\(\int \frac {(c+d x)^2}{\sqrt {e x} (a+b x^2)^2} \, dx\) [411]

Optimal result

Integrand size = 24, antiderivative size = 303 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=-\frac {\sqrt {e x} \left (a d^2-b c (c+2 d x)\right )}{2 a b e \left (a+b x^2\right )}-\frac {\left (3 b c^2+2 \sqrt {a} \sqrt {b} c d+a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4} \sqrt {e}}+\frac {\left (3 b c^2+2 \sqrt {a} \sqrt {b} c d+a d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{4 \sqrt {2} a^{7/4} b^{5/4} \sqrt {e}}+\frac {\left (3 b c^2-2 \sqrt {a} \sqrt {b} c d+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{7/4} b^{5/4} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.68 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {x} \left (\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-a d^2+b c (c+2 d x)\right )}{a+b x^2}-\sqrt {2} \left (3 b c^2+2 \sqrt {a} \sqrt {b} c d+a d^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt {2} \left (3 b c^2-2 \sqrt {a} \sqrt {b} c d+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{8 a^{7/4} b^{5/4} \sqrt {e x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.23, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {558, 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.

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

\(\Big \downarrow \) 558

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 554

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(Sqrt[e*x]*(c^2 - (a*d^2)/b + 2*c*d*x))/(2*a*e*(a + b*x^2)) + (((3*b*c^2 + 
 2*Sqrt[a]*Sqrt[b]*c*d + a*d^2)*(-(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[a]) + ((3*b*c^2 - 2*Sqrt[a]*Sqrt[b]*c*d + a*d^2)*(-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[a]) 
)/(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 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 558

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, a + b*x^2, x], f = 
 Coeff[PolynomialRemainder[(c + d*x)^n, a + b*x^2, x], x, 0], g = Coeff[Pol 
ynomialRemainder[(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(-(e*x)^(m + 1))* 
(f + g*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1))), x] + Simp[1/(2*a*(p + 1)) 
  Int[(e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + f*(m + 2*p + 
 3) + g*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && IGt 
Q[n, 1] &&  !IntegerQ[m] && LtQ[p, -1]
 

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 = 0.43 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {\frac {\sqrt {2}\, \left (a \,d^{2}+3 b \,c^{2}\right ) \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right ) \left (b \,x^{2}+a \right ) \sqrt {\frac {a \,e^{2}}{b}}}{2}+\left (-4 \left (a \,d^{2}-b c \left (2 d x +c \right )\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}+d e \sqrt {2}\, \left (b \,x^{2}+a \right ) c \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )\right ) a}{8 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} e \,a^{2} b \left (b \,x^{2}+a \right )}\)	\(356\)
derivativedivides	\(2 e \left (\frac {\frac {c d \left (e x \right )^{\frac {3}{2}}}{2 a e}-\frac {\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {e x}}{4 b a}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\frac {\left (a \,d^{2} e +3 b \,c^{2} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {d c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{4 a e b}\right )\)	\(367\)
default	\(2 e \left (\frac {\frac {c d \left (e x \right )^{\frac {3}{2}}}{2 a e}-\frac {\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {e x}}{4 b a}}{b \,e^{2} x^{2}+a \,e^{2}}+\frac {\frac {\left (a \,d^{2} e +3 b \,c^{2} e \right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a \,e^{2}}+\frac {d c \sqrt {2}\, \left (\ln \left (\frac {e x -\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}{e x +\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x}\, \sqrt {2}+\sqrt {\frac {a \,e^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e x}}{\left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}}{4 a e b}\right )\)	\(367\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1716 vs. \(2 (223) = 446\).

