\(\int \frac {(d+e x)^2 (f+g x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [289]

Optimal result

Integrand size = 44, antiderivative size = 297 \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (c d f-a e g)^3 (d+e x)^2}{3 c^3 d^3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 (c d f-a e g)^2 \left (7 a e^2 g+c d (2 e f-9 d g)\right ) (d+e x)}{3 c^3 d^3 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {g^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 e}-\frac {g^2 \left (5 a e^2 g-c d (6 e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{7/2} d^{7/2} e^{3/2}} \] Output:

-2/3*(-a*e*g+c*d*f)^3*(e*x+d)^2/c^3/d^3/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2 
)*x+c*d*e*x^2)^(3/2)+2/3*(-a*e*g+c*d*f)^2*(7*a*e^2*g+c*d*(-9*d*g+2*e*f))*( 
e*x+d)/c^3/d^3/(-a*e^2+c*d^2)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+g^ 
3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3/e-g^2*(5*a*e^2*g-c*d*(-d 
*g+6*e*f))*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 = 1.30 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {c} \sqrt {d} \sqrt {e} (d+e x) \left (15 a^4 e^6 g^3-2 a^3 c d e^4 g^2 (9 e f+11 d g-10 e g x)+3 a^2 c^2 d^2 e^2 g^2 \left (d^2 g+10 d e (f-g x)+e^2 x (-8 f+g x)\right )+c^4 d^4 \left (4 e^2 f^3 x+3 d^2 g^3 x^2-2 d e f^2 (f+9 g x)\right )+6 a c^3 d^3 e \left (d^2 g^3 x+e^2 f^2 (f+g x)-d e g \left (2 f^2-6 f g x+g^2 x^2\right )\right )\right )}{\left (c d^2-a e^2\right )^2 (a e+c d x)}-3 g^2 \left (5 a e^2 g+c d (-6 e f+d g)\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{3 c^{7/2} d^{7/2} e^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

((Sqrt[c]*Sqrt[d]*Sqrt[e]*(d + e*x)*(15*a^4*e^6*g^3 - 2*a^3*c*d*e^4*g^2*(9 
*e*f + 11*d*g - 10*e*g*x) + 3*a^2*c^2*d^2*e^2*g^2*(d^2*g + 10*d*e*(f - g*x 
) + e^2*x*(-8*f + g*x)) + c^4*d^4*(4*e^2*f^3*x + 3*d^2*g^3*x^2 - 2*d*e*f^2 
*(f + 9*g*x)) + 6*a*c^3*d^3*e*(d^2*g^3*x + e^2*f^2*(f + g*x) - d*e*g*(2*f^ 
2 - 6*f*g*x + g^2*x^2))))/((c*d^2 - a*e^2)^2*(a*e + c*d*x)) - 3*g^2*(5*a*e 
^2*g + c*d*(-6*e*f + d*g))*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])])/(3*c^(7/2)*d^(7/2)* 
e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1242, 27, 2165, 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.

\(\displaystyle \int \frac {(d+e x)^2 (f+g x)^3}{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1242

\(\displaystyle -\frac {2 \int \frac {(d+e x) \left (-\frac {3 \left (c d^2-a e^2\right )^2 x^2 g^3}{c d}-\frac {3 \left (c d^2-a e^2\right )^2 (3 c d f-a e g) x g^2}{c^2 d^2}+\frac {\left (c d^2-a e^2\right ) \left (a^3 g^3 e^4-3 a^2 c d g^2 (e f+d g) e^2+3 a c^2 d^2 f g (e f+3 d g) e+c^3 d^3 f^2 (2 e f-9 d g)\right )}{c^3 d^3}\right )}{2 \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )^2}-\frac {2 (d+e x)^2 (c d f-a e g)^3}{3 c^3 d^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(d+e x) \left (-\frac {3 \left (c d^2-a e^2\right )^2 x^2 g^3}{c d}-\frac {3 \left (c d^2-a e^2\right )^2 (3 c d f-a e g) x g^2}{c^2 d^2}+\frac {\left (c d^2-a e^2\right ) \left (a^3 g^3 e^4-3 a^2 c d g^2 (e f+d g) e^2+3 a c^2 d^2 f g (e f+3 d g) e+c^3 d^3 f^2 (2 e f-9 d g)\right )}{c^3 d^3}\right )}{\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}dx}{3 \left (c d^2-a e^2\right )^2}-\frac {2 (d+e x)^2 (c d f-a e g)^3}{3 c^3 d^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2165

