\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x) (f+g x)^3} \, dx\) [273]

Optimal result

Integrand size = 44, antiderivative size = 364 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=-\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^2 (f+g x)}-\frac {\left (c d^2 f+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (e f-d g) (f+g x)^2}+\frac {2 c^{3/2} d^{3/2} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{g^3}+\frac {\left (a^2 e^4 g^2+2 a c d e^2 g (2 e f-3 d g)-c^2 d^2 \left (8 e^2 f^2-12 d e f g+3 d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 g^3 (e f-d g)^{3/2} \sqrt {c d f-a e g}} \] Output:

-c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^2/(g*x+f)-1/4*(c*d^2*f+a*e* 
(-2*d*g+e*f)-(a*e^2*g-c*d*(-d*g+2*e*f))*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^ 
2)^(1/2)/g/(-d*g+e*f)/(g*x+f)^2+2*c^(3/2)*d^(3/2)*e^(1/2)*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))/g^3+1/4*( 
a^2*e^4*g^2+2*a*c*d*e^2*g*(-3*d*g+2*e*f)-c^2*d^2*(3*d^2*g^2-12*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))/g^3/(-d*g+e*f)^(3/2)/(-a*e*g+c*d*f)^(1/2)
 

Mathematica [A] (verified)

Time = 2.57 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {g (a e g (-e f+2 d g+e g x)+c d (-2 e f (2 f+3 g x)+d g (3 f+5 g x)))}{(e f-d g) (f+g x)^2}+\frac {\left (a^2 e^4 g^2+2 a c d e^2 g (2 e f-3 d g)+c^2 d^2 \left (-8 e^2 f^2+12 d e f g-3 d^2 g^2\right )\right ) \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )}{(-e f+d g)^{3/2} \sqrt {c d f-a e g} \sqrt {a e+c d x} \sqrt {d+e x}}+\frac {8 c^{3/2} d^{3/2} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{4 g^3} \] Input:

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

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((g*(a*e*g*(-(e*f) + 2*d*g + e*g*x) + c*d*( 
-2*e*f*(2*f + 3*g*x) + d*g*(3*f + 5*g*x))))/((e*f - d*g)*(f + g*x)^2) + (( 
a^2*e^4*g^2 + 2*a*c*d*e^2*g*(2*e*f - 3*d*g) + c^2*d^2*(-8*e^2*f^2 + 12*d*e 
*f*g - 3*d^2*g^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)*Sqrt[c*d*f - a*e*g]*Sqr 
t[a*e + c*d*x]*Sqrt[d + e*x]) + (8*c^(3/2)*d^(3/2)*Sqrt[e]*ArcTanh[(Sqrt[c 
]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]* 
Sqrt[d + e*x])))/(4*g^3)
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.23, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1215, 1229, 27, 1269, 1092, 219, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1229