Time = 0.14 (sec) , antiderivative size = 1716, normalized size of antiderivative = 5.66 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*((a*b^2*e*x^2 + a^2*b*e)*sqrt(-(a^3*b^2*e*sqrt(-(81*b^4*c^8 + 36*a*b^3 
*c^6*d^2 + 22*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)/(a^7*b^5*e^2)) 
+ 12*b*c^3*d + 4*a*c*d^3)/(a^3*b^2*e))*log((81*b^4*c^8 + 108*a*b^3*c^6*d^2 
 + 38*a^2*b^2*c^4*d^4 + 12*a^3*b*c^2*d^6 + a^4*d^8)*sqrt(e*x) + (2*a^6*b^4 
*c*d*e^2*sqrt(-(81*b^4*c^8 + 36*a*b^3*c^6*d^2 + 22*a^2*b^2*c^4*d^4 + 4*a^3 
*b*c^2*d^6 + a^4*d^8)/(a^7*b^5*e^2)) + (27*a^2*b^4*c^6 + 15*a^3*b^3*c^4*d^ 
2 + 5*a^4*b^2*c^2*d^4 + a^5*b*d^6)*e)*sqrt(-(a^3*b^2*e*sqrt(-(81*b^4*c^8 + 
 36*a*b^3*c^6*d^2 + 22*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)/(a^7*b 
^5*e^2)) + 12*b*c^3*d + 4*a*c*d^3)/(a^3*b^2*e))) - (a*b^2*e*x^2 + a^2*b*e) 
*sqrt(-(a^3*b^2*e*sqrt(-(81*b^4*c^8 + 36*a*b^3*c^6*d^2 + 22*a^2*b^2*c^4*d^ 
4 + 4*a^3*b*c^2*d^6 + a^4*d^8)/(a^7*b^5*e^2)) + 12*b*c^3*d + 4*a*c*d^3)/(a 
^3*b^2*e))*log((81*b^4*c^8 + 108*a*b^3*c^6*d^2 + 38*a^2*b^2*c^4*d^4 + 12*a 
^3*b*c^2*d^6 + a^4*d^8)*sqrt(e*x) - (2*a^6*b^4*c*d*e^2*sqrt(-(81*b^4*c^8 + 
 36*a*b^3*c^6*d^2 + 22*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)/(a^7*b 
^5*e^2)) + (27*a^2*b^4*c^6 + 15*a^3*b^3*c^4*d^2 + 5*a^4*b^2*c^2*d^4 + a^5* 
b*d^6)*e)*sqrt(-(a^3*b^2*e*sqrt(-(81*b^4*c^8 + 36*a*b^3*c^6*d^2 + 22*a^2*b 
^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8)/(a^7*b^5*e^2)) + 12*b*c^3*d + 4*a* 
c*d^3)/(a^3*b^2*e))) - (a*b^2*e*x^2 + a^2*b*e)*sqrt((a^3*b^2*e*sqrt(-(81*b 
^4*c^8 + 36*a*b^3*c^6*d^2 + 22*a^2*b^2*c^4*d^4 + 4*a^3*b*c^2*d^6 + a^4*d^8 
)/(a^7*b^5*e^2)) - 12*b*c^3*d - 4*a*c*d^3)/(a^3*b^2*e))*log((81*b^4*c^8...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^2/(e*x)^(1/2)/(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. 466 vs. \(2 (223) = 446\).

