*sqrt(c*d*e*f^2 - 
 c*d^2*f*g - a*e^2*f*g + a*d*e*g^2)*a*c^2*d^2*e^3*f*g^2 + sqrt(c*d*e*f^2 - 
 c*d^2*f*g - a*e^2*f*g + a*d*e*g^2)*c^3*d^5*g^3 + 6*sqrt(c*d*e*f^2 - c*d^2 
*f*g - a*e^2*f*g + a*d*e*g^2)*a*c^2*d^3*e^2*g^3 + sqrt(c*d*e*f^2 - c*d^2*f 
*g - a*e^2*f*g + a*d*e*g^2)*a^2*c*d*e^4*g^3)) - 3*c*d^2*g^4*log(abs(-8*sqr 
t(c*d*e)*c^3*d^3*e^2*f^3*abs(g) + 12*sqrt(c*d*e)*c^3*d^4*e*f^2*g*abs(g) + 
12*sqrt(c*d*e)*a*c^2*d^2*e^3*f^2*g*abs(g) - 4*sqrt(c*d*e)*c^3*d^5*f*g^2*ab 
s(g) - 16*sqrt(c*d*e)*a*c^2*d^3*e^2*f*g^2*abs(g) - 4*sqrt(c*d*e)*a^2*c*d*e 
^4*f*g^2*abs(g) + 4*sqrt(c*d*e)*a*c^2*d^4*e*g^3*abs(g) + 4*sqrt(c*d*e)*a^2 
*c*d^2*e^3*g^3*abs(g) + 8*sqrt(c*d*e*f^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g 
^2)*c^3*d^3*e^2*f^2*g - 8*sqrt(c*d*e*f^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g 
^2)*c^3*d^4*e*f*g^2 - 8*sqrt(c*d*e*f^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g^2 
)*a*c^2*d^2*e^3*f*g^2 + sqrt(c*d*e*f^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g^2 
)*c^3*d^5*g^3 + 6*sqrt(c*d*e*f^2 - c*d^2*f*g - a*e^2*f*g + a*d*e*g^2)*a...
 

Mupad [F(-1)]

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

Reduce [B] (verification not implemented)

Time = 8.60 (sec) , antiderivative size = 4829, normalized size of antiderivative = 23.00 \[ \int \frac {(d+e x)^2}{(f+g x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] 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)^(3/2),x)
 

Output:

(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*e**4*f*g**2 + 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*e**4*g**3*x - 2*sqr 
t(d*g - e*f)*sqrt(a*e + c*d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqr 
t(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)*sqr 
t(d + e*x))*a*c*d*e**3*f**2*g - 2*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*c*d*e**3*f*g**2*x - 9*s 
qrt(d*g - e*f)*sqrt(a*e + c*d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*s 
qrt(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)*s 
qrt(d + e*x))*c**2*d**4*f*g**2 - 9*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 -...