\(\int \frac {c+d x}{(e x)^{3/2} (a+b x^2)} \, dx\) [376]

Optimal result

Integrand size = 22, antiderivative size = 235 \[ \int \frac {c+d x}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=-\frac {2 c}{a e \sqrt {e x}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{5/4} \sqrt [4]{b} e^{3/2}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} a^{5/4} \sqrt [4]{b} e^{3/2}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{\sqrt {2} a^{5/4} \sqrt [4]{b} e^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.69 \[ \int \frac {c+d x}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {x \left (-4 \sqrt [4]{a} \sqrt [4]{b} c+\sqrt {2} \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {x} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{2 a^{5/4} \sqrt [4]{b} (e x)^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.36, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {553, 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.

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

\(\Big \downarrow \) 553

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 554

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

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

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 553

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp 
[1/(a*e*(m + 1))   Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 
2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[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 = 0.37 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e}-\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{a e}-\frac {2 c}{a e \sqrt {e x}}\)	\(300\)
default	\(\frac {\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e}-\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{a e}-\frac {2 c}{a e \sqrt {e x}}\)	\(300\)
risch	\(-\frac {2 c}{a e \sqrt {e x}}-\frac {-\frac {d \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 e}+\frac {c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{a e}\)	\(300\)
pseudoelliptic	\(\frac {-\frac {\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) c \sqrt {2}\, e \sqrt {e x}}{2}+\frac {\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right ) d \sqrt {\frac {a \,e^{2}}{b}}\, \sqrt {2}\, \sqrt {e x}}{2}+\sqrt {2}\, \sqrt {e x}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d -c e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}-\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )+\sqrt {2}\, \sqrt {e x}\, \left (\sqrt {\frac {a \,e^{2}}{b}}\, d -c e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}+\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}\right )-4 c e \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{2 \sqrt {e x}\, \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} a \,e^{2}}\)	\(332\)

Input:

int((d*x+c)/(e*x)^(3/2)/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

2/a/e*(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)))-2*c/a/e/(e*x 
)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (165) = 330\).

Time = 0.12 (sec) , antiderivative size = 910, normalized size of antiderivative = 3.87 \[ \int \frac {c+d x}{(e x)^{3/2} \left (a+b x^2\right )} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)/(e*x)^(3/2)/(b*x^2+a),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {c+d x}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\int \frac {c + d x}{\left (e x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )}\, dx \] Input:

integrate((d*x+c)/(e*x)**(3/2)/(b*x**2+a),x)
 

Output:

Integral((c + d*x)/((e*x)**(3/2)*(a + b*x**2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {c+d x}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)/(e*x)^(3/2)/(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. 331 vs. \(2 (165) = 330\).

Time = 0.13 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.41 \[ \int \frac {c+d x}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=-\frac {\frac {8 \, c}{\sqrt {e x} a} - \frac {2 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{2} e^{2}} - \frac {2 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e - \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{2} e^{2}} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{a^{2} b^{2} e^{2}} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d e + \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{a^{2} b^{2} e^{2}}}{4 \, e} \] Input:

integrate((d*x+c)/(e*x)^(3/2)/(b*x^2+a),x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 717, normalized size of antiderivative = 3.05 \[ \int \frac {c+d x}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=2\,\mathrm {atanh}\left (\frac {32\,a^4\,b^4\,c^2\,e^5\,\sqrt {e\,x}\,\sqrt {\frac {c\,d}{2\,a^2\,e^3}-\frac {c^2\,\sqrt {-a^5\,b}}{4\,a^5\,e^3}+\frac {d^2\,\sqrt {-a^5\,b}}{4\,a^4\,b\,e^3}}}{16\,a^3\,b^4\,c^3\,e^4-16\,a^2\,b^2\,d^3\,e^4\,\sqrt {-a^5\,b}-16\,a^4\,b^3\,c\,d^2\,e^4+16\,a\,b^3\,c^2\,d\,e^4\,\sqrt {-a^5\,b}}-\frac {32\,a^5\,b^3\,d^2\,e^5\,\sqrt {e\,x}\,\sqrt {\frac {c\,d}{2\,a^2\,e^3}-\frac {c^2\,\sqrt {-a^5\,b}}{4\,a^5\,e^3}+\frac {d^2\,\sqrt {-a^5\,b}}{4\,a^4\,b\,e^3}}}{16\,a^3\,b^4\,c^3\,e^4-16\,a^2\,b^2\,d^3\,e^4\,\sqrt {-a^5\,b}-16\,a^4\,b^3\,c\,d^2\,e^4+16\,a\,b^3\,c^2\,d\,e^4\,\sqrt {-a^5\,b}}\right )\,\sqrt {\frac {a\,d^2\,\sqrt {-a^5\,b}-b\,c^2\,\sqrt {-a^5\,b}+2\,a^3\,b\,c\,d}{4\,a^5\,b\,e^3}}+2\,\mathrm {atanh}\left (\frac {32\,a^4\,b^4\,c^2\,e^5\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,\sqrt {-a^5\,b}}{4\,a^5\,e^3}+\frac {c\,d}{2\,a^2\,e^3}-\frac {d^2\,\sqrt {-a^5\,b}}{4\,a^4\,b\,e^3}}}{16\,a^3\,b^4\,c^3\,e^4+16\,a^2\,b^2\,d^3\,e^4\,\sqrt {-a^5\,b}-16\,a^4\,b^3\,c\,d^2\,e^4-16\,a\,b^3\,c^2\,d\,e^4\,\sqrt {-a^5\,b}}-\frac {32\,a^5\,b^3\,d^2\,e^5\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,\sqrt {-a^5\,b}}{4\,a^5\,e^3}+\frac {c\,d}{2\,a^2\,e^3}-\frac {d^2\,\sqrt {-a^5\,b}}{4\,a^4\,b\,e^3}}}{16\,a^3\,b^4\,c^3\,e^4+16\,a^2\,b^2\,d^3\,e^4\,\sqrt {-a^5\,b}-16\,a^4\,b^3\,c\,d^2\,e^4-16\,a\,b^3\,c^2\,d\,e^4\,\sqrt {-a^5\,b}}\right )\,\sqrt {\frac {b\,c^2\,\sqrt {-a^5\,b}-a\,d^2\,\sqrt {-a^5\,b}+2\,a^3\,b\,c\,d}{4\,a^5\,b\,e^3}}-\frac {2\,c}{a\,e\,\sqrt {e\,x}} \] Input:

