\(\int \frac {1}{\sqrt {e x} (c+d x)^2 (a+b x^2)^3} \, dx\) [455]

Optimal result

Integrand size = 24, antiderivative size = 746 \[ \int \frac {1}{\sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=-\frac {d^4 \left (b c^2-2 a d^2\right ) \sqrt {e x}}{2 a c \left (b c^2+a d^2\right )^3 e (c+d x)}+\frac {b \sqrt {e x} (c-d x)}{4 a \left (b c^2+a d^2\right ) e (c+d x) \left (a+b x^2\right )^2}-\frac {b \sqrt {e x} \left (15 a^2 d^4-2 a b c d^2 (8 c-17 d x)-b^2 c^3 (7 c-10 d x)\right )}{16 a^2 \left (b c^2+a d^2\right )^3 e \left (a+b x^2\right )}+\frac {d^{11/2} \left (13 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right )}{c^{3/2} \left (b c^2+a d^2\right )^4 \sqrt {e}}-\frac {b^{3/4} \left (21 b^3 c^6-10 \sqrt {a} b^{5/2} c^5 d+93 a b^2 c^4 d^2-52 a^{3/2} b^{3/2} c^3 d^3+187 a^2 b c^2 d^4-234 a^{5/2} \sqrt {b} c d^5-77 a^3 d^6\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^4 \sqrt {e}}+\frac {b^{3/4} \left (21 b^3 c^6-10 \sqrt {a} b^{5/2} c^5 d+93 a b^2 c^4 d^2-52 a^{3/2} b^{3/2} c^3 d^3+187 a^2 b c^2 d^4-234 a^{5/2} \sqrt {b} c d^5-77 a^3 d^6\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^4 \sqrt {e}}+\frac {b^{3/4} \left (21 b^3 c^6+10 \sqrt {a} b^{5/2} c^5 d+93 a b^2 c^4 d^2+52 a^{3/2} b^{3/2} c^3 d^3+187 a^2 b c^2 d^4+234 a^{5/2} \sqrt {b} c d^5-77 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}}{\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^4 \sqrt {e}} \] Output:

-1/2*d^4*(-2*a*d^2+b*c^2)*(e*x)^(1/2)/a/c/(a*d^2+b*c^2)^3/e/(d*x+c)+1/4*b* 
(e*x)^(1/2)*(-d*x+c)/a/(a*d^2+b*c^2)/e/(d*x+c)/(b*x^2+a)^2-1/16*b*(e*x)^(1 
/2)*(15*a^2*d^4-2*a*b*c*d^2*(-17*d*x+8*c)-b^2*c^3*(-10*d*x+7*c))/a^2/(a*d^ 
2+b*c^2)^3/e/(b*x^2+a)+d^(11/2)*(a*d^2+13*b*c^2)*arctan(d^(1/2)*(e*x)^(1/2 
)/c^(1/2)/e^(1/2))/c^(3/2)/(a*d^2+b*c^2)^4/e^(1/2)-1/64*b^(3/4)*(21*b^3*c^ 
6-10*a^(1/2)*b^(5/2)*c^5*d+93*a*b^2*c^4*d^2-52*a^(3/2)*b^(3/2)*c^3*d^3+187 
*a^2*b*c^2*d^4-234*a^(5/2)*b^(1/2)*c*d^5-77*a^3*d^6)*arctan(1-2^(1/2)*b^(1 
/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(11/4)/(a*d^2+b*c^2)^4/e^(1/2)+ 
1/64*b^(3/4)*(21*b^3*c^6-10*a^(1/2)*b^(5/2)*c^5*d+93*a*b^2*c^4*d^2-52*a^(3 
/2)*b^(3/2)*c^3*d^3+187*a^2*b*c^2*d^4-234*a^(5/2)*b^(1/2)*c*d^5-77*a^3*d^6 
)*arctan(1+2^(1/2)*b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))*2^(1/2)/a^(11/4)/( 
a*d^2+b*c^2)^4/e^(1/2)+1/64*b^(3/4)*(21*b^3*c^6+10*a^(1/2)*b^(5/2)*c^5*d+9 
3*a*b^2*c^4*d^2+52*a^(3/2)*b^(3/2)*c^3*d^3+187*a^2*b*c^2*d^4+234*a^(5/2)*b 
^(1/2)*c*d^5-77*a^3*d^6)*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^(11/4)/(a*d^2+b*c^2)^4/e^(1/2)
 

Mathematica [A] (verified)

