\(\int \frac {(e x)^{3/2} (c+d x)}{(a+b x^2)^2} \, dx\) [404]

Optimal result

Integrand size = 22, antiderivative size = 256 \[ \int \frac {(e x)^{3/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}-\frac {\left (\sqrt {b} c+3 \sqrt {a} d\right ) e^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\left (\sqrt {b} c+3 \sqrt {a} d\right ) e^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{3/4} b^{7/4}}+\frac {\left (c-\frac {3 \sqrt {a} d}{\sqrt {b}}\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{3/4} b^{5/4}} \] Output:

-1/2*e*(e*x)^(1/2)*(d*x+c)/b/(b*x^2+a)-1/8*(b^(1/2)*c+3*a^(1/2)*d)*e^(3/2) 
*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^( 
7/4)+1/8*(b^(1/2)*c+3*a^(1/2)*d)*e^(3/2)*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^(7/4)+1/8*(c-3*a^(1/2)*d/b^(1/2))*e 
^(3/2)*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^(3/4)/b^(5/4)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.69 \[ \int \frac {(e x)^{3/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\frac {(e x)^{3/2} \left (-\frac {4 b^{3/4} \sqrt {x} (c+d x)}{a+b x^2}-\frac {\sqrt {2} \left (\sqrt {b} c+3 \sqrt {a} d\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4}}+\frac {\sqrt {2} \left (\sqrt {b} c-3 \sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4}}\right )}{8 b^{7/4} x^{3/2}} \] Input:

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

Output:

((e*x)^(3/2)*((-4*b^(3/4)*Sqrt[x]*(c + d*x))/(a + b*x^2) - (Sqrt[2]*(Sqrt[ 
b]*c + 3*Sqrt[a]*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)* 
Sqrt[x])])/a^(3/4) + (Sqrt[2]*(Sqrt[b]*c - 3*Sqrt[a]*d)*ArcTanh[(Sqrt[2]*a 
^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(3/4)))/(8*b^(7/4)*x^(3/ 
2))
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.32, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {549, 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.

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

\(\Big \downarrow \) 549

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \int \frac {c+3 d x}{\sqrt {e x} \left (b x^2+a\right )}dx}{4 b}-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 554

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

\(\displaystyle \frac {e^2 \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-3 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+3 d\right ) \left (\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {b}}\right )}{2 \sqrt {b}}\right )}{2 b}-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {e^2 \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-3 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+3 d\right ) \left (\frac {\int \frac {1}{-e x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int \frac {1}{-e x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{2 b}-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {e^2 \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-3 d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{b x^2 e^2+a e^2}d\sqrt {e x}}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+3 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{2 b}-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {e^2 \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-3 d\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+3 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{2 b}-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-3 d\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}\right )}{\sqrt [4]{b} \left (x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}\right )}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+3 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{2 b}-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-3 d\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {e}-2 \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {e}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {e}+\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{x e+\frac {\sqrt {a} e}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {e x} \sqrt {e}}{\sqrt [4]{b}}}d\sqrt {e x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+3 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{2 b}-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {e^2 \left (\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}+3 d\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}+\frac {\left (\frac {\sqrt {b} c}{\sqrt {a}}-3 d\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e} \sqrt {e x}+\sqrt {a} e+\sqrt {b} e x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e}}\right )}{2 \sqrt {b}}\right )}{2 b}-\frac {e \sqrt {e x} (c+d x)}{2 b \left (a+b x^2\right )}\)

Input:

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

Output:

-1/2*(e*Sqrt[e*x]*(c + d*x))/(b*(a + b*x^2)) + (e^2*((((Sqrt[b]*c)/Sqrt[a] 
 + 3*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] - 3*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)*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*Sqrt[b])))/(2*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 549