\(\displaystyle -\frac {-\frac {2 \int \frac {3 \left (c d^2-a e^2\right )^4 g^2 (3 c d f-2 a e g+c d g x)}{2 c^3 d^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{\left (c d^2-a e^2\right )^2}-\frac {2 (d+e x) \left (7 a e^2 g+c d (2 e f-9 d g)\right ) (c d f-a e g)^2}{c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 \left (c d^2-a e^2\right )^2}-\frac {2 (d+e x)^2 (c d f-a e g)^3}{3 c^3 d^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 g^2 \left (c d^2-a e^2\right )^2 \int \frac {3 c d f-2 a e g+c d g x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c^3 d^3}-\frac {2 (d+e x) (c d f-a e g)^2 \left (7 a e^2 g+c d (2 e f-9 d g)\right )}{c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 \left (c d^2-a e^2\right )^2}-\frac {2 (d+e x)^2 (c d f-a e g)^3}{3 c^3 d^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle -\frac {-\frac {3 g^2 \left (c d^2-a e^2\right )^2 \left (\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {\left (5 a e^2 g-c d (6 e f-d g)\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 e}\right )}{c^3 d^3}-\frac {2 (d+e x) (c d f-a e g)^2 \left (7 a e^2 g+c d (2 e f-9 d g)\right )}{c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 \left (c d^2-a e^2\right )^2}-\frac {2 (d+e x)^2 (c d f-a e g)^3}{3 c^3 d^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1092

\(\displaystyle -\frac {-\frac {3 g^2 \left (c d^2-a e^2\right )^2 \left (\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {\left (5 a e^2 g-c d (6 e f-d g)\right ) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{e}\right )}{c^3 d^3}-\frac {2 (d+e x) (c d f-a e g)^2 \left (7 a e^2 g+c d (2 e f-9 d g)\right )}{c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 \left (c d^2-a e^2\right )^2}-\frac {2 (d+e x)^2 (c d f-a e g)^3}{3 c^3 d^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {3 g^2 \left (c d^2-a e^2\right )^2 \left (\frac {g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e}-\frac {\text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right ) \left (5 a e^2 g-c d (6 e f-d g)\right )}{2 \sqrt {c} \sqrt {d} e^{3/2}}\right )}{c^3 d^3}-\frac {2 (d+e x) (c d f-a e g)^2 \left (7 a e^2 g+c d (2 e f-9 d g)\right )}{c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}}{3 \left (c d^2-a e^2\right )^2}-\frac {2 (d+e x)^2 (c d f-a e g)^3}{3 c^3 d^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

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)^(5/2 
),x]
 

Output:

(-2*(c*d*f - a*e*g)^3*(d + e*x)^2)/(3*c^3*d^3*(c*d^2 - a*e^2)*(a*d*e + (c* 
d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - ((-2*(c*d*f - a*e*g)^2*(7*a*e^2*g + c 
*d*(2*e*f - 9*d*g))*(d + e*x))/(c^3*d^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 - ((5*a*e^2*g - c*d*(6*e*f - d*g))*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))))/(c^3*d^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 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 1242

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, a*e + c*d*x, x], R = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, 
 Simp[R*(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^ 
2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*( 
a + b*x + c*x^2)^(p + 1)*ExpandToSum[d*e*(p + 1)*(b^2 - 4*a*c)*Q - R*(2*c*d 
 - b*e)*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IG 
tQ[n, 1] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

rule 2165

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4500\) vs. \(2(273)=546\).

Time = 3.34 (sec) , antiderivative size = 4501, normalized size of antiderivative = 15.15

method	result	size
default	\(\text {Expression too large to display}\)	\(4501\)

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)^(5/2),x,method=_ 
RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (273) = 546\).

