\(\int \frac {\sqrt {e x} (c+d x)}{(a+b x^2)^2} \, dx\) [405]

Optimal result

Integrand size = 22, antiderivative size = 261 \[ \int \frac {\sqrt {e x} (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {e x} (a d-b c x)}{2 a b \left (a+b x^2\right )}-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{5/4} b^{5/4}} \] Output:

-1/2*(e*x)^(1/2)*(-b*c*x+a*d)/a/b/(b*x^2+a)-1/8*(b^(1/2)*c+a^(1/2)*d)*e^(1 
/2)*arctan(1-2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(5/4)/ 
b^(5/4)+1/8*(b^(1/2)*c+a^(1/2)*d)*e^(1/2)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^( 
1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(5/4)/b^(5/4)-1/8*(b^(1/2)*c-a^(1/2)*d)*e^ 
(1/2)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(e*x)^(1/2)/e^(1/2)/(a^(1/2)+b^(1/2) 
*x))*2^(1/2)/a^(5/4)/b^(5/4)
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {e x} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {e x} \left (\frac {4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} (-a d+b c x)}{a+b x^2}-\sqrt {2} \left (\sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (-\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{8 a^{5/4} b^{5/4} \sqrt {x}} \] Input:

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

Output:

(Sqrt[e*x]*((4*a^(1/4)*b^(1/4)*Sqrt[x]*(-(a*d) + b*c*x))/(a + b*x^2) - Sqr 
t[2]*(Sqrt[b]*c + Sqrt[a]*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4) 
*b^(1/4)*Sqrt[x])] + Sqrt[2]*(-(Sqrt[b]*c) + Sqrt[a]*d)*ArcTanh[(Sqrt[2]*a 
^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(8*a^(5/4)*b^(5/4)*Sqrt[x 
])
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.31, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {550, 27, 554, 1482, 27, 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.

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

\(\Big \downarrow \) 550

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 554

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

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 550

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

rule 554

Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] 
 :> Simp[2   Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] 
 /; FreeQ[{a, b, c, d, e}, 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]
 

rule 1482

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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(2 e^{2} \left (\frac {\frac {c \left (e x \right )^{\frac {3}{2}}}{4 a e}-\frac {d \sqrt {e x}}{4 b}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{4 a e b}\right )\)	\(337\)
default	\(2 e^{2} \left (\frac {\frac {c \left (e x \right )^{\frac {3}{2}}}{4 a e}-\frac {d \sqrt {e x}}{4 b}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{4 a e b}\right )\)	\(337\)
pseudoelliptic	\(\frac {\frac {\sqrt {2}\, c e \left (b \,x^{2}+a \right ) \ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}+\frac {\sqrt {2}\, d \sqrt {\frac {a \,e^{2}}{b}}\, \left (b \,x^{2}+a \right ) \ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )}{2}+\sqrt {2}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d +c e \right ) \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+\sqrt {2}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d +c e \right ) \left (b \,x^{2}+a \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )-4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \left (-c b x +a d \right )}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} a b \left (b \,x^{2}+a \right )}\)	\(354\)

Input:

int((e*x)^(1/2)*(d*x+c)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

2*e^2*((1/4*c/a/e*(e*x)^(3/2)-1/4*d*(e*x)^(1/2)/b)/(b*e^2*x^2+a*e^2)+1/4/a 
/e/b*(1/8*d/e*(a*e^2/b)^(1/4)*2^(1/2)*(ln((e*x+(a*e^2/b)^(1/4)*(e*x)^(1/2) 
*2^(1/2)+(a*e^2/b)^(1/2))/(e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/ 
b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2 
)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))+1/8*c/(a*e^2/b)^(1/4)*2^(1/2)*(ln((e*x-( 
a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e*x+(a*e^2/b)^(1/4)*( 
e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x 
)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 973 vs. \(2 (181) = 362\).

Time = 0.16 (sec) , antiderivative size = 973, normalized size of antiderivative = 3.73 \[ \int \frac {\sqrt {e x} (c+d x)}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*((a*b^2*x^2 + a^2*b)*sqrt(-(a^2*b^2*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a 
^2*d^4)*e^2/(a^5*b^5)) + 2*c*d*e)/(a^2*b^2))*log(-(b^2*c^4 - a^2*d^4)*sqrt 
(e*x)*e + (a^4*b^4*c*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^ 
5)) - (a^2*b^2*c^2*d - a^3*b*d^3)*e)*sqrt(-(a^2*b^2*sqrt(-(b^2*c^4 - 2*a*b 
*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^5)) + 2*c*d*e)/(a^2*b^2))) - (a*b^2*x^2 + a 
^2*b)*sqrt(-(a^2*b^2*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^ 
5)) + 2*c*d*e)/(a^2*b^2))*log(-(b^2*c^4 - a^2*d^4)*sqrt(e*x)*e - (a^4*b^4* 
c*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^5)) - (a^2*b^2*c^2* 
d - a^3*b*d^3)*e)*sqrt(-(a^2*b^2*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4) 
*e^2/(a^5*b^5)) + 2*c*d*e)/(a^2*b^2))) - (a*b^2*x^2 + a^2*b)*sqrt((a^2*b^2 
*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^5)) - 2*c*d*e)/(a^2* 
b^2))*log(-(b^2*c^4 - a^2*d^4)*sqrt(e*x)*e + (a^4*b^4*c*sqrt(-(b^2*c^4 - 2 
*a*b*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^5)) + (a^2*b^2*c^2*d - a^3*b*d^3)*e)*sq 
rt((a^2*b^2*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^5)) - 2*c 
*d*e)/(a^2*b^2))) + (a*b^2*x^2 + a^2*b)*sqrt((a^2*b^2*sqrt(-(b^2*c^4 - 2*a 
*b*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^5)) - 2*c*d*e)/(a^2*b^2))*log(-(b^2*c^4 - 
 a^2*d^4)*sqrt(e*x)*e - (a^4*b^4*c*sqrt(-(b^2*c^4 - 2*a*b*c^2*d^2 + a^2*d^ 
4)*e^2/(a^5*b^5)) + (a^2*b^2*c^2*d - a^3*b*d^3)*e)*sqrt((a^2*b^2*sqrt(-(b^ 
2*c^4 - 2*a*b*c^2*d^2 + a^2*d^4)*e^2/(a^5*b^5)) - 2*c*d*e)/(a^2*b^2))) + 4 
*(b*c*x - a*d)*sqrt(e*x))/(a*b^2*x^2 + a^2*b)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 29.41 (sec) , antiderivative size = 1896, normalized size of antiderivative = 7.26 \[ \int \frac {\sqrt {e x} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

c*(-3*a**(7/4)*sqrt(e)*x**(3/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4) 
/a**(1/4))*gamma(3/4)/(32*a**3*b**(3/4)*x**(3/2)*exp(3*I*pi/4)*gamma(7/4) 
+ 32*a**2*b**(7/4)*x**(7/2)*exp(3*I*pi/4)*gamma(7/4)) - 3*I*a**(7/4)*sqrt( 
e)*x**(3/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(3 
/4)/(32*a**3*b**(3/4)*x**(3/2)*exp(3*I*pi/4)*gamma(7/4) + 32*a**2*b**(7/4) 
*x**(7/2)*exp(3*I*pi/4)*gamma(7/4)) + 3*a**(7/4)*sqrt(e)*x**(3/2)*log(1 - 
b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(3/4)/(32*a**3*b**(3/4 
)*x**(3/2)*exp(3*I*pi/4)*gamma(7/4) + 32*a**2*b**(7/4)*x**(7/2)*exp(3*I*pi 
/4)*gamma(7/4)) + 3*I*a**(7/4)*sqrt(e)*x**(3/2)*log(1 - b**(1/4)*sqrt(x)*e 
xp_polar(7*I*pi/4)/a**(1/4))*gamma(3/4)/(32*a**3*b**(3/4)*x**(3/2)*exp(3*I 
*pi/4)*gamma(7/4) + 32*a**2*b**(7/4)*x**(7/2)*exp(3*I*pi/4)*gamma(7/4)) - 
3*a**(3/4)*b*sqrt(e)*x**(7/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi/4)/a 
**(1/4))*gamma(3/4)/(32*a**3*b**(3/4)*x**(3/2)*exp(3*I*pi/4)*gamma(7/4) + 
32*a**2*b**(7/4)*x**(7/2)*exp(3*I*pi/4)*gamma(7/4)) - 3*I*a**(3/4)*b*sqrt( 
e)*x**(7/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gamma(3 
/4)/(32*a**3*b**(3/4)*x**(3/2)*exp(3*I*pi/4)*gamma(7/4) + 32*a**2*b**(7/4) 
*x**(7/2)*exp(3*I*pi/4)*gamma(7/4)) + 3*a**(3/4)*b*sqrt(e)*x**(7/2)*log(1 
- b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(3/4)/(32*a**3*b**(3 
/4)*x**(3/2)*exp(3*I*pi/4)*gamma(7/4) + 32*a**2*b**(7/4)*x**(7/2)*exp(3*I* 
pi/4)*gamma(7/4)) + 3*I*a**(3/4)*b*sqrt(e)*x**(7/2)*log(1 - b**(1/4)*sq...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {e x} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x)^(1/2)*(d*x+c)/(b*x^2+a)^2,x, algorithm="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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (181) = 362\).

Time = 0.14 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {e x} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {e x} b c e^{2} x - \sqrt {e x} a d e^{2}}{2 \, {\left (b e^{2} x^{2} + a e^{2}\right )} a b} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{3} e} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{3} e} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{16 \, a^{2} b^{3} e} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{16 \, a^{2} b^{3} e} \] Input:

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

Output:

1/2*(sqrt(e*x)*b*c*e^2*x - sqrt(e*x)*a*d*e^2)/((b*e^2*x^2 + a*e^2)*a*b) + 
1/8*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e + (a*b^3*e^2)^(3/4)*c)*arctan(1/2*s 
qrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(a^2*b^3*e 
) + 1/8*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e + (a*b^3*e^2)^(3/4)*c)*arctan(- 
1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(a^2* 
b^3*e) + 1/16*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e - (a*b^3*e^2)^(3/4)*c)*lo 
g(e*x + sqrt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(a^2*b^3*e) - 1 
/16*sqrt(2)*((a*b^3*e^2)^(1/4)*a*b*d*e - (a*b^3*e^2)^(3/4)*c)*log(e*x - sq 
rt(2)*(a*e^2/b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(a^2*b^3*e)
 

Mupad [B] (verification not implemented)

Time = 7.20 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.89 \[ \int \frac {\sqrt {e x} (c+d x)}{\left (a+b x^2\right )^2} \, dx=2\,\mathrm {atanh}\left (\frac {2\,b^2\,c^2\,e^4\,\sqrt {e\,x}\,\sqrt {\frac {d^2\,e\,\sqrt {-a^5\,b^5}}{64\,a^4\,b^5}-\frac {c^2\,e\,\sqrt {-a^5\,b^5}}{64\,a^5\,b^4}-\frac {c\,d\,e}{32\,a^2\,b^2}}}{\frac {c\,d^2\,e^5}{4}-\frac {b\,c^3\,e^5}{4\,a}-\frac {d^3\,e^5\,\sqrt {-a^5\,b^5}}{4\,a^2\,b^3}+\frac {c^2\,d\,e^5\,\sqrt {-a^5\,b^5}}{4\,a^3\,b^2}}+\frac {2\,b\,d^2\,e^4\,\sqrt {e\,x}\,\sqrt {\frac {d^2\,e\,\sqrt {-a^5\,b^5}}{64\,a^4\,b^5}-\frac {c^2\,e\,\sqrt {-a^5\,b^5}}{64\,a^5\,b^4}-\frac {c\,d\,e}{32\,a^2\,b^2}}}{\frac {b\,c^3\,e^5}{4\,a^2}-\frac {c\,d^2\,e^5}{4\,a}+\frac {d^3\,e^5\,\sqrt {-a^5\,b^5}}{4\,a^3\,b^3}-\frac {c^2\,d\,e^5\,\sqrt {-a^5\,b^5}}{4\,a^4\,b^2}}\right )\,\sqrt {-\frac {b\,c^2\,e\,\sqrt {-a^5\,b^5}-a\,d^2\,e\,\sqrt {-a^5\,b^5}+2\,a^3\,b^3\,c\,d\,e}{64\,a^5\,b^5}}+2\,\mathrm {atanh}\left (\frac {2\,b^2\,c^2\,e^4\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,e\,\sqrt {-a^5\,b^5}}{64\,a^5\,b^4}-\frac {c\,d\,e}{32\,a^2\,b^2}-\frac {d^2\,e\,\sqrt {-a^5\,b^5}}{64\,a^4\,b^5}}}{\frac {c\,d^2\,e^5}{4}-\frac {b\,c^3\,e^5}{4\,a}+\frac {d^3\,e^5\,\sqrt {-a^5\,b^5}}{4\,a^2\,b^3}-\frac {c^2\,d\,e^5\,\sqrt {-a^5\,b^5}}{4\,a^3\,b^2}}+\frac {2\,b\,d^2\,e^4\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,e\,\sqrt {-a^5\,b^5}}{64\,a^5\,b^4}-\frac {c\,d\,e}{32\,a^2\,b^2}-\frac {d^2\,e\,\sqrt {-a^5\,b^5}}{64\,a^4\,b^5}}}{\frac {b\,c^3\,e^5}{4\,a^2}-\frac {c\,d^2\,e^5}{4\,a}-\frac {d^3\,e^5\,\sqrt {-a^5\,b^5}}{4\,a^3\,b^3}+\frac {c^2\,d\,e^5\,\sqrt {-a^5\,b^5}}{4\,a^4\,b^2}}\right )\,\sqrt {-\frac {a\,d^2\,e\,\sqrt {-a^5\,b^5}-b\,c^2\,e\,\sqrt {-a^5\,b^5}+2\,a^3\,b^3\,c\,d\,e}{64\,a^5\,b^5}}+\frac {\frac {c\,e\,{\left (e\,x\right )}^{3/2}}{2\,a}-\frac {d\,e^2\,\sqrt {e\,x}}{2\,b}}{b\,e^2\,x^2+a\,e^2} \] Input:

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

Output:

2*atanh((2*b^2*c^2*e^4*(e*x)^(1/2)*((d^2*e*(-a^5*b^5)^(1/2))/(64*a^4*b^5) 
- (c^2*e*(-a^5*b^5)^(1/2))/(64*a^5*b^4) - (c*d*e)/(32*a^2*b^2))^(1/2))/((c 
*d^2*e^5)/4 - (b*c^3*e^5)/(4*a) - (d^3*e^5*(-a^5*b^5)^(1/2))/(4*a^2*b^3) + 
 (c^2*d*e^5*(-a^5*b^5)^(1/2))/(4*a^3*b^2)) + (2*b*d^2*e^4*(e*x)^(1/2)*((d^ 
2*e*(-a^5*b^5)^(1/2))/(64*a^4*b^5) - (c^2*e*(-a^5*b^5)^(1/2))/(64*a^5*b^4) 
 - (c*d*e)/(32*a^2*b^2))^(1/2))/((b*c^3*e^5)/(4*a^2) - (c*d^2*e^5)/(4*a) + 
 (d^3*e^5*(-a^5*b^5)^(1/2))/(4*a^3*b^3) - (c^2*d*e^5*(-a^5*b^5)^(1/2))/(4* 
a^4*b^2)))*(-(b*c^2*e*(-a^5*b^5)^(1/2) - a*d^2*e*(-a^5*b^5)^(1/2) + 2*a^3* 
b^3*c*d*e)/(64*a^5*b^5))^(1/2) + 2*atanh((2*b^2*c^2*e^4*(e*x)^(1/2)*((c^2* 
e*(-a^5*b^5)^(1/2))/(64*a^5*b^4) - (c*d*e)/(32*a^2*b^2) - (d^2*e*(-a^5*b^5 
)^(1/2))/(64*a^4*b^5))^(1/2))/((c*d^2*e^5)/4 - (b*c^3*e^5)/(4*a) + (d^3*e^ 
5*(-a^5*b^5)^(1/2))/(4*a^2*b^3) - (c^2*d*e^5*(-a^5*b^5)^(1/2))/(4*a^3*b^2) 
) + (2*b*d^2*e^4*(e*x)^(1/2)*((c^2*e*(-a^5*b^5)^(1/2))/(64*a^5*b^4) - (c*d 
*e)/(32*a^2*b^2) - (d^2*e*(-a^5*b^5)^(1/2))/(64*a^4*b^5))^(1/2))/((b*c^3*e 
^5)/(4*a^2) - (c*d^2*e^5)/(4*a) - (d^3*e^5*(-a^5*b^5)^(1/2))/(4*a^3*b^3) + 
 (c^2*d*e^5*(-a^5*b^5)^(1/2))/(4*a^4*b^2)))*(-(a*d^2*e*(-a^5*b^5)^(1/2) - 
b*c^2*e*(-a^5*b^5)^(1/2) + 2*a^3*b^3*c*d*e)/(64*a^5*b^5))^(1/2) + ((c*e*(e 
*x)^(3/2))/(2*a) - (d*e^2*(e*x)^(1/2))/(2*b))/(a*e^2 + b*e^2*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.36 \[ \int \frac {\sqrt {e x} (c+d x)}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*( - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 
 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*c - 2*b**(1/4)*a**(3/ 
4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)* 
a**(1/4)*sqrt(2)))*b**2*c*x**2 - 2*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4 
)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2* 
d - 2*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x 
)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*d*x**2 + 2*b**(1/4)*a**(3/4)*s 
qrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**( 
1/4)*sqrt(2)))*a*b*c + 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4) 
*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*c*x**2 + 2 
*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqr 
t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*d + 2*b**(3/4)*a**(1/4)*sqrt(2)*at 
an((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt 
(2)))*a*b*d*x**2 + b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1 
/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b*c + b**(1/4)*a**(3/4)*sqrt(2)*log( 
- sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b**2*c*x**2 - b 
**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + 
 sqrt(b)*x)*a*b*c - b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4 
)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b**2*c*x**2 - b**(3/4)*a**(1/4)*sqrt(2)*l 
og( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**2*d -...