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[e*(e*x)^(m - 1)*(c + d*x)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), 
 x] - Simp[e^2/(2*b*(p + 1))   Int[(e*x)^(m - 2)*(c*(m - 1) + d*m*x)*(a + b 
*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[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 = 0.39 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(-\frac {\left (-\frac {b \sqrt {2}\, \left (b \,x^{2}+a \right ) c \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \sqrt {\frac {a \,e^{2}}{b}}}{8}+\left (\left (d x +c \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} b \sqrt {e x}-\frac {3 d e \sqrt {2}\, \left (b \,x^{2}+a \right ) \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8}\right ) a \right ) e}{2 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} b^{2} \left (b \,x^{2}+a \right ) a}\)	\(333\)
derivativedivides	\(2 e^{2} \left (\frac {-\frac {d \left (e x \right )^{\frac {3}{2}}}{4 b}-\frac {c e \sqrt {e x}}{4 b}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e a}+\frac {3 d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{4 b}\right )\)	\(335\)
default	\(2 e^{2} \left (\frac {-\frac {d \left (e x \right )^{\frac {3}{2}}}{4 b}-\frac {c e \sqrt {e x}}{4 b}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\frac {c \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 e a}+\frac {3 d \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{4 b}\right )\)	\(335\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (176) = 352\).

Time = 0.20 (sec) , antiderivative size = 997, normalized size of antiderivative = 3.89 \[ \int \frac {(e x)^{3/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/8*((b^2*x^2 + a*b)*sqrt(-(6*c*d*e^3 + a*b^3*sqrt(-(b^2*c^4 - 18*a*b*c^2 
*d^2 + 81*a^2*d^4)*e^6/(a^3*b^7)))/(a*b^3))*log(-(b^2*c^4 - 81*a^2*d^4)*sq 
rt(e*x)*e^4 + (3*a^3*b^5*d*sqrt(-(b^2*c^4 - 18*a*b*c^2*d^2 + 81*a^2*d^4)*e 
^6/(a^3*b^7)) + (a*b^3*c^3 - 9*a^2*b^2*c*d^2)*e^3)*sqrt(-(6*c*d*e^3 + a*b^ 
3*sqrt(-(b^2*c^4 - 18*a*b*c^2*d^2 + 81*a^2*d^4)*e^6/(a^3*b^7)))/(a*b^3))) 
- (b^2*x^2 + a*b)*sqrt(-(6*c*d*e^3 + a*b^3*sqrt(-(b^2*c^4 - 18*a*b*c^2*d^2 
 + 81*a^2*d^4)*e^6/(a^3*b^7)))/(a*b^3))*log(-(b^2*c^4 - 81*a^2*d^4)*sqrt(e 
*x)*e^4 - (3*a^3*b^5*d*sqrt(-(b^2*c^4 - 18*a*b*c^2*d^2 + 81*a^2*d^4)*e^6/( 
a^3*b^7)) + (a*b^3*c^3 - 9*a^2*b^2*c*d^2)*e^3)*sqrt(-(6*c*d*e^3 + a*b^3*sq 
rt(-(b^2*c^4 - 18*a*b*c^2*d^2 + 81*a^2*d^4)*e^6/(a^3*b^7)))/(a*b^3))) - (b 
^2*x^2 + a*b)*sqrt(-(6*c*d*e^3 - a*b^3*sqrt(-(b^2*c^4 - 18*a*b*c^2*d^2 + 8 
1*a^2*d^4)*e^6/(a^3*b^7)))/(a*b^3))*log(-(b^2*c^4 - 81*a^2*d^4)*sqrt(e*x)* 
e^4 + (3*a^3*b^5*d*sqrt(-(b^2*c^4 - 18*a*b*c^2*d^2 + 81*a^2*d^4)*e^6/(a^3* 
b^7)) - (a*b^3*c^3 - 9*a^2*b^2*c*d^2)*e^3)*sqrt(-(6*c*d*e^3 - a*b^3*sqrt(- 
(b^2*c^4 - 18*a*b*c^2*d^2 + 81*a^2*d^4)*e^6/(a^3*b^7)))/(a*b^3))) + (b^2*x 
^2 + a*b)*sqrt(-(6*c*d*e^3 - a*b^3*sqrt(-(b^2*c^4 - 18*a*b*c^2*d^2 + 81*a^ 
2*d^4)*e^6/(a^3*b^7)))/(a*b^3))*log(-(b^2*c^4 - 81*a^2*d^4)*sqrt(e*x)*e^4 
- (3*a^3*b^5*d*sqrt(-(b^2*c^4 - 18*a*b*c^2*d^2 + 81*a^2*d^4)*e^6/(a^3*b^7) 
) - (a*b^3*c^3 - 9*a^2*b^2*c*d^2)*e^3)*sqrt(-(6*c*d*e^3 - a*b^3*sqrt(-(b^2 
*c^4 - 18*a*b*c^2*d^2 + 81*a^2*d^4)*e^6/(a^3*b^7)))/(a*b^3))) + 4*(d*e*...
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 33.24 (sec) , antiderivative size = 1911, normalized size of antiderivative = 7.46 \[ \int \frac {(e x)^{3/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