Time = 0.15 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.54 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x} b c d e x + \sqrt {e x} b c^{2} e - \sqrt {e x} a d^{2} e}{2 \, {\left (b e^{2} x^{2} + a e^{2}\right )} a b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e + \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e + 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{3} e^{2}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e + \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e + 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {e x}\right )}}{2 \, \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{3} e^{2}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e + \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e - 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \log \left (e x + \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{16 \, a^{2} b^{3} e^{2}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3} e^{2}\right )^{\frac {1}{4}} b^{2} c^{2} e + \left (a b^{3} e^{2}\right )^{\frac {1}{4}} a b d^{2} e - 2 \, \left (a b^{3} e^{2}\right )^{\frac {3}{4}} c d\right )} \log \left (e x - \sqrt {2} \left (\frac {a e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {e x} + \sqrt {\frac {a e^{2}}{b}}\right )}{16 \, a^{2} b^{3} e^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 7.71 (sec) , antiderivative size = 1843, normalized size of antiderivative = 6.08 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((c*d*(e*x)^(3/2))/a - (e*(e*x)^(1/2)*(a*d^2 - b*c^2))/(2*a*b))/(a*e^2 + b 
*e^2*x^2) - 2*atanh((18*b^3*c^4*e^2*(e*x)^(1/2)*(- (9*c^4*(-a^7*b^5)^(1/2) 
)/(64*a^7*b^3*e) - (d^4*(-a^7*b^5)^(1/2))/(64*a^5*b^5*e) - (c*d^3)/(16*a^2 
*b^2*e) - (3*c^3*d)/(16*a^3*b*e) - (c^2*d^2*(-a^7*b^5)^(1/2))/(32*a^6*b^4* 
e))^(1/2))/(b*c^3*d^3*e^2 + (a*c*d^5*e^2)/2 + (27*c^6*e^2*(-a^7*b^5)^(1/2) 
)/(4*a^5) + (d^6*e^2*(-a^7*b^5)^(1/2))/(4*a^2*b^3) + (9*b^2*c^5*d*e^2)/(2* 
a) + (5*c^2*d^4*e^2*(-a^7*b^5)^(1/2))/(4*a^3*b^2) + (15*c^4*d^2*e^2*(-a^7* 
b^5)^(1/2))/(4*a^4*b)) + (2*b*d^4*e^2*(e*x)^(1/2)*(- (9*c^4*(-a^7*b^5)^(1/ 
2))/(64*a^7*b^3*e) - (d^4*(-a^7*b^5)^(1/2))/(64*a^5*b^5*e) - (c*d^3)/(16*a 
^2*b^2*e) - (3*c^3*d)/(16*a^3*b*e) - (c^2*d^2*(-a^7*b^5)^(1/2))/(32*a^6*b^ 
4*e))^(1/2))/((c*d^5*e^2)/(2*a) + (27*c^6*e^2*(-a^7*b^5)^(1/2))/(4*a^7) + 
(d^6*e^2*(-a^7*b^5)^(1/2))/(4*a^4*b^3) + (b*c^3*d^3*e^2)/a^2 + (9*b^2*c^5* 
d*e^2)/(2*a^3) + (5*c^2*d^4*e^2*(-a^7*b^5)^(1/2))/(4*a^5*b^2) + (15*c^4*d^ 
2*e^2*(-a^7*b^5)^(1/2))/(4*a^6*b)) + (4*b^2*c^2*d^2*e^2*(e*x)^(1/2)*(- (9* 
c^4*(-a^7*b^5)^(1/2))/(64*a^7*b^3*e) - (d^4*(-a^7*b^5)^(1/2))/(64*a^5*b^5* 
e) - (c*d^3)/(16*a^2*b^2*e) - (3*c^3*d)/(16*a^3*b*e) - (c^2*d^2*(-a^7*b^5) 
^(1/2))/(32*a^6*b^4*e))^(1/2))/((c*d^5*e^2)/2 + (27*c^6*e^2*(-a^7*b^5)^(1/ 
2))/(4*a^6) + (d^6*e^2*(-a^7*b^5)^(1/2))/(4*a^3*b^3) + (b*c^3*d^3*e^2)/a + 
 (9*b^2*c^5*d*e^2)/(2*a^2) + (5*c^2*d^4*e^2*(-a^7*b^5)^(1/2))/(4*a^4*b^2) 
+ (15*c^4*d^2*e^2*(-a^7*b^5)^(1/2))/(4*a^5*b)))*(-(a^2*d^4*(-a^7*b^5)^(...
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 964, normalized size of antiderivative = 3.18 \[ \int \frac {(c+d x)^2}{\sqrt {e x} \left (a+b x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(e)*( - 4*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*d - 4*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*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 
**2*d**2 - 6*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*c**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*b*d**2*x**2 - 6*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* 
*2*c**2*x**2 + 4*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*d + 4*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*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**2*d**2 + 6*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*c**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*b*d**2*x**2 + 6*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)...