\(\int \frac {c+d x}{\sqrt {e x} (a+b x^2)} \, dx\) [375]

Optimal result

Integrand size = 22, antiderivative size = 218 \[ \int \frac {c+d x}{\sqrt {e x} \left (a+b x^2\right )} \, dx=-\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{3/4} b^{3/4} \sqrt {e}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{3/4} b^{3/4} \sqrt {e}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} a^{3/4} b^{3/4} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.62 \[ \int \frac {c+d x}{\sqrt {e x} \left (a+b x^2\right )} \, dx=-\frac {\sqrt {x} \left (\left (\sqrt {b} c+\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\left (-\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{\sqrt {2} a^{3/4} b^{3/4} \sqrt {e x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.37, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {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.

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

\(\Big \downarrow \) 554

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

2*((((Sqrt[b]*c)/Sqrt[a] + d)*(-(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]) + (((Sqrt[b]*c)/Sqrt[a] - d)*(-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)*Sqr 
t[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*Sqrt[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 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 = 0.33 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(\frac {\sqrt {2}\, \left (\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) d a e}{2}+\frac {\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) c \sqrt {\frac {a \,e^{2}}{b}}\, b}{2}+\left (\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )+\arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )\right ) \left (a d e +c \sqrt {\frac {a \,e^{2}}{b}}\, b \right )\right )}{2 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} a e b}\)	\(251\)
derivativedivides	\(\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e a}+\frac {d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\)	\(283\)
default	\(\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e a}+\frac {d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\)	\(283\)

Input:

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

Output:

1/2*2^(1/2)*(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)))*d*a*e+1/2*l 
n((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)))*c*(a*e^2/b)^(1/2)*b+(arctan(2 
^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2)-1)+arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x) 
^(1/2)+1))*(a*d*e+c*(a*e^2/b)^(1/2)*b))/(a*e^2/b)^(1/4)/a/e/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (150) = 300\).

Time = 0.27 (sec) , antiderivative size = 877, normalized size of antiderivative = 4.02 \[ \int \frac {c+d x}{\sqrt {e x} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)/(e*x)^(1/2)/(b*x^2+a),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. 320 vs. \(2 (150) = 300\).

Time = 0.14 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.47 \[ \int \frac {c+d x}{\sqrt {e x} \left (a+b x^2\right )} \, dx=\frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{3} e^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{2 \, a b^{3} e^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{4 \, a b^{3} e^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{4 \, a b^{3} e^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 717, normalized size of antiderivative = 3.29 \[ \int \frac {c+d x}{\sqrt {e x} \left (a+b x^2\right )} \, dx=-2\,\mathrm {atanh}\left (\frac {32\,b^3\,c^2\,e^2\,\sqrt {e\,x}\,\sqrt {\frac {d^2\,\sqrt {-a^3\,b^3}}{4\,a^2\,b^3\,e}-\frac {c^2\,\sqrt {-a^3\,b^3}}{4\,a^3\,b^2\,e}-\frac {c\,d}{2\,a\,b\,e}}}{16\,b^2\,c^2\,d\,e^2-16\,a\,b\,d^3\,e^2+\frac {16\,b\,c^3\,e^2\,\sqrt {-a^3\,b^3}}{a^2}-\frac {16\,c\,d^2\,e^2\,\sqrt {-a^3\,b^3}}{a}}-\frac {32\,a\,b^2\,d^2\,e^2\,\sqrt {e\,x}\,\sqrt {\frac {d^2\,\sqrt {-a^3\,b^3}}{4\,a^2\,b^3\,e}-\frac {c^2\,\sqrt {-a^3\,b^3}}{4\,a^3\,b^2\,e}-\frac {c\,d}{2\,a\,b\,e}}}{16\,b^2\,c^2\,d\,e^2-16\,a\,b\,d^3\,e^2+\frac {16\,b\,c^3\,e^2\,\sqrt {-a^3\,b^3}}{a^2}-\frac {16\,c\,d^2\,e^2\,\sqrt {-a^3\,b^3}}{a}}\right )\,\sqrt {-\frac {b\,c^2\,\sqrt {-a^3\,b^3}-a\,d^2\,\sqrt {-a^3\,b^3}+2\,a^2\,b^2\,c\,d}{4\,a^3\,b^3\,e}}-2\,\mathrm {atanh}\left (\frac {32\,b^3\,c^2\,e^2\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,\sqrt {-a^3\,b^3}}{4\,a^3\,b^2\,e}-\frac {c\,d}{2\,a\,b\,e}-\frac {d^2\,\sqrt {-a^3\,b^3}}{4\,a^2\,b^3\,e}}}{16\,b^2\,c^2\,d\,e^2-16\,a\,b\,d^3\,e^2-\frac {16\,b\,c^3\,e^2\,\sqrt {-a^3\,b^3}}{a^2}+\frac {16\,c\,d^2\,e^2\,\sqrt {-a^3\,b^3}}{a}}-\frac {32\,a\,b^2\,d^2\,e^2\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,\sqrt {-a^3\,b^3}}{4\,a^3\,b^2\,e}-\frac {c\,d}{2\,a\,b\,e}-\frac {d^2\,\sqrt {-a^3\,b^3}}{4\,a^2\,b^3\,e}}}{16\,b^2\,c^2\,d\,e^2-16\,a\,b\,d^3\,e^2-\frac {16\,b\,c^3\,e^2\,\sqrt {-a^3\,b^3}}{a^2}+\frac {16\,c\,d^2\,e^2\,\sqrt {-a^3\,b^3}}{a}}\right )\,\sqrt {-\frac {a\,d^2\,\sqrt {-a^3\,b^3}-b\,c^2\,\sqrt {-a^3\,b^3}+2\,a^2\,b^2\,c\,d}{4\,a^3\,b^3\,e}} \] Input:

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

Output:

- 2*atanh((32*b^3*c^2*e^2*(e*x)^(1/2)*((d^2*(-a^3*b^3)^(1/2))/(4*a^2*b^3*e 
) - (c^2*(-a^3*b^3)^(1/2))/(4*a^3*b^2*e) - (c*d)/(2*a*b*e))^(1/2))/(16*b^2 
*c^2*d*e^2 - 16*a*b*d^3*e^2 + (16*b*c^3*e^2*(-a^3*b^3)^(1/2))/a^2 - (16*c* 
d^2*e^2*(-a^3*b^3)^(1/2))/a) - (32*a*b^2*d^2*e^2*(e*x)^(1/2)*((d^2*(-a^3*b 
^3)^(1/2))/(4*a^2*b^3*e) - (c^2*(-a^3*b^3)^(1/2))/(4*a^3*b^2*e) - (c*d)/(2 
*a*b*e))^(1/2))/(16*b^2*c^2*d*e^2 - 16*a*b*d^3*e^2 + (16*b*c^3*e^2*(-a^3*b 
^3)^(1/2))/a^2 - (16*c*d^2*e^2*(-a^3*b^3)^(1/2))/a))*(-(b*c^2*(-a^3*b^3)^( 
1/2) - a*d^2*(-a^3*b^3)^(1/2) + 2*a^2*b^2*c*d)/(4*a^3*b^3*e))^(1/2) - 2*at 
anh((32*b^3*c^2*e^2*(e*x)^(1/2)*((c^2*(-a^3*b^3)^(1/2))/(4*a^3*b^2*e) - (c 
*d)/(2*a*b*e) - (d^2*(-a^3*b^3)^(1/2))/(4*a^2*b^3*e))^(1/2))/(16*b^2*c^2*d 
*e^2 - 16*a*b*d^3*e^2 - (16*b*c^3*e^2*(-a^3*b^3)^(1/2))/a^2 + (16*c*d^2*e^ 
2*(-a^3*b^3)^(1/2))/a) - (32*a*b^2*d^2*e^2*(e*x)^(1/2)*((c^2*(-a^3*b^3)^(1 
/2))/(4*a^3*b^2*e) - (c*d)/(2*a*b*e) - (d^2*(-a^3*b^3)^(1/2))/(4*a^2*b^3*e 
))^(1/2))/(16*b^2*c^2*d*e^2 - 16*a*b*d^3*e^2 - (16*b*c^3*e^2*(-a^3*b^3)^(1 
/2))/a^2 + (16*c*d^2*e^2*(-a^3*b^3)^(1/2))/a))*(-(a*d^2*(-a^3*b^3)^(1/2) - 
 b*c^2*(-a^3*b^3)^(1/2) + 2*a^2*b^2*c*d)/(4*a^3*b^3*e))^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x}{\sqrt {e x} \left (a+b x^2\right )} \, dx=\frac {\sqrt {e}\, \sqrt {2}\, \left (-2 \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d -2 \sqrt {b}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c +2 \sqrt {a}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d +2 \sqrt {b}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c +\sqrt {a}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d -\sqrt {a}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d -\sqrt {b}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c +\sqrt {b}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c \right )}{4 b^{\frac {3}{4}} a^{\frac {3}{4}} e} \] Input:

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

Output:

(sqrt(e)*b**(1/4)*a**(1/4)*sqrt(2)*( - 2*sqrt(a)*atan((b**(1/4)*a**(1/4)*s 
qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*d - 2*sqrt(b)*ata 
n((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt( 
2)))*c + 2*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b 
**(1/4)*a**(1/4)*sqrt(2)))*d + 2*sqrt(b)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 
 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*c + sqrt(a)*log( - sqrt(x 
)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*d - sqrt(a)*log(sqrt(x) 
*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*d - sqrt(b)*log( - sqrt( 
x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*c + sqrt(b)*log(sqrt(x 
)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*c))/(4*a*b*e)