\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx\) [266]

Optimal result

Integrand size = 44, antiderivative size = 400 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (e f-d g) (f+g x)^3}-\frac {\left (5 a e^2 g-c d (4 e f+d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (e f-d g)^2 (c d f-a e g) (f+g x)^2}+\frac {\left (3 a e^2 g-c d (4 e f-d g)\right ) \left (5 a e^2 g-c d (2 e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 (e f-d g)^3 (c d f-a e g)^2 (f+g x)}-\frac {\left (c d^2-a e^2\right ) \left (5 a^2 e^4 g^2-2 a c d e^2 g (6 e f-d g)+c^2 d^2 \left (8 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 (e f-d g)^{7/2} (c d f-a e g)^{5/2}} \] Output:

1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-d*g+e*f)/(g*x+f)^3-1/12*(5*a 
*e^2*g-c*d*(d*g+4*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-d*g+e*f) 
^2/(-a*e*g+c*d*f)/(g*x+f)^2+1/24*(3*a*e^2*g-c*d*(-d*g+4*e*f))*(5*a*e^2*g-c 
*d*(3*d*g+2*e*f))*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-d*g+e*f)^3/(-a 
*e*g+c*d*f)^2/(g*x+f)-1/8*(-a*e^2+c*d^2)*(5*a^2*e^4*g^2-2*a*c*d*e^2*g*(-d* 
g+6*e*f)+c^2*d^2*(d^2*g^2-4*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))/(-d*g 
+e*f)^(7/2)/(-a*e*g+c*d*f)^(5/2)
 

Mathematica [A] (verified)

Time = 11.22 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-8 g (a e+c d x)-\frac {2 g \left (5 a e^2 g+c d (-8 e f+3 d g)\right ) (a e+c d x) (f+g x)}{(e f-d g) (-c d f+a e g)}+\frac {3 \left (5 a^2 e^4 g^2+2 a c d e^2 g (-6 e f+d g)+c^2 d^2 \left (8 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) (f+g x)^2 \left (\sqrt {e f-d g} \sqrt {c d f-a e g} \sqrt {a e+c d x} \sqrt {d+e x}-\left (c d^2-a e^2\right ) (f+g x) \text {arctanh}\left (\frac {\sqrt {e f-d g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )\right )}{(e f-d g)^{5/2} (c d f-a e g)^{3/2} \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{24 (-e f+d g) (-c d f+a e g) (f+g x)^3} \] Input:

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/((d + e*x)*(f + g*x) 
^4),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1215, 1237, 27, 1237, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1237

\(\displaystyle \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (f+g x)^3 (e f-d g)}-\frac {\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (5 a e^2 g-c d (d g+4 e f)\right )}{2 (f+g x)^2 (e f-d g) (c d f-a e g)}-\frac {\int -\frac {15 a^2 g e^4-4 a c d (4 e f+d g) e^2+2 c d \left (5 a e^2 g-c d (4 e f+d g)\right ) x e+c^2 d^3 (8 e f-3 d g)}{2 (f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (e f-d g) (c d f-a e g)}}{6 (e f-d g)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (f+g x)^3 (e f-d g)}-\frac {\frac {\int \frac {15 a^2 g e^4-4 a c d (4 e f+d g) e^2+2 c d \left (5 a e^2 g-c d (4 e f+d g)\right ) x e+c^2 d^3 (8 e f-3 d g)}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (e f-d g) (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (5 a e^2 g-c d (d g+4 e f)\right )}{2 (f+g x)^2 (e f-d g) (c d f-a e g)}}{6 (e f-d g)}\)

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/((d + e*x)*(f + g*x)^4),x]
 

Output:

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*(e*f - d*g)*(f + g*x)^3) - 
(((5*a*e^2*g - c*d*(4*e*f + d*g))*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x 
^2])/(2*(e*f - d*g)*(c*d*f - a*e*g)*(f + g*x)^2) + (-(((2*c*d*e*f + 3*c*d^ 
2*g - 5*a*e^2*g)*(4*c*d*e*f - c*d^2*g - 3*a*e^2*g)*Sqrt[a*d*e + (c*d^2 + a 
*e^2)*x + c*d*e*x^2])/((e*f - d*g)*(c*d*f - a*e*g)*(f + g*x))) + (3*(c*d^2 
 - a*e^2)*(5*a^2*e^4*g^2 - 2*a*c*d*e^2*g*(6*e*f - d*g) + c^2*d^2*(8*e^2*f^ 
2 - 4*d*e*f*g + 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])])/(2*(e*f - d*g)^(3/2)*(c*d*f - a*e*g)^( 
3/2)))/(4*(e*f - d*g)*(c*d*f - a*e*g)))/(6*(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 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 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 1237

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)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5580\) vs. \(2(376)=752\).