\(\displaystyle -\frac {\int -\frac {(c d f-a e g) \left (a^2 g e^4+2 a c d (2 e f-3 d g) e^2+8 c^2 d^2 (e f-d g) x e+c^2 d^3 (4 e f-3 d g)\right )}{2 (f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 g^2 (e f-d g) (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left ((c d f-a e g) \left (-2 a d e g^2+a e^2 f g-3 c d^2 f g+4 c d e f^2\right )+g x (c d f-a e g) \left (-a e^2 g-5 c d^2 g+6 c d e f\right )\right )}{4 g^2 (f+g x)^2 (e f-d g) (c d f-a e g)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a^2 g e^4+2 a c d (2 e f-3 d g) e^2+8 c^2 d^2 (e f-d g) x e+c^2 d^3 (4 e f-3 d g)}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 g^2 (e f-d g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left ((c d f-a e g) \left (-2 a d e g^2+a e^2 f g-3 c d^2 f g+4 c d e f^2\right )+g x (c d f-a e g) \left (-a e^2 g-5 c d^2 g+6 c d e f\right )\right )}{4 g^2 (f+g x)^2 (e f-d g) (c d f-a e g)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\left (a^2 e^4 g^2+2 a c d e^2 g (2 e f-3 d g)-c^2 d^2 \left (3 d^2 g^2-12 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{g}+\frac {8 c^2 d^2 e (e f-d g) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{g}}{8 g^2 (e f-d g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left ((c d f-a e g) \left (-2 a d e g^2+a e^2 f g-3 c d^2 f g+4 c d e f^2\right )+g x (c d f-a e g) \left (-a e^2 g-5 c d^2 g+6 c d e f\right )\right )}{4 g^2 (f+g x)^2 (e f-d g) (c d f-a e g)}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\left (a^2 e^4 g^2+2 a c d e^2 g (2 e f-3 d g)-c^2 d^2 \left (3 d^2 g^2-12 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{g}+\frac {16 c^2 d^2 e (e f-d g) \int \frac {1}{4 c d e-\frac {\left (c d^2+2 c e x d+a e^2\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {c d^2+2 c e x d+a e^2}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{g}}{8 g^2 (e f-d g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left ((c d f-a e g) \left (-2 a d e g^2+a e^2 f g-3 c d^2 f g+4 c d e f^2\right )+g x (c d f-a e g) \left (-a e^2 g-5 c d^2 g+6 c d e f\right )\right )}{4 g^2 (f+g x)^2 (e f-d g) (c d f-a e g)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (a^2 e^4 g^2+2 a c d e^2 g (2 e f-3 d g)-c^2 d^2 \left (3 d^2 g^2-12 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{g}+\frac {8 c^{3/2} d^{3/2} \sqrt {e} (e f-d g) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{g}}{8 g^2 (e f-d g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left ((c d f-a e g) \left (-2 a d e g^2+a e^2 f g-3 c d^2 f g+4 c d e f^2\right )+g x (c d f-a e g) \left (-a e^2 g-5 c d^2 g+6 c d e f\right )\right )}{4 g^2 (f+g x)^2 (e f-d g) (c d f-a e g)}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {8 c^{3/2} d^{3/2} \sqrt {e} (e f-d g) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{g}-\frac {2 \left (a^2 e^4 g^2+2 a c d e^2 g (2 e f-3 d g)-c^2 d^2 \left (3 d^2 g^2-12 d e f g+8 e^2 f^2\right )\right ) \int \frac {1}{4 (e f-d g) (c d f-a e g)-\frac {\left (c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\left (-\frac {c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{g}}{8 g^2 (e f-d g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left ((c d f-a e g) \left (-2 a d e g^2+a e^2 f g-3 c d^2 f g+4 c d e f^2\right )+g x (c d f-a e g) \left (-a e^2 g-5 c d^2 g+6 c d e f\right )\right )}{4 g^2 (f+g x)^2 (e f-d g) (c d f-a e g)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (a^2 e^4 g^2+2 a c d e^2 g (2 e f-3 d g)-c^2 d^2 \left (3 d^2 g^2-12 d e f g+8 e^2 f^2\right )\right ) \text {arctanh}\left (\frac {-x \left (a e^2 g-c d (2 e f-d g)\right )+a e (e f-2 d g)+c d^2 f}{2 \sqrt {e f-d g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt {c d f-a e g}}\right )}{g \sqrt {e f-d g} \sqrt {c d f-a e g}}+\frac {8 c^{3/2} d^{3/2} \sqrt {e} (e f-d g) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{g}}{8 g^2 (e f-d g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left ((c d f-a e g) \left (-2 a d e g^2+a e^2 f g-3 c d^2 f g+4 c d e f^2\right )+g x (c d f-a e g) \left (-a e^2 g-5 c d^2 g+6 c d e f\right )\right )}{4 g^2 (f+g x)^2 (e f-d g) (c d f-a e g)}\)

Input:

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

Output:

-1/4*(((c*d*f - a*e*g)*(4*c*d*e*f^2 - 3*c*d^2*f*g + a*e^2*f*g - 2*a*d*e*g^ 
2) + g*(c*d*f - a*e*g)*(6*c*d*e*f - 5*c*d^2*g - a*e^2*g)*x)*Sqrt[a*d*e + ( 
c*d^2 + a*e^2)*x + c*d*e*x^2])/(g^2*(e*f - d*g)*(c*d*f - a*e*g)*(f + g*x)^ 
2) + ((8*c^(3/2)*d^(3/2)*Sqrt[e]*(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])])/g + ((a^2*e^4*g^2 + 2*a*c*d*e^2*g*(2*e*f - 3*d*g) - c^2*d^2*(8*e^2* 
f^2 - 12*d*e*f*g + 3*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])])/(g*Sqrt[e*f - d*g]*Sqrt[c*d*f - a 
*e*g]))/(8*g^2*(e*f - d*g))
 

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 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 1215

Int[(((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/( 
(d_) + (e_.)*(x_)), x_Symbol] :> Int[(a/d + c*(x/e))*(f + g*x)^n*(a + b*x + 
 c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[c*d^2 - 
 b*d*e + a*e^2, 0] && GtQ[p, 0]
 

rule 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6008\) vs. \(2(332)=664\).

Time = 3.05 (sec) , antiderivative size = 6009, normalized size of antiderivative = 16.51

method	result	size
default	\(\text {Expression too large to display}\)	\(6009\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)/(g*x+f)^3,x, alg 
orithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2052 vs. \(2 (332) = 664\).

Time = 1.18 (sec) , antiderivative size = 2052, normalized size of antiderivative = 5.64 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-c^2*d^2*e*log(abs(-c*d^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c* 
d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))))/(sqrt(c*d*e)*g^3) - 1/4*(8*c^2*d^2 
*e^2*f^2 - 12*c^2*d^3*e*f*g - 4*a*c*d*e^3*f*g + 3*c^2*d^4*g^2 + 6*a*c*d^2* 
e^2*g^2 - a^2*e^4*g^2)*arctan(-((sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x 
+ a*e^2*x + a*d*e))*g + sqrt(c*d*e)*f)/sqrt(-c*d*e*f^2 + c*d^2*f*g + a*e^2 
*f*g - a*d*e*g^2))/(sqrt(-c*d*e*f^2 + c*d^2*f*g + a*e^2*f*g - a*d*e*g^2)*( 
e*f*g^3 - d*g^4)) - 1/4*(6*c^4*d^6*e^2*f^3 + 12*a*c^3*d^4*e^4*f^3 + 6*a^2* 
c^2*d^2*e^6*f^3 - 5*c^4*d^7*e*f^2*g - 27*a*c^3*d^5*e^3*f^2*g - 23*a^2*c^2* 
d^3*e^5*f^2*g - a^3*c*d*e^7*f^2*g + 13*a*c^3*d^6*e^2*f*g^2 + 26*a^2*c^2*d^ 
4*e^4*f*g^2 + a^3*c*d^2*e^6*f*g^2 - 8*a^2*c^2*d^5*e^3*g^3 + 24*(sqrt(c*d*e 
)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*c^3*d^3*e^3*f^3 - 20* 
(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*c^3*d^4*e^ 
2*f^2*g - 4*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^ 
2*a*c^2*d^2*e^4*f^2*g - (sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2* 
x + a*d*e))^2*c^3*d^5*e*f*g^2 - 18*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2 
*x + a*e^2*x + a*d*e))^2*a*c^2*d^3*e^3*f*g^2 - 5*(sqrt(c*d*e)*x - sqrt(c*d 
*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^2*c*d*e^5*f*g^2 + 16*(sqrt(c*d*e) 
*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a*c^2*d^4*e^2*g^3 + 8* 
(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^2*c*d^2* 
e^4*g^3 + 24*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 9.33 (sec) , antiderivative size = 14401, normalized size of antiderivative = 39.56 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x) (f+g x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - sqrt(d*g - e*f)*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*f**2*g**3 - 2*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*s 
qrt(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)*s 
qrt(c)*sqrt(d + e*x))*a**3*e**6*f*g**4*x - sqrt(d*g - e*f)*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 + 5*sqrt(d*g - 
 e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*s 
qrt(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 + 10*sqrt(d*g - e*f)*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*g**4*x + 5*sqrt(d*g - e*f)*sqrt(a*e*g - 
 c*d*f)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqr 
t(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*g**5*x**2 -...