\(\int \frac {(d+e x)^2 (f+g x)^2}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [283]

Optimal result

Integrand size = 44, antiderivative size = 290 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 (c d f-a e g)^2 (d+e x)}{c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {g \left (7 a e^2 g-c d (8 e f+d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^3 d^3 e}+\frac {g^2 x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c^2 d^2}+\frac {\left (15 a^2 e^4 g^2-6 a c d e^2 g (4 e f+d g)+c^2 d^2 \left (8 e^2 f^2+8 d e f g-d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{7/2} d^{7/2} e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {c} \sqrt {d} \sqrt {e} (d+e x) \left (-15 a^2 e^3 g^2+a c d e g (24 e f+d g-5 e g x)+c^2 d^2 \left (d g^2 x+e \left (-8 f^2+8 f g x+2 g^2 x^2\right )\right )\right )-\left (-15 a^2 e^4 g^2+6 a c d e^2 g (4 e f+d g)+c^2 d^2 \left (-8 e^2 f^2-8 d e f g+d^2 g^2\right )\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{4 c^{7/2} d^{7/2} e^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] 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 
)^(3/2),x]
 

Output:

(Sqrt[c]*Sqrt[d]*Sqrt[e]*(d + e*x)*(-15*a^2*e^3*g^2 + a*c*d*e*g*(24*e*f + 
d*g - 5*e*g*x) + c^2*d^2*(d*g^2*x + e*(-8*f^2 + 8*f*g*x + 2*g^2*x^2))) - ( 
-15*a^2*e^4*g^2 + 6*a*c*d*e^2*g*(4*e*f + d*g) + c^2*d^2*(-8*e^2*f^2 - 8*d* 
e*f*g + d^2*g^2))*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[c]*Sqrt[d] 
*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(4*c^(7/2)*d^(7/2)*e^(3/2)*S 
qrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1211, 2192, 27, 1160, 1092, 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.

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

\(\Big \downarrow \) 1211

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

\(\Big \downarrow \) 2192

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

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

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1211

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*(2*c*d - b*e)^(m - 2)*(c*( 
e*f + d*g) - b*e*g)^n*((d + e*x)/(c^(m + n - 1)*e^(n - 1)*Sqrt[a + b*x + c* 
x^2])), x] + Simp[1/(c^(m + n - 1)*e^(n - 2))   Int[ExpandToSum[((2*c*d - b 
*e)^(m - 1)*(c*(e*f + d*g) - b*e*g)^n - c^(m + n - 1)*e^n*(d + e*x)^(m - 1) 
*(f + g*x)^n)/(c*d - b*e - c*e*x), x]/Sqrt[a + b*x + c*x^2], x], x] /; Free 
Q[{a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[m, 0] 
&& IGtQ[n, 0]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1799\) vs. \(2(264)=528\).

Time = 2.52 (sec) , antiderivative size = 1800, normalized size of antiderivative = 6.21

method	result	size
default	\(\text {Expression too large to display}\)	\(1800\)

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

Fricas [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.01 \[ \int \frac {(d+e x)^2 (f+g x)^2}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] 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:

