\(\int \frac {a+b (e+f x)^2}{\sqrt {e g+f g x} (c+d (e+f x)^2)} \, dx\) [581]

Optimal result

Integrand size = 37, antiderivative size = 255 \[ \int \frac {a+b (e+f x)^2}{\sqrt {e g+f g x} \left (c+d (e+f x)^2\right )} \, dx=\frac {2 b \sqrt {e g+f g x}}{d f g}+\frac {(b c-a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e g+f g x}}{\sqrt [4]{c} \sqrt {g}}\right )}{\sqrt {2} c^{3/4} d^{5/4} f \sqrt {g}}-\frac {(b c-a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e g+f g x}}{\sqrt [4]{c} \sqrt {g}}\right )}{\sqrt {2} c^{3/4} d^{5/4} f \sqrt {g}}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {e g+f g x}}{\sqrt {c} \sqrt {g}+\sqrt {d} \sqrt {g} (e+f x)}\right )}{\sqrt {2} c^{3/4} d^{5/4} f \sqrt {g}} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.72 \[ \int \frac {a+b (e+f x)^2}{\sqrt {e g+f g x} \left (c+d (e+f x)^2\right )} \, dx=\frac {4 b c^{3/4} \sqrt [4]{d} (e+f x)+\sqrt {2} (b c-a d) \sqrt {e+f x} \arctan \left (\frac {\sqrt {c}-\sqrt {d} (e+f x)}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {e+f x}}\right )-\sqrt {2} (b c-a d) \sqrt {e+f x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt {c}+\sqrt {d} (e+f x)}\right )}{2 c^{3/4} d^{5/4} f \sqrt {g (e+f x)}} \] Input:

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

Output:

(4*b*c^(3/4)*d^(1/4)*(e + f*x) + Sqrt[2]*(b*c - a*d)*Sqrt[e + f*x]*ArcTan[ 
(Sqrt[c] - Sqrt[d]*(e + f*x))/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[e + f*x])] - S 
qrt[2]*(b*c - a*d)*Sqrt[e + f*x]*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[e + 
 f*x])/(Sqrt[c] + Sqrt[d]*(e + f*x))])/(2*c^(3/4)*d^(5/4)*f*Sqrt[g*(e + f* 
x)])
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {1014, 363, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

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

\(\Big \downarrow \) 1014

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\sqrt {e+f x} \left (\frac {2 b \sqrt {e+f x}}{d}-\frac {2 (b c-a d) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{d} \left (e+f x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {e+f x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {e+f x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (e+f x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {e+f x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {e+f x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{f \sqrt {e g+f g x}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {e+f x} \left (\frac {2 b \sqrt {e+f x}}{d}-\frac {2 (b c-a d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{d} \left (e+f x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {e+f x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {e+f x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (e+f x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {e+f x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {e+f x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{f \sqrt {e g+f g x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {e+f x} \left (\frac {2 b \sqrt {e+f x}}{d}-\frac {2 (b c-a d) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {e+f x}}{e+f x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {e+f x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {e+f x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}+\sqrt [4]{c}}{e+f x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {e+f x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {e+f x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{f \sqrt {e g+f g x}}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\sqrt {e+f x} \left (\frac {2 b \sqrt {e+f x}}{d}-\frac {2 (b c-a d) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {e+f x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {e+f x}+\sqrt {c}+\sqrt {d} (e+f x)\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {e+f x}+\sqrt {c}+\sqrt {d} (e+f x)\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )}{d}\right )}{f \sqrt {e g+f g x}}\)

Input:

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

Output:

(Sqrt[e + f*x]*((2*b*Sqrt[e + f*x])/d - (2*(b*c - a*d)*((-(ArcTan[1 - (Sqr 
t[2]*d^(1/4)*Sqrt[e + f*x])/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4))) + ArcTan[1 
 + (Sqrt[2]*d^(1/4)*Sqrt[e + f*x])/c^(1/4)]/(Sqrt[2]*c^(1/4)*d^(1/4)))/(2* 
Sqrt[c]) + (-1/2*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[e + f*x] + Sqr 
t[d]*(e + f*x)]/(Sqrt[2]*c^(1/4)*d^(1/4)) + Log[Sqrt[c] + Sqrt[2]*c^(1/4)* 
d^(1/4)*Sqrt[e + f*x] + Sqrt[d]*(e + f*x)]/(2*Sqrt[2]*c^(1/4)*d^(1/4)))/(2 
*Sqrt[c])))/d))/(f*Sqrt[e*g + f*g*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1014