int((c + d*x)/((e*x)^(3/2)*(a + b*x^2)),x)
 

Output:

2*atanh((32*a^4*b^4*c^2*e^5*(e*x)^(1/2)*((c*d)/(2*a^2*e^3) - (c^2*(-a^5*b) 
^(1/2))/(4*a^5*e^3) + (d^2*(-a^5*b)^(1/2))/(4*a^4*b*e^3))^(1/2))/(16*a^3*b 
^4*c^3*e^4 - 16*a^2*b^2*d^3*e^4*(-a^5*b)^(1/2) - 16*a^4*b^3*c*d^2*e^4 + 16 
*a*b^3*c^2*d*e^4*(-a^5*b)^(1/2)) - (32*a^5*b^3*d^2*e^5*(e*x)^(1/2)*((c*d)/ 
(2*a^2*e^3) - (c^2*(-a^5*b)^(1/2))/(4*a^5*e^3) + (d^2*(-a^5*b)^(1/2))/(4*a 
^4*b*e^3))^(1/2))/(16*a^3*b^4*c^3*e^4 - 16*a^2*b^2*d^3*e^4*(-a^5*b)^(1/2) 
- 16*a^4*b^3*c*d^2*e^4 + 16*a*b^3*c^2*d*e^4*(-a^5*b)^(1/2)))*((a*d^2*(-a^5 
*b)^(1/2) - b*c^2*(-a^5*b)^(1/2) + 2*a^3*b*c*d)/(4*a^5*b*e^3))^(1/2) + 2*a 
tanh((32*a^4*b^4*c^2*e^5*(e*x)^(1/2)*((c^2*(-a^5*b)^(1/2))/(4*a^5*e^3) + ( 
c*d)/(2*a^2*e^3) - (d^2*(-a^5*b)^(1/2))/(4*a^4*b*e^3))^(1/2))/(16*a^3*b^4* 
c^3*e^4 + 16*a^2*b^2*d^3*e^4*(-a^5*b)^(1/2) - 16*a^4*b^3*c*d^2*e^4 - 16*a* 
b^3*c^2*d*e^4*(-a^5*b)^(1/2)) - (32*a^5*b^3*d^2*e^5*(e*x)^(1/2)*((c^2*(-a^ 
5*b)^(1/2))/(4*a^5*e^3) + (c*d)/(2*a^2*e^3) - (d^2*(-a^5*b)^(1/2))/(4*a^4* 
b*e^3))^(1/2))/(16*a^3*b^4*c^3*e^4 + 16*a^2*b^2*d^3*e^4*(-a^5*b)^(1/2) - 1 
6*a^4*b^3*c*d^2*e^4 - 16*a*b^3*c^2*d*e^4*(-a^5*b)^(1/2)))*((b*c^2*(-a^5*b) 
^(1/2) - a*d^2*(-a^5*b)^(1/2) + 2*a^3*b*c*d)/(4*a^5*b*e^3))^(1/2) - (2*c)/ 
(a*e*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.34 \[ \int \frac {c+d x}{(e x)^{3/2} \left (a+b x^2\right )} \, dx=\frac {\sqrt {e}\, \left (2 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c -2 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d -2 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) c +2 \sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d -\sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c +\sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) c -\sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d +\sqrt {x}\, b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) d -8 a b c \right )}{4 \sqrt {x}\, a^{2} b \,e^{2}} \] Input:

int((d*x+c)/(e*x)^(3/2)/(b*x^2+a),x)
 

Output:

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