[1/16*((8*a*c^2*d^2*e^3*f^2 + 8*(a*c^2*d^3*e^2 - 3*a^2*c*d*e^4)*f*g - (a*c 
^2*d^4*e + 6*a^2*c*d^2*e^3 - 15*a^3*e^5)*g^2 + (8*c^3*d^3*e^2*f^2 + 8*(c^3 
*d^4*e - 3*a*c^2*d^2*e^3)*f*g - (c^3*d^5 + 6*a*c^2*d^3*e^2 - 15*a^2*c*d*e^ 
4)*g^2)*x)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a 
^2*e^4 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 
+ a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*(2*c^3*d^3*e^2*g^2 
*x^2 - 8*c^3*d^3*e^2*f^2 + 24*a*c^2*d^2*e^3*f*g + (a*c^2*d^3*e^2 - 15*a^2* 
c*d*e^4)*g^2 + (8*c^3*d^3*e^2*f*g + (c^3*d^4*e - 5*a*c^2*d^2*e^3)*g^2)*x)* 
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^2*x + a*c^4*d^4*e^ 
3), -1/8*((8*a*c^2*d^2*e^3*f^2 + 8*(a*c^2*d^3*e^2 - 3*a^2*c*d*e^4)*f*g - ( 
a*c^2*d^4*e + 6*a^2*c*d^2*e^3 - 15*a^3*e^5)*g^2 + (8*c^3*d^3*e^2*f^2 + 8*( 
c^3*d^4*e - 3*a*c^2*d^2*e^3)*f*g - (c^3*d^5 + 6*a*c^2*d^3*e^2 - 15*a^2*c*d 
*e^4)*g^2)*x)*sqrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a* 
e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c^2*d^2*e^2*x^2 + a*c*d^ 
2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) - 2*(2*c^3*d^3*e^2*g^2*x^2 - 8*c^3*d^3 
*e^2*f^2 + 24*a*c^2*d^2*e^3*f*g + (a*c^2*d^3*e^2 - 15*a^2*c*d*e^4)*g^2 + ( 
8*c^3*d^3*e^2*f*g + (c^3*d^4*e - 5*a*c^2*d^2*e^3)*g^2)*x)*sqrt(c*d*e*x^2 + 
 a*d*e + (c*d^2 + a*e^2)*x))/(c^5*d^5*e^2*x + a*c^4*d^4*e^3)]
 

Sympy [F]

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

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

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-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

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

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,[4,4,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,0, 
0,0]%%%}+
 

Mupad [F(-1)]

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

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

Output:

int(((f + g*x)^2*(d + e*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 = 0.66 (sec) , antiderivative size = 725, normalized size of antiderivative = 2.50 \[ \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 {15 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4} g^{2}-6 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{2} g^{2}-24 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c d \,e^{3} f g -\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4} g^{2}+8 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{3} e f g +8 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{2} e^{2} f^{2}-10 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} e^{4} g^{2}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2} g^{2}+18 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d \,e^{3} f g -2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{3} e f g -8 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{2} f^{2}-15 \sqrt {e x +d}\, a^{2} c d \,e^{4} g^{2}+\sqrt {e x +d}\, a \,c^{2} d^{3} e^{2} g^{2}+24 \sqrt {e x +d}\, a \,c^{2} d^{2} e^{3} f g -5 \sqrt {e x +d}\, a \,c^{2} d^{2} e^{3} g^{2} x +\sqrt {e x +d}\, c^{3} d^{4} e \,g^{2} x -8 \sqrt {e x +d}\, c^{3} d^{3} e^{2} f^{2}+8 \sqrt {e x +d}\, c^{3} d^{3} e^{2} f g x +2 \sqrt {e x +d}\, c^{3} d^{3} e^{2} g^{2} x^{2}}{4 \sqrt {c d x +a e}\, c^{4} d^{4} e^{2}} \] 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:

(15*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c*d* 
x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*e**4*g**2 
- 6*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c*d* 
x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a*c*d**2*e**2*g 
**2 - 24*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + 
 c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a*c*d*e**3 
*f*g - sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c 
*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c**2*d**4*g* 
*2 + 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c 
*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c**2*d**3*e* 
f*g + 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + 
c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c**2*d**2*e 
**2*f**2 - 10*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*e**4*g**2 + 2 
*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d**2*e**2*g**2 + 18*sqrt(e) 
*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d*e**3*f*g - 2*sqrt(e)*sqrt(d)*sqrt 
(c)*sqrt(a*e + c*d*x)*c**2*d**3*e*f*g - 8*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e 
 + c*d*x)*c**2*d**2*e**2*f**2 - 15*sqrt(d + e*x)*a**2*c*d*e**4*g**2 + sqrt 
(d + e*x)*a*c**2*d**3*e**2*g**2 + 24*sqrt(d + e*x)*a*c**2*d**2*e**3*f*g - 
5*sqrt(d + e*x)*a*c**2*d**2*e**3*g**2*x + sqrt(d + e*x)*c**3*d**4*e*g**2*x 
 - 8*sqrt(d + e*x)*c**3*d**3*e**2*f**2 + 8*sqrt(d + e*x)*c**3*d**3*e**2...