Time = 4.63 (sec) , antiderivative size = 1718, normalized size of antiderivative = 5.78 \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Too large to display} \] 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)^(5/2),x, a 
lgorithm="fricas")
 

Output:

[-1/12*(3*(6*(a^2*c^3*d^5*e^3 - 2*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*f*g^2 - ( 
a^2*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 - 9*a^4*c*d^2*e^6 + 5*a^5*e^8)*g^3 + ( 
6*(c^5*d^7*e - 2*a*c^4*d^5*e^3 + a^2*c^3*d^3*e^5)*f*g^2 - (c^5*d^8 + 3*a*c 
^4*d^6*e^2 - 9*a^2*c^3*d^4*e^4 + 5*a^3*c^2*d^2*e^6)*g^3)*x^2 + 2*(6*(a*c^4 
*d^6*e^2 - 2*a^2*c^3*d^4*e^4 + a^3*c^2*d^2*e^6)*f*g^2 - (a*c^4*d^7*e + 3*a 
^2*c^3*d^5*e^3 - 9*a^3*c^2*d^3*e^5 + 5*a^4*c*d*e^7)*g^3)*x)*sqrt(c*d*e)*lo 
g(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*(12*a*c^4*d^5*e^3*f^2*g - 3*(c^5*d^7*e - 2* 
a*c^4*d^5*e^3 + a^2*c^3*d^3*e^5)*g^3*x^2 + 2*(c^5*d^6*e^2 - 3*a*c^4*d^4*e^ 
4)*f^3 - 6*(5*a^2*c^3*d^4*e^4 - 3*a^3*c^2*d^2*e^6)*f*g^2 - (3*a^2*c^3*d^5* 
e^3 - 22*a^3*c^2*d^3*e^5 + 15*a^4*c*d*e^7)*g^3 - 2*(2*c^5*d^5*e^3*f^3 - 3* 
(3*c^5*d^6*e^2 - a*c^4*d^4*e^4)*f^2*g + 6*(3*a*c^4*d^5*e^3 - 2*a^2*c^3*d^3 
*e^5)*f*g^2 + (3*a*c^4*d^6*e^2 - 15*a^2*c^3*d^4*e^4 + 10*a^3*c^2*d^2*e^6)* 
g^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^2*c^6*d^8*e^4 - 2* 
a^3*c^5*d^6*e^6 + a^4*c^4*d^4*e^8 + (c^8*d^10*e^2 - 2*a*c^7*d^8*e^4 + a^2* 
c^6*d^6*e^6)*x^2 + 2*(a*c^7*d^9*e^3 - 2*a^2*c^6*d^7*e^5 + a^3*c^5*d^5*e^7) 
*x), -1/6*(3*(6*(a^2*c^3*d^5*e^3 - 2*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*f*g^2 
- (a^2*c^3*d^6*e^2 + 3*a^3*c^2*d^4*e^4 - 9*a^4*c*d^2*e^6 + 5*a^5*e^8)*g^3 
+ (6*(c^5*d^7*e - 2*a*c^4*d^5*e^3 + a^2*c^3*d^3*e^5)*f*g^2 - (c^5*d^8 +...
 

Sympy [F]

\[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{2} \left (f + g x\right )^{3}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \] 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)**(5/2 
),x)
 

Output:

Integral((d + e*x)**2*(f + g*x)**3/((d + e*x)*(a*e + c*d*x))**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] 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)^(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-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] 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)^(5/2),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%%%{%%%{2,[5,5,9]%%%},[4,4]%%%}+%%%{%%%{-8,[6,7,7]%%%},[4,3]%% 
%}+%%%{%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,{\left (d+e\,x\right )}^2}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \] Input:

int(((f + g*x)^3*(d + e*x)^2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2 
),x)
 

Output:

int(((f + g*x)^3*(d + e*x)^2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2 
), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.62 (sec) , antiderivative size = 2000, normalized size of antiderivative = 6.73 \[ \int \frac {(d+e x)^2 (f+g x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] 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)^(5/2),x)
 

Output:

( - 30*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**4*e**7*g* 
*3 + 54*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**3*c*d**2 
*e**5*g**3 + 36*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqr 
t(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a** 
3*c*d*e**6*f*g**2 - 30*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**3*c*d*e**6*g**3*x - 18*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*lo 
g((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*c**2*d**4*e**3*g**3 - 72*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))/s 
qrt(a*e**2 - c*d**2))*a**2*c**2*d**3*e**4*f*g**2 + 54*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)*sqr 
t(d + e*x))/sqrt(a*e**2 - c*d**2))*a**2*c**2*d**3*e**4*g**3*x + 36*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*c**2*d**2*e**5*f*g**2 
*x - 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**3*d**6* 
e*g**3 + 36*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt...