\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x) (f+g x)^3} \, dx\) [265]

Optimal result

Integrand size = 44, antiderivative size = 258 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^3} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 (e f-d g) (f+g x)^2}-\frac {\left (3 a e^2 g-c d (2 e f+d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 (e f-d g)^2 (c d f-a e g) (f+g x)}+\frac {\left (c d^2-a e^2\right ) \left (3 a e^2 g-c d (4 e f-d g)\right ) \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 (e f-d g)^{5/2} (c d f-a e g)^{3/2}} \] Output:

1/2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-d*g+e*f)/(g*x+f)^2-1/4*(3*a* 
e^2*g-c*d*(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)^ 
2/(-a*e*g+c*d*f)/(g*x+f)+1/4*(-a*e^2+c*d^2)*(3*a*e^2*g-c*d*(-d*g+4*e*f))*a 
rctanh((-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)^(5/2)/(-a*e*g+c*d*f)^(3/2)
 

Mathematica [A] (verified)

Time = 10.48 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^3} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {g (a e+c d x)}{(f+g x)^2}-\frac {\left (3 a e^2 g+c d (-4 e f+d g)\right ) \left (\frac {\sqrt {a e+c d x} \sqrt {d+e x}}{(e f-d g) (f+g x)}+\frac {\left (-c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e f-d g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{(e f-d g)^{3/2} \sqrt {c d f-a e g}}\right )}{2 \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{2 (-e f+d g) (-c d f+a e g)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1215, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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. \(3573\) vs. \(2(238)=476\).

Time = 3.05 (sec) , antiderivative size = 3574, normalized size of antiderivative = 13.85

method	result	size
default	\(\text {Expression too large to display}\)	\(3574\)

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)^3,x,method=_RE 
TURNVERBOSE)
 

Output:

1/g^2/(d*g-e*f)*(-1/2/(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 
)^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)^(3/2)-1/4*(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/(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)^(3/2)+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)*((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)+1/2*(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*ln((1/2*(a*e^2*g+c* 
d^2*g-2*c*d*e*f)/g+d*e*c*(x+f/g))/(d*e*c)^(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))/(d*e*c)^(1/2)-(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)))+2*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*(1 
/4*(2*d*e*c*(x+f/g)+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g)/d/e/c*(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)+1/8*(4*d*e*c*(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (238) = 476\).

Time = 14.73 (sec) , antiderivative size = 2441, normalized size of antiderivative = 9.46 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^3} \, dx=\text {Too large to display} \] 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)^3,x, alg 
orithm="fricas")
 

Output:

[-1/16*(sqrt(c*d*e*f^2 + a*d*e*g^2 - (c*d^2 + a*e^2)*f*g)*(4*(c^2*d^3*e - 
a*c*d*e^3)*f^3 - (c^2*d^4 + 2*a*c*d^2*e^2 - 3*a^2*e^4)*f^2*g + (4*(c^2*d^3 
*e - a*c*d*e^3)*f*g^2 - (c^2*d^4 + 2*a*c*d^2*e^2 - 3*a^2*e^4)*g^3)*x^2 + 2 
*(4*(c^2*d^3*e - a*c*d*e^3)*f^2*g - (c^2*d^4 + 2*a*c*d^2*e^2 - 3*a^2*e^4)* 
f*g^2)*x)*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*sqrt(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*(4*c^2*d^2*e^2*f^4 + 2*a^2*d^2*e^2*g^4 - (5*c^2*d^3*e + 9*a*c*d*e^3)*f 
^3*g + (c^2*d^4 + 12*a*c*d^2*e^2 + 5*a^2*e^4)*f^2*g^2 - (3*a*c*d^3*e + 7*a 
^2*d*e^3)*f*g^3 + (2*c^2*d^2*e^2*f^3*g - (c^2*d^3*e + 5*a*c*d*e^3)*f^2*g^2 
 - (c^2*d^4 - 4*a*c*d^2*e^2 - 3*a^2*e^4)*f*g^3 + (a*c*d^3*e - 3*a^2*d*e^3) 
*g^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^2*d^2*e^3*f^7 - a 
^2*d^3*e^2*f^2*g^5 - (3*c^2*d^3*e^2 + 2*a*c*d*e^4)*f^6*g + (3*c^2*d^4*e + 
6*a*c*d^2*e^3 + a^2*e^5)*f^5*g^2 - (c^2*d^5 + 6*a*c*d^3*e^2 + 3*a^2*d*e^4) 
*f^4*g^3 + (2*a*c*d^4*e + 3*a^2*d^2*e^3)*f^3*g^4 + (c^2*d^2*e^3*f^5*g^2 - 
a^2*d^3*e^2*g^7 - (3*c^2*d^3*e^2 + 2*a*c*d*e^4)*f^4*g^3 + (3*c^2*d^4*e ...
 

Sympy [F]

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

Output:

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

Maxima [F]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1848 vs. \(2 (238) = 476\).

Time = 0.39 (sec) , antiderivative size = 1848, normalized size of antiderivative = 7.16 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^3} \, dx=\text {Too large to display} \] 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)^3,x, alg 
orithm="giac")
 

Output:

-1/4*(4*c^2*d^3*e*f - 4*a*c*d*e^3*f - c^2*d^4*g - 2*a*c*d^2*e^2*g + 3*a^2* 
e^4*g)*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*d*e^2*f^3 - 2*c*d^2*e*f^2*g - a*e^3*f^2*g + c*d^3*f*g^2 + 2*a*d*e^2 
*f*g^2 - a*d^2*e*g^3)*sqrt(-c*d*e*f^2 + c*d^2*f*g + a*e^2*f*g - a*d*e*g^2) 
) + 1/4*(8*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*c 
^3*d^4*e^2*f^3 + 8*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a 
*d*e))*a*c^2*d^2*e^4*f^3 - 32*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + 
a*e^2*x + a*d*e))*a*c^2*d^3*e^3*f^2*g - 16*(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*d*e^5*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))*c^3*d^6*f*g^2 + 7*(sqrt(c*d*e)*x - 
sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a*c^2*d^4*e^2*f*g^2 + 35*(sqr 
t(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^2*c*d^2*e^4*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))*a^3* 
e^6*f*g^2 - (sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))* 
a*c^2*d^5*e*g^3 - 10*(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*d^3*e^3*g^3 - 5*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + 
 a*e^2*x + a*d*e))*a^3*d*e^5*g^3 + 2*sqrt(c*d*e)*c^3*d^5*e*f^3 + 4*sqrt(c* 
d*e)*a*c^2*d^3*e^3*f^3 + 2*sqrt(c*d*e)*a^2*c*d*e^5*f^3 + sqrt(c*d*e)*c^3*d 
^6*f^2*g - 9*sqrt(c*d*e)*a*c^2*d^4*e^2*f^2*g - 13*sqrt(c*d*e)*a^2*c*d^2...
 

Mupad [F(-1)]

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

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/((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)^(1/2)/((f + g*x)^3*(d + e*x)), 
 x)
 

Reduce [B] (verification not implemented)

Time = 2.22 (sec) , antiderivative size = 11755, normalized size of antiderivative = 45.56 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)^3} \, dx =\text {Too large to display} \] 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)^3,x)
 

Output:

(3*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 + 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*e**6*f*g**4*x + 3*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 + 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*sq 
rt(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 + 2*sqrt(d*g - e*f)*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))*a**2*c*d**2*e**4*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**2*c*d**2*e**4*g**5*x**2 - 10*...