Time = 3.74 (sec) , antiderivative size = 5581, normalized size of antiderivative = 13.95

method	result	size
default	\(\text {Expression too large to display}\)	\(5581\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)/(g*x+f)^4,x,method=_RE 
TURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)/(g*x+f)^4,x, alg 
orithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)/(g*x+f)**4,x 
)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx=\int { \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )} {\left (g x + f\right )}^{4}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)/(g*x+f)^4,x, alg 
orithm="maxima")
 

Output:

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((e*x + d)*(g*x + f) 
^4), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6024 vs. \(2 (376) = 752\).

Time = 0.55 (sec) , antiderivative size = 6024, normalized size of antiderivative = 15.06 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)/(g*x+f)^4,x, alg 
orithm="giac")
 

Output:

-1/8*(8*c^3*d^4*e^2*f^2 - 8*a*c^2*d^2*e^4*f^2 - 4*c^3*d^5*e*f*g - 8*a*c^2* 
d^3*e^3*f*g + 12*a^2*c*d*e^5*f*g + c^3*d^6*g^2 + a*c^2*d^4*e^2*g^2 + 3*a^2 
*c*d^2*e^4*g^2 - 5*a^3*e^6*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))/((c^2*d^2*e^3*f^5 - 3*c^2*d^3*e^2*f^4*g - 2*a* 
c*d*e^4*f^4*g + 3*c^2*d^4*e*f^3*g^2 + 6*a*c*d^2*e^3*f^3*g^2 + a^2*e^5*f^3* 
g^2 - c^2*d^5*f^2*g^3 - 6*a*c*d^3*e^2*f^2*g^3 - 3*a^2*d*e^4*f^2*g^3 + 2*a* 
c*d^4*e*f*g^4 + 3*a^2*d^2*e^3*f*g^4 - a^2*d^3*e^2*g^5)*sqrt(-c*d*e*f^2 + c 
*d^2*f*g + a*e^2*f*g - a*d*e*g^2)) + 1/24*(48*(sqrt(c*d*e)*x - sqrt(c*d*e* 
x^2 + c*d^2*x + a*e^2*x + a*d*e))*c^5*d^7*e^3*f^5 + 96*(sqrt(c*d*e)*x - sq 
rt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a*c^4*d^5*e^5*f^5 + 48*(sqrt(c* 
d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^2*c^3*d^3*e^7*f^5 
+ 36*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*c^5*d^8 
*e^2*f^4*g - 300*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d 
*e))*a*c^4*d^6*e^4*f^4*g - 516*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + 
 a*e^2*x + a*d*e))*a^2*c^3*d^4*e^6*f^4*g - 180*(sqrt(c*d*e)*x - sqrt(c*d*e 
*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^3*c^2*d^2*e^8*f^4*g - 6*(sqrt(c*d*e)* 
x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*c^5*d^9*e*f^3*g^2 - 12*(s 
qrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a*c^4*d^7*e^3* 
f^3*g^2 + 864*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx=\int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (f+g\,x\right )}^4\,\left (d+e\,x\right )} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/((f + g*x)^4*(d + e*x)), 
x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/((f + g*x)^4*(d + e*x)), 
 x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 17.70 (sec) , antiderivative size = 25163, normalized size of antiderivative = 62.91 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^4} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)/(g*x+f)^4,x)
 

Output:

( - 15*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**4*e**8*f**3*g**4 - 45*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**4*e**8*f**2*g**5*x - 45*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**4*e**8*f*g**6*x**2 - 
15*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**4*e**8*g**7*x**3 - 6*sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sq 
rt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*s 
qrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sq 
rt(c)*sqrt(d + e*x))*a**3*c*d**2*e**6*f**3*g**4 - 18*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*c*d**2*e**6*f**2...