c*(-5*a**(9/4)*b*e**(3/2)*x**(9/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(I*pi 
/4)/a**(1/4))*gamma(5/4)/(32*a**3*b**(9/4)*x**(9/2)*exp(I*pi/4)*gamma(9/4) 
 + 32*a**2*b**(13/4)*x**(13/2)*exp(I*pi/4)*gamma(9/4)) + 5*I*a**(9/4)*b*e* 
*(3/2)*x**(9/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1/4))*gam 
ma(5/4)/(32*a**3*b**(9/4)*x**(9/2)*exp(I*pi/4)*gamma(9/4) + 32*a**2*b**(13 
/4)*x**(13/2)*exp(I*pi/4)*gamma(9/4)) + 5*a**(9/4)*b*e**(3/2)*x**(9/2)*log 
(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(5/4)/(32*a**3*b* 
*(9/4)*x**(9/2)*exp(I*pi/4)*gamma(9/4) + 32*a**2*b**(13/4)*x**(13/2)*exp(I 
*pi/4)*gamma(9/4)) - 5*I*a**(9/4)*b*e**(3/2)*x**(9/2)*log(1 - b**(1/4)*sqr 
t(x)*exp_polar(7*I*pi/4)/a**(1/4))*gamma(5/4)/(32*a**3*b**(9/4)*x**(9/2)*e 
xp(I*pi/4)*gamma(9/4) + 32*a**2*b**(13/4)*x**(13/2)*exp(I*pi/4)*gamma(9/4) 
) - 5*a**(5/4)*b**2*e**(3/2)*x**(13/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar( 
I*pi/4)/a**(1/4))*gamma(5/4)/(32*a**3*b**(9/4)*x**(9/2)*exp(I*pi/4)*gamma( 
9/4) + 32*a**2*b**(13/4)*x**(13/2)*exp(I*pi/4)*gamma(9/4)) + 5*I*a**(5/4)* 
b**2*e**(3/2)*x**(13/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(3*I*pi/4)/a**(1 
/4))*gamma(5/4)/(32*a**3*b**(9/4)*x**(9/2)*exp(I*pi/4)*gamma(9/4) + 32*a** 
2*b**(13/4)*x**(13/2)*exp(I*pi/4)*gamma(9/4)) + 5*a**(5/4)*b**2*e**(3/2)*x 
**(13/2)*log(1 - b**(1/4)*sqrt(x)*exp_polar(5*I*pi/4)/a**(1/4))*gamma(5/4) 
/(32*a**3*b**(9/4)*x**(9/2)*exp(I*pi/4)*gamma(9/4) + 32*a**2*b**(13/4)*x** 
(13/2)*exp(I*pi/4)*gamma(9/4)) - 5*I*a**(5/4)*b**2*e**(3/2)*x**(13/2)*l...
 

Maxima [F(-2)]

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

integrate((e*x)^(3/2)*(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. 367 vs. \(2 (176) = 352\).

Time = 0.14 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.43 \[ \int \frac {(e x)^{3/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=-\frac {1}{16} \, e {\left (\frac {8 \, {\left (\sqrt {e x} d e^{2} x + \sqrt {e x} c e^{2}\right )}}{{\left (b e^{2} x^{2} + a e^{2}\right )} b} - \frac {2 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e + 3 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4} e} - \frac {2 \, \sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e + 3 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{4} e} - \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e - 3 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{a b^{4} e} + \frac {\sqrt {2} {\left (\left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c e - 3 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} d\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{a b^{4} e}\right )} \] Input:

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

Output:

-1/16*e*(8*(sqrt(e*x)*d*e^2*x + sqrt(e*x)*c*e^2)/((b*e^2*x^2 + a*e^2)*b) - 
 2*sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c*e + 3*(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^4*e) 
 - 2*sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c*e + 3*(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^4 
*e) - sqrt(2)*((a*b^3*e^2)^(1/4)*b^2*c*e - 3*(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^4*e) + sqrt(2)*( 
(a*b^3*e^2)^(1/4)*b^2*c*e - 3*(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^4*e))
 

Mupad [B] (verification not implemented)

Time = 7.48 (sec) , antiderivative size = 795, normalized size of antiderivative = 3.11 \[ \int \frac {(e x)^{3/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx=-2\,\mathrm {atanh}\left (\frac {18\,a\,d^2\,e^6\,\sqrt {e\,x}\,\sqrt {\frac {9\,d^2\,e^3\,\sqrt {-a^3\,b^7}}{64\,a^2\,b^7}-\frac {c^2\,e^3\,\sqrt {-a^3\,b^7}}{64\,a^3\,b^6}-\frac {3\,c\,d\,e^3}{32\,a\,b^3}}}{\frac {27\,a\,d^3\,e^8}{4\,b^2}-\frac {3\,c^2\,d\,e^8}{4\,b}-\frac {c^3\,e^8\,\sqrt {-a^3\,b^7}}{4\,a^2\,b^4}+\frac {9\,c\,d^2\,e^8\,\sqrt {-a^3\,b^7}}{4\,a\,b^5}}-\frac {2\,b\,c^2\,e^6\,\sqrt {e\,x}\,\sqrt {\frac {9\,d^2\,e^3\,\sqrt {-a^3\,b^7}}{64\,a^2\,b^7}-\frac {c^2\,e^3\,\sqrt {-a^3\,b^7}}{64\,a^3\,b^6}-\frac {3\,c\,d\,e^3}{32\,a\,b^3}}}{\frac {27\,a\,d^3\,e^8}{4\,b^2}-\frac {3\,c^2\,d\,e^8}{4\,b}-\frac {c^3\,e^8\,\sqrt {-a^3\,b^7}}{4\,a^2\,b^4}+\frac {9\,c\,d^2\,e^8\,\sqrt {-a^3\,b^7}}{4\,a\,b^5}}\right )\,\sqrt {-\frac {b\,c^2\,e^3\,\sqrt {-a^3\,b^7}-9\,a\,d^2\,e^3\,\sqrt {-a^3\,b^7}+6\,a^2\,b^4\,c\,d\,e^3}{64\,a^3\,b^7}}-2\,\mathrm {atanh}\left (\frac {18\,a\,d^2\,e^6\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,e^3\,\sqrt {-a^3\,b^7}}{64\,a^3\,b^6}-\frac {3\,c\,d\,e^3}{32\,a\,b^3}-\frac {9\,d^2\,e^3\,\sqrt {-a^3\,b^7}}{64\,a^2\,b^7}}}{\frac {27\,a\,d^3\,e^8}{4\,b^2}-\frac {3\,c^2\,d\,e^8}{4\,b}+\frac {c^3\,e^8\,\sqrt {-a^3\,b^7}}{4\,a^2\,b^4}-\frac {9\,c\,d^2\,e^8\,\sqrt {-a^3\,b^7}}{4\,a\,b^5}}-\frac {2\,b\,c^2\,e^6\,\sqrt {e\,x}\,\sqrt {\frac {c^2\,e^3\,\sqrt {-a^3\,b^7}}{64\,a^3\,b^6}-\frac {3\,c\,d\,e^3}{32\,a\,b^3}-\frac {9\,d^2\,e^3\,\sqrt {-a^3\,b^7}}{64\,a^2\,b^7}}}{\frac {27\,a\,d^3\,e^8}{4\,b^2}-\frac {3\,c^2\,d\,e^8}{4\,b}+\frac {c^3\,e^8\,\sqrt {-a^3\,b^7}}{4\,a^2\,b^4}-\frac {9\,c\,d^2\,e^8\,\sqrt {-a^3\,b^7}}{4\,a\,b^5}}\right )\,\sqrt {-\frac {9\,a\,d^2\,e^3\,\sqrt {-a^3\,b^7}-b\,c^2\,e^3\,\sqrt {-a^3\,b^7}+6\,a^2\,b^4\,c\,d\,e^3}{64\,a^3\,b^7}}-\frac {\frac {c\,e^3\,\sqrt {e\,x}}{2\,b}+\frac {d\,e^2\,{\left (e\,x\right )}^{3/2}}{2\,b}}{b\,e^2\,x^2+a\,e^2} \] Input:

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

Output:

- 2*atanh((18*a*d^2*e^6*(e*x)^(1/2)*((9*d^2*e^3*(-a^3*b^7)^(1/2))/(64*a^2* 
b^7) - (c^2*e^3*(-a^3*b^7)^(1/2))/(64*a^3*b^6) - (3*c*d*e^3)/(32*a*b^3))^( 
1/2))/((27*a*d^3*e^8)/(4*b^2) - (3*c^2*d*e^8)/(4*b) - (c^3*e^8*(-a^3*b^7)^ 
(1/2))/(4*a^2*b^4) + (9*c*d^2*e^8*(-a^3*b^7)^(1/2))/(4*a*b^5)) - (2*b*c^2* 
e^6*(e*x)^(1/2)*((9*d^2*e^3*(-a^3*b^7)^(1/2))/(64*a^2*b^7) - (c^2*e^3*(-a^ 
3*b^7)^(1/2))/(64*a^3*b^6) - (3*c*d*e^3)/(32*a*b^3))^(1/2))/((27*a*d^3*e^8 
)/(4*b^2) - (3*c^2*d*e^8)/(4*b) - (c^3*e^8*(-a^3*b^7)^(1/2))/(4*a^2*b^4) + 
 (9*c*d^2*e^8*(-a^3*b^7)^(1/2))/(4*a*b^5)))*(-(b*c^2*e^3*(-a^3*b^7)^(1/2) 
- 9*a*d^2*e^3*(-a^3*b^7)^(1/2) + 6*a^2*b^4*c*d*e^3)/(64*a^3*b^7))^(1/2) - 
2*atanh((18*a*d^2*e^6*(e*x)^(1/2)*((c^2*e^3*(-a^3*b^7)^(1/2))/(64*a^3*b^6) 
 - (3*c*d*e^3)/(32*a*b^3) - (9*d^2*e^3*(-a^3*b^7)^(1/2))/(64*a^2*b^7))^(1/ 
2))/((27*a*d^3*e^8)/(4*b^2) - (3*c^2*d*e^8)/(4*b) + (c^3*e^8*(-a^3*b^7)^(1 
/2))/(4*a^2*b^4) - (9*c*d^2*e^8*(-a^3*b^7)^(1/2))/(4*a*b^5)) - (2*b*c^2*e^ 
6*(e*x)^(1/2)*((c^2*e^3*(-a^3*b^7)^(1/2))/(64*a^3*b^6) - (3*c*d*e^3)/(32*a 
*b^3) - (9*d^2*e^3*(-a^3*b^7)^(1/2))/(64*a^2*b^7))^(1/2))/((27*a*d^3*e^8)/ 
(4*b^2) - (3*c^2*d*e^8)/(4*b) + (c^3*e^8*(-a^3*b^7)^(1/2))/(4*a^2*b^4) - ( 
9*c*d^2*e^8*(-a^3*b^7)^(1/2))/(4*a*b^5)))*(-(9*a*d^2*e^3*(-a^3*b^7)^(1/2) 
- b*c^2*e^3*(-a^3*b^7)^(1/2) + 6*a^2*b^4*c*d*e^3)/(64*a^3*b^7))^(1/2) - (( 
c*e^3*(e*x)^(1/2))/(2*b) + (d*e^2*(e*x)^(3/2))/(2*b))/(a*e^2 + b*e^2*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.40 \[ \int \frac {(e x)^{3/2} (c+d x)}{\left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*e*( - 6*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*d - 6*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*d*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*c - 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)))*b*c*x**2 + 6*b**(1/4)*a**(3/4)*sqrt(2)*a 
tan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr 
t(2)))*a*d + 6*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*d*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*c + 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)))*b*c*x**2 + 
 3*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sq 
rt(a) + sqrt(b)*x)*a*d + 3*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*d*x**2 - 3*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*d - 
 3*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*d*x**2 - 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*c - b**(3/4)*a**(1/4)*sqrt...