\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x) (f+g x)} \, dx\) [263]

Optimal result

Integrand size = 44, antiderivative size = 165 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)} \, dx=\frac {2 \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {e} g}-\frac {2 \sqrt {c d f-a e g} \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{g \sqrt {e f-d g}} \] Output:

2*c^(1/2)*d^(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))/e^(1/2)/g-2*(-a*e*g+c*d*f)^(1/2)*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/(-d*g+e*f)^(1/2)
 

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)} \, dx=\frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {e} \sqrt {c d f-a e g} \arctan \left (\frac {\sqrt {c} \sqrt {d} \left (-\sqrt {\frac {e}{c d}} g \sqrt {a e+c d x} \sqrt {d+e x}+e (f+g x)\right )}{\sqrt {e} \sqrt {-e f+d g} \sqrt {c d f-a e g}}\right )-\sqrt {c} \sqrt {d} \sqrt {-e f+d g} \log \left (-\sqrt {\frac {e}{c d}} \sqrt {a e+c d x}+\sqrt {d+e x}\right )\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {e}{c d}} g \sqrt {-e f+d g} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[e]*Sqrt[c*d*f - a*e*g]*ArcTan[(Sq 
rt[c]*Sqrt[d]*(-(Sqrt[e/(c*d)]*g*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]) + e*(f + 
 g*x)))/(Sqrt[e]*Sqrt[-(e*f) + d*g]*Sqrt[c*d*f - a*e*g])] - Sqrt[c]*Sqrt[d 
]*Sqrt[-(e*f) + d*g]*Log[-(Sqrt[e/(c*d)]*Sqrt[a*e + c*d*x]) + Sqrt[d + e*x 
]]))/(Sqrt[c]*Sqrt[d]*Sqrt[e/(c*d)]*g*Sqrt[-(e*f) + d*g]*Sqrt[(a*e + c*d*x 
)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1215, 1268, 140, 27, 66, 104, 221}

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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*((2*Sqrt[c]*Sqrt[d]*ArcTanh[(Sqrt[e]*Sqrt 
[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[e]*g) + (2*(a*e - ( 
c*d*f)/g)*ArcTanh[(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])/(Sqrt[e*f - d*g]*Sqr 
t[a*e + c*d*x])])/(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]
 

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 140

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*d^(m + n)*f^p   Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] 
, x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x 
)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 
0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n, -1]))
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 1268

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(659\) vs. \(2(141)=282\).

Time = 2.35 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.00

method	result	size
default	\(\frac {\sqrt {c d \left (x +\frac {f}{g}\right )^{2} e +\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}+\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \ln \left (\frac {\frac {a \,e^{2} g +c \,d^{2} g -2 c d e f}{2 g}+d e c \left (x +\frac {f}{g}\right )}{\sqrt {d e c}}+\sqrt {c d \left (x +\frac {f}{g}\right )^{2} e +\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}\right )}{2 g \sqrt {d e c}}-\frac {\left (a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}\right ) \ln \left (\frac {\frac {2 a d e \,g^{2}-2 a \,e^{2} f g -2 c \,d^{2} f g +2 c d e \,f^{2}}{g^{2}}+\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}\, \sqrt {c d \left (x +\frac {f}{g}\right )^{2} e +\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}}{x +\frac {f}{g}}\right )}{g^{2} \sqrt {\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}}}{d g -e f}-\frac {\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}}{d g -e f}\)	\(660\)

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),x,method=_RETU 
RNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (141) = 282\).

Time = 8.91 (sec) , antiderivative size = 1689, normalized size of antiderivative = 10.24 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)} \, dx=\text {Too large to display} \] 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),x, algor 
ithm="fricas")
 

Output:

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

Sympy [F]

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)} \, dx=\text {Exception raised: ValueError} \] 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),x, algor 
ithm="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(d*g-e*f>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)} \, dx=-\frac {2 \, {\left (c d f - a e g\right )} \arctan \left (-\frac {{\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} g + \sqrt {c d e} f}{\sqrt {-c d e f^{2} + c d^{2} f g + a e^{2} f g - a d e g^{2}}}\right )}{\sqrt {-c d e f^{2} + c d^{2} f g + a e^{2} f g - a d e g^{2}} g} - \frac {\sqrt {c d e} \log \left ({\left | -2 \, {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} c d e - \sqrt {c d e} c d^{2} - \sqrt {c d e} a e^{2} \right |}\right )}{e g} \] 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),x, algor 
ithm="giac")
 

Output:

-2*(c*d*f - a*e*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))/(sqrt(-c*d*e*f^2 + c*d^2*f*g + a*e^2*f*g - a*d*e*g^2)*g) - 
sqrt(c*d*e)*log(abs(-2*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x 
 + a*d*e))*c*d*e - sqrt(c*d*e)*c*d^2 - sqrt(c*d*e)*a*e^2))/(e*g)
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.48 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x) (f+g x)} \, dx=\frac {\sqrt {d g -e f}\, \sqrt {a e g -c d f}\, \mathrm {log}\left (\sqrt {g}\, \sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {d g -e f}\, \sqrt {a e g -c d f}+a \,e^{2} g +c \,d^{2} g -2 c d e f}+\sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) e +\sqrt {d g -e f}\, \sqrt {a e g -c d f}\, \mathrm {log}\left (\sqrt {g}\, \sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {d g -e f}\, \sqrt {a e g -c d f}+a \,e^{2} g +c \,d^{2} g -2 c d e f}+\sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) e -\sqrt {d g -e f}\, \sqrt {a e g -c d f}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {d g -e f}\, \sqrt {a e g -c d f}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}\, g +2 c d e f +2 c d e g x \right ) e +2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) d g -2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) e f}{e g \left (d g -e f \right )} \] 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),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))*e 
+ 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))*e 
 - sqrt(d*g - e*f)*sqrt(a*e*g - c*d*f)*log(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt( 
d*g - e*f)*sqrt(a*e*g - c*d*f) + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*g + 2*c*d*e*f + 2*c*d*e*g*x)*e + 2*sqrt(e)*sqrt(d)*sqrt(c 
)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e 
**2 - c*d**2))*d*g - 2*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d 
*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*e*f)/(e*g*(d*g 
 - e*f))