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^(n_))^(p_.)*((c_.) + (d_.)*(v_)^(n_))^(q 
_.), x_Symbol] :> Simp[u^m/(Coefficient[v, x, 1]*v^m)   Subst[Int[x^m*(a + 
b*x^n)^p*(c + d*x^n)^q, x], x, v], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] 
 && LinearPairQ[u, v, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(\frac {\frac {2 b \sqrt {f g x +e g}}{d}+\frac {\left (a d -c b \right ) \left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {f g x +e g +\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {f g x +e g}\, \sqrt {2}+\sqrt {\frac {c \,g^{2}}{d}}}{f g x +e g -\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {f g x +e g}\, \sqrt {2}+\sqrt {\frac {c \,g^{2}}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f g x +e g}}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {f g x +e g}}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d c}}{f g}\)	\(203\)
default	\(\frac {\frac {2 b \sqrt {f g x +e g}}{d}+\frac {\left (a d -c b \right ) \left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {f g x +e g +\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {f g x +e g}\, \sqrt {2}+\sqrt {\frac {c \,g^{2}}{d}}}{f g x +e g -\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {f g x +e g}\, \sqrt {2}+\sqrt {\frac {c \,g^{2}}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f g x +e g}}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {f g x +e g}}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d c}}{f g}\)	\(203\)
risch	\(\frac {2 b \left (f x +e \right )}{d f \sqrt {g \left (f x +e \right )}}+\frac {\left (a d -c b \right ) \left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {f g x +e g +\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {f g x +e g}\, \sqrt {2}+\sqrt {\frac {c \,g^{2}}{d}}}{f g x +e g -\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {f g x +e g}\, \sqrt {2}+\sqrt {\frac {c \,g^{2}}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {f g x +e g}}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {f g x +e g}}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d f g c}\)	\(209\)
pseudoelliptic	\(\frac {\left (\ln \left (\frac {-\sqrt {\frac {c \,g^{2}}{d}}-\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {g \left (f x +e \right )}\, \sqrt {2}+\left (-f x -e \right ) g}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}} \sqrt {g \left (f x +e \right )}\, \sqrt {2}-\sqrt {\frac {c \,g^{2}}{d}}+\left (-f x -e \right ) g}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {g \left (f x +e \right )}}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}-1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {g \left (f x +e \right )}+\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}{\left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}}\right )\right ) \left (a d -c b \right ) \sqrt {2}\, \left (\frac {c \,g^{2}}{d}\right )^{\frac {1}{4}}+8 b \sqrt {g \left (f x +e \right )}\, c}{4 d f g c}\)	\(215\)

Input:

int((a+b*(f*x+e)^2)/(f*g*x+e*g)^(1/2)/(c+d*(f*x+e)^2),x,method=_RETURNVERB 
OSE)
 

Output:

2/g/f*(b/d*(f*g*x+e*g)^(1/2)+1/8*(a*d-b*c)/d*(c*g^2/d)^(1/4)/c*2^(1/2)*(ln 
((f*g*x+e*g+(c*g^2/d)^(1/4)*(f*g*x+e*g)^(1/2)*2^(1/2)+(c*g^2/d)^(1/2))/(f* 
g*x+e*g-(c*g^2/d)^(1/4)*(f*g*x+e*g)^(1/2)*2^(1/2)+(c*g^2/d)^(1/2)))+2*arct 
an(2^(1/2)/(c*g^2/d)^(1/4)*(f*g*x+e*g)^(1/2)+1)-2*arctan(-2^(1/2)/(c*g^2/d 
)^(1/4)*(f*g*x+e*g)^(1/2)+1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 675, normalized size of antiderivative = 2.65 \[ \int \frac {a+b (e+f x)^2}{\sqrt {e g+f g x} \left (c+d (e+f x)^2\right )} \, dx=\frac {d f g \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5} f^{4} g^{2}}\right )^{\frac {1}{4}} \log \left (c d f g \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5} f^{4} g^{2}}\right )^{\frac {1}{4}} - \sqrt {f g x + e g} {\left (b c - a d\right )}\right ) + i \, d f g \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5} f^{4} g^{2}}\right )^{\frac {1}{4}} \log \left (i \, c d f g \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5} f^{4} g^{2}}\right )^{\frac {1}{4}} - \sqrt {f g x + e g} {\left (b c - a d\right )}\right ) - i \, d f g \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5} f^{4} g^{2}}\right )^{\frac {1}{4}} \log \left (-i \, c d f g \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5} f^{4} g^{2}}\right )^{\frac {1}{4}} - \sqrt {f g x + e g} {\left (b c - a d\right )}\right ) - d f g \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5} f^{4} g^{2}}\right )^{\frac {1}{4}} \log \left (-c d f g \left (-\frac {b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}}{c^{3} d^{5} f^{4} g^{2}}\right )^{\frac {1}{4}} - \sqrt {f g x + e g} {\left (b c - a d\right )}\right ) + 4 \, \sqrt {f g x + e g} b}{2 \, d f g} \] Input:

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

Output:

1/2*(d*f*g*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 
+ a^4*d^4)/(c^3*d^5*f^4*g^2))^(1/4)*log(c*d*f*g*(-(b^4*c^4 - 4*a*b^3*c^3*d 
 + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(c^3*d^5*f^4*g^2))^(1/4) - 
 sqrt(f*g*x + e*g)*(b*c - a*d)) + I*d*f*g*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a 
^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(c^3*d^5*f^4*g^2))^(1/4)*log(I*c 
*d*f*g*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^ 
4*d^4)/(c^3*d^5*f^4*g^2))^(1/4) - sqrt(f*g*x + e*g)*(b*c - a*d)) - I*d*f*g 
*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) 
/(c^3*d^5*f^4*g^2))^(1/4)*log(-I*c*d*f*g*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^ 
2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(c^3*d^5*f^4*g^2))^(1/4) - sqrt(f 
*g*x + e*g)*(b*c - a*d)) - d*f*g*(-(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^ 
2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(c^3*d^5*f^4*g^2))^(1/4)*log(-c*d*f*g*(-( 
b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)/(c^ 
3*d^5*f^4*g^2))^(1/4) - sqrt(f*g*x + e*g)*(b*c - a*d)) + 4*sqrt(f*g*x + e* 
g)*b)/(d*f*g)
 

Sympy [F]

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

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

Output:

Integral((a + b*e**2 + 2*b*e*f*x + b*f**2*x**2)/(sqrt(g*(e + f*x))*(c + d* 
e**2 + 2*d*e*f*x + d*f**2*x**2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.14 \[ \int \frac {a+b (e+f x)^2}{\sqrt {e g+f g x} \left (c+d (e+f x)^2\right )} \, dx=-\frac {\frac {{\left (\frac {\sqrt {2} g^{2} \log \left (\sqrt {2} \left (c g^{2}\right )^{\frac {1}{4}} \sqrt {f g x + e g} d^{\frac {1}{4}} + {\left (f g x + e g\right )} \sqrt {d} + \sqrt {c} g\right )}{\left (c g^{2}\right )^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} g^{2} \log \left (-\sqrt {2} \left (c g^{2}\right )^{\frac {1}{4}} \sqrt {f g x + e g} d^{\frac {1}{4}} + {\left (f g x + e g\right )} \sqrt {d} + \sqrt {c} g\right )}{\left (c g^{2}\right )^{\frac {3}{4}} d^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} g \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (c g^{2}\right )^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {f g x + e g} \sqrt {d}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d} g}}\right )}{\sqrt {\sqrt {c} \sqrt {d} g} \sqrt {c}} + \frac {2 \, \sqrt {2} g \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (c g^{2}\right )^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {f g x + e g} \sqrt {d}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d} g}}\right )}{\sqrt {\sqrt {c} \sqrt {d} g} \sqrt {c}}\right )} {\left (b c - a d\right )}}{d} - \frac {8 \, \sqrt {f g x + e g} b}{d}}{4 \, f g} \] Input:

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

Output:

-1/4*((sqrt(2)*g^2*log(sqrt(2)*(c*g^2)^(1/4)*sqrt(f*g*x + e*g)*d^(1/4) + ( 
f*g*x + e*g)*sqrt(d) + sqrt(c)*g)/((c*g^2)^(3/4)*d^(1/4)) - sqrt(2)*g^2*lo 
g(-sqrt(2)*(c*g^2)^(1/4)*sqrt(f*g*x + e*g)*d^(1/4) + (f*g*x + e*g)*sqrt(d) 
 + sqrt(c)*g)/((c*g^2)^(3/4)*d^(1/4)) + 2*sqrt(2)*g*arctan(1/2*sqrt(2)*(sq 
rt(2)*(c*g^2)^(1/4)*d^(1/4) + 2*sqrt(f*g*x + e*g)*sqrt(d))/sqrt(sqrt(c)*sq 
rt(d)*g))/(sqrt(sqrt(c)*sqrt(d)*g)*sqrt(c)) + 2*sqrt(2)*g*arctan(-1/2*sqrt 
(2)*(sqrt(2)*(c*g^2)^(1/4)*d^(1/4) - 2*sqrt(f*g*x + e*g)*sqrt(d))/sqrt(sqr 
t(c)*sqrt(d)*g))/(sqrt(sqrt(c)*sqrt(d)*g)*sqrt(c)))*(b*c - a*d)/d - 8*sqrt 
(f*g*x + e*g)*b/d)/(f*g)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.47 \[ \int \frac {a+b (e+f x)^2}{\sqrt {e g+f g x} \left (c+d (e+f x)^2\right )} \, dx=\frac {2 \, \sqrt {f g x + e g} b}{d f g} - \frac {\sqrt {2} {\left (\left (c d^{3} g^{2}\right )^{\frac {1}{4}} b c - \left (c d^{3} g^{2}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c g^{2}}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {f g x + e g}\right )}}{2 \, \left (\frac {c g^{2}}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c d^{2} f g} - \frac {\sqrt {2} {\left (\left (c d^{3} g^{2}\right )^{\frac {1}{4}} b c - \left (c d^{3} g^{2}\right )^{\frac {1}{4}} a d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c g^{2}}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {f g x + e g}\right )}}{2 \, \left (\frac {c g^{2}}{d}\right )^{\frac {1}{4}}}\right )}{2 \, c d^{2} f g} - \frac {\sqrt {2} {\left (\left (c d^{3} g^{2}\right )^{\frac {1}{4}} b c - \left (c d^{3} g^{2}\right )^{\frac {1}{4}} a d\right )} \log \left (f g x + e g + \sqrt {2} \sqrt {f g x + e g} \left (\frac {c g^{2}}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c g^{2}}{d}}\right )}{4 \, c d^{2} f g} + \frac {\sqrt {2} {\left (\left (c d^{3} g^{2}\right )^{\frac {1}{4}} b c - \left (c d^{3} g^{2}\right )^{\frac {1}{4}} a d\right )} \log \left (f g x + e g - \sqrt {2} \sqrt {f g x + e g} \left (\frac {c g^{2}}{d}\right )^{\frac {1}{4}} + \sqrt {\frac {c g^{2}}{d}}\right )}{4 \, c d^{2} f g} \] Input:

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

Output:

2*sqrt(f*g*x + e*g)*b/(d*f*g) - 1/2*sqrt(2)*((c*d^3*g^2)^(1/4)*b*c - (c*d^ 
3*g^2)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(c*g^2/d)^(1/4) + 2*sqrt(f*g 
*x + e*g))/(c*g^2/d)^(1/4))/(c*d^2*f*g) - 1/2*sqrt(2)*((c*d^3*g^2)^(1/4)*b 
*c - (c*d^3*g^2)^(1/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c*g^2/d)^(1/4) - 
 2*sqrt(f*g*x + e*g))/(c*g^2/d)^(1/4))/(c*d^2*f*g) - 1/4*sqrt(2)*((c*d^3*g 
^2)^(1/4)*b*c - (c*d^3*g^2)^(1/4)*a*d)*log(f*g*x + e*g + sqrt(2)*sqrt(f*g* 
x + e*g)*(c*g^2/d)^(1/4) + sqrt(c*g^2/d))/(c*d^2*f*g) + 1/4*sqrt(2)*((c*d^ 
3*g^2)^(1/4)*b*c - (c*d^3*g^2)^(1/4)*a*d)*log(f*g*x + e*g - sqrt(2)*sqrt(f 
*g*x + e*g)*(c*g^2/d)^(1/4) + sqrt(c*g^2/d))/(c*d^2*f*g)
 

Mupad [B] (verification not implemented)

Time = 3.86 (sec) , antiderivative size = 1044, normalized size of antiderivative = 4.09 \[ \int \frac {a+b (e+f x)^2}{\sqrt {e g+f g x} \left (c+d (e+f x)^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*b*(e*g + f*g*x)^(1/2))/(d*f*g) - (atan((((a*d - b*c)*((16*(e*g + f*g*x) 
^(1/2)*(a^2*d^3*g^2 + b^2*c^2*d*g^2 - 2*a*b*c*d^2*g^2))/f^2 - (16*(b*c^2*d 
^2*g^3 - a*c*d^3*g^3)*(a*d - b*c))/((-c)^(3/4)*d^(5/4)*f^2*g^(1/2)))*1i)/( 
2*(-c)^(3/4)*d^(5/4)*f*g^(1/2)) + ((a*d - b*c)*((16*(e*g + f*g*x)^(1/2)*(a 
^2*d^3*g^2 + b^2*c^2*d*g^2 - 2*a*b*c*d^2*g^2))/f^2 + (16*(b*c^2*d^2*g^3 - 
a*c*d^3*g^3)*(a*d - b*c))/((-c)^(3/4)*d^(5/4)*f^2*g^(1/2)))*1i)/(2*(-c)^(3 
/4)*d^(5/4)*f*g^(1/2)))/(((a*d - b*c)*((16*(e*g + f*g*x)^(1/2)*(a^2*d^3*g^ 
2 + b^2*c^2*d*g^2 - 2*a*b*c*d^2*g^2))/f^2 - (16*(b*c^2*d^2*g^3 - a*c*d^3*g 
^3)*(a*d - b*c))/((-c)^(3/4)*d^(5/4)*f^2*g^(1/2))))/(2*(-c)^(3/4)*d^(5/4)* 
f*g^(1/2)) - ((a*d - b*c)*((16*(e*g + f*g*x)^(1/2)*(a^2*d^3*g^2 + b^2*c^2* 
d*g^2 - 2*a*b*c*d^2*g^2))/f^2 + (16*(b*c^2*d^2*g^3 - a*c*d^3*g^3)*(a*d - b 
*c))/((-c)^(3/4)*d^(5/4)*f^2*g^(1/2))))/(2*(-c)^(3/4)*d^(5/4)*f*g^(1/2)))) 
*(a*d - b*c)*1i)/((-c)^(3/4)*d^(5/4)*f*g^(1/2)) - (atan((((a*d - b*c)*((16 
*(e*g + f*g*x)^(1/2)*(a^2*d^3*g^2 + b^2*c^2*d*g^2 - 2*a*b*c*d^2*g^2))/f^2 
- ((b*c^2*d^2*g^3 - a*c*d^3*g^3)*(a*d - b*c)*16i)/((-c)^(3/4)*d^(5/4)*f^2* 
g^(1/2))))/(2*(-c)^(3/4)*d^(5/4)*f*g^(1/2)) + ((a*d - b*c)*((16*(e*g + f*g 
*x)^(1/2)*(a^2*d^3*g^2 + b^2*c^2*d*g^2 - 2*a*b*c*d^2*g^2))/f^2 + ((b*c^2*d 
^2*g^3 - a*c*d^3*g^3)*(a*d - b*c)*16i)/((-c)^(3/4)*d^(5/4)*f^2*g^(1/2))))/ 
(2*(-c)^(3/4)*d^(5/4)*f*g^(1/2)))/(((a*d - b*c)*((16*(e*g + f*g*x)^(1/2)*( 
a^2*d^3*g^2 + b^2*c^2*d*g^2 - 2*a*b*c*d^2*g^2))/f^2 - ((b*c^2*d^2*g^3 -...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.40 \[ \int \frac {a+b (e+f x)^2}{\sqrt {e g+f g x} \left (c+d (e+f x)^2\right )} \, dx=\frac {\sqrt {g}\, \left (2 d^{\frac {7}{4}} c^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {d}\, \sqrt {f x +e}-d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a -2 d^{\frac {3}{4}} c^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {d}\, \sqrt {f x +e}-d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) b +2 d^{\frac {7}{4}} c^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {d}\, \sqrt {f x +e}+d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) a -2 d^{\frac {3}{4}} c^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {2 \sqrt {d}\, \sqrt {f x +e}+d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}{d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {2}}\right ) b +8 \sqrt {f x +e}\, b c d -d^{\frac {7}{4}} c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {f x +e}\, \sqrt {2}+\sqrt {c}+\sqrt {d}\, e +\sqrt {d}\, f x \right ) a +d^{\frac {3}{4}} c^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {f x +e}\, \sqrt {2}+\sqrt {c}+\sqrt {d}\, e +\sqrt {d}\, f x \right ) b +d^{\frac {7}{4}} c^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {f x +e}\, \sqrt {2}+\sqrt {c}+\sqrt {d}\, e +\sqrt {d}\, f x \right ) a -d^{\frac {3}{4}} c^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (d^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {f x +e}\, \sqrt {2}+\sqrt {c}+\sqrt {d}\, e +\sqrt {d}\, f x \right ) b \right )}{4 c \,d^{2} f g} \] Input:

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

Output:

(sqrt(g)*(2*d**(3/4)*c**(1/4)*sqrt(2)*atan((2*sqrt(d)*sqrt(e + f*x) - d**( 
1/4)*c**(1/4)*sqrt(2))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*d - 2*d**(3/4)*c**(1 
/4)*sqrt(2)*atan((2*sqrt(d)*sqrt(e + f*x) - d**(1/4)*c**(1/4)*sqrt(2))/(d* 
*(1/4)*c**(1/4)*sqrt(2)))*b*c + 2*d**(3/4)*c**(1/4)*sqrt(2)*atan((2*sqrt(d 
)*sqrt(e + f*x) + d**(1/4)*c**(1/4)*sqrt(2))/(d**(1/4)*c**(1/4)*sqrt(2)))* 
a*d - 2*d**(3/4)*c**(1/4)*sqrt(2)*atan((2*sqrt(d)*sqrt(e + f*x) + d**(1/4) 
*c**(1/4)*sqrt(2))/(d**(1/4)*c**(1/4)*sqrt(2)))*b*c + 8*sqrt(e + f*x)*b*c* 
d - d**(3/4)*c**(1/4)*sqrt(2)*log( - d**(1/4)*c**(1/4)*sqrt(e + f*x)*sqrt( 
2) + sqrt(c) + sqrt(d)*e + sqrt(d)*f*x)*a*d + d**(3/4)*c**(1/4)*sqrt(2)*lo 
g( - d**(1/4)*c**(1/4)*sqrt(e + f*x)*sqrt(2) + sqrt(c) + sqrt(d)*e + sqrt( 
d)*f*x)*b*c + d**(3/4)*c**(1/4)*sqrt(2)*log(d**(1/4)*c**(1/4)*sqrt(e + f*x 
)*sqrt(2) + sqrt(c) + sqrt(d)*e + sqrt(d)*f*x)*a*d - d**(3/4)*c**(1/4)*sqr 
t(2)*log(d**(1/4)*c**(1/4)*sqrt(e + f*x)*sqrt(2) + sqrt(c) + sqrt(d)*e + s 
qrt(d)*f*x)*b*c))/(4*c*d**2*f*g)