Time = 3.77 (sec) , antiderivative size = 563, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-\frac {4 \left (b c^2+a d^2\right ) \sqrt {x} \left (-16 a^4 d^6+a^3 b d^4 \left (19 c^2+19 c d x-32 d^2 x^2\right )+b^4 c^4 x^2 \left (-7 c^2+3 c d x+10 d^2 x^2\right )+a^2 b^2 d^2 \left (-24 c^4+26 c^3 d x+65 c^2 d^2 x^2+15 c d^3 x^3-16 d^4 x^4\right )+a b^3 c^2 \left (-11 c^4+7 c^3 d x-6 c^2 d^2 x^2+18 c d^3 x^3+42 d^4 x^4\right )\right )}{a^2 c (c+d x) \left (a+b x^2\right )^2}+\frac {64 d^{11/2} \left (13 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {\sqrt {2} b^{3/4} \left (-21 b^3 c^6+10 \sqrt {a} b^{5/2} c^5 d-93 a b^2 c^4 d^2+52 a^{3/2} b^{3/2} c^3 d^3-187 a^2 b c^2 d^4+234 a^{5/2} \sqrt {b} c d^5+77 a^3 d^6\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{11/4}}+\frac {\sqrt {2} b^{3/4} \left (21 b^3 c^6+10 \sqrt {a} b^{5/2} c^5 d+93 a b^2 c^4 d^2+52 a^{3/2} b^{3/2} c^3 d^3+187 a^2 b c^2 d^4+234 a^{5/2} \sqrt {b} c d^5-77 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{11/4}}\right )}{64 \left (b c^2+a d^2\right )^4 \sqrt {e x}} \] Input:

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

Output:

(Sqrt[x]*((-4*(b*c^2 + a*d^2)*Sqrt[x]*(-16*a^4*d^6 + a^3*b*d^4*(19*c^2 + 1 
9*c*d*x - 32*d^2*x^2) + b^4*c^4*x^2*(-7*c^2 + 3*c*d*x + 10*d^2*x^2) + a^2* 
b^2*d^2*(-24*c^4 + 26*c^3*d*x + 65*c^2*d^2*x^2 + 15*c*d^3*x^3 - 16*d^4*x^4 
) + a*b^3*c^2*(-11*c^4 + 7*c^3*d*x - 6*c^2*d^2*x^2 + 18*c*d^3*x^3 + 42*d^4 
*x^4)))/(a^2*c*(c + d*x)*(a + b*x^2)^2) + (64*d^(11/2)*(13*b*c^2 + a*d^2)* 
ArcTan[(Sqrt[d]*Sqrt[x])/Sqrt[c]])/c^(3/2) + (Sqrt[2]*b^(3/4)*(-21*b^3*c^6 
 + 10*Sqrt[a]*b^(5/2)*c^5*d - 93*a*b^2*c^4*d^2 + 52*a^(3/2)*b^(3/2)*c^3*d^ 
3 - 187*a^2*b*c^2*d^4 + 234*a^(5/2)*Sqrt[b]*c*d^5 + 77*a^3*d^6)*ArcTan[(Sq 
rt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/a^(11/4) + (Sqrt[2] 
*b^(3/4)*(21*b^3*c^6 + 10*Sqrt[a]*b^(5/2)*c^5*d + 93*a*b^2*c^4*d^2 + 52*a^ 
(3/2)*b^(3/2)*c^3*d^3 + 187*a^2*b*c^2*d^4 + 234*a^(5/2)*Sqrt[b]*c*d^5 - 77 
*a^3*d^6)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)] 
)/a^(11/4)))/(64*(b*c^2 + a*d^2)^4*Sqrt[e*x])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1594\) vs. \(2(746)=1492\).

Time = 3.91 (sec) , antiderivative size = 1594, normalized size of antiderivative = 2.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {615, 2009}

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.

\(\displaystyle \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^3 (c+d x)^2} \, dx\)

\(\Big \downarrow \) 615

\(\displaystyle \int \left (-\frac {b d^2 \left (a d^2-3 b c^2+4 b c d x\right )}{\sqrt {e x} \left (a+b x^2\right )^2 \left (a d^2+b c^2\right )^3}+\frac {b \left (-a d^2+b c^2-2 b c d x\right )}{\sqrt {e x} \left (a+b x^2\right )^3 \left (a d^2+b c^2\right )^2}+\frac {6 b c d^6}{\sqrt {e x} (c+d x) \left (a d^2+b c^2\right )^4}+\frac {d^6}{\sqrt {e x} (c+d x)^2 \left (a d^2+b c^2\right )^3}-\frac {b d^4 \left (a d^2-5 b c^2+6 b c d x\right )}{\sqrt {e x} \left (a+b x^2\right ) \left (a d^2+b c^2\right )^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {e x} d^6}{c \left (b c^2+a d^2\right )^3 e (c+d x)}+\frac {\arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{11/2}}{c^{3/2} \left (b c^2+a d^2\right )^3 \sqrt {e}}+\frac {12 b \sqrt {c} \arctan \left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {c} \sqrt {e}}\right ) d^{11/2}}{\left (b c^2+a d^2\right )^4 \sqrt {e}}-\frac {b^{3/4} \left (5 b c^2-6 \sqrt {a} \sqrt {b} d c-a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^4}{\sqrt {2} a^{3/4} \left (b c^2+a d^2\right )^4 \sqrt {e}}+\frac {b^{3/4} \left (5 b c^2-6 \sqrt {a} \sqrt {b} d c-a d^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^4}{\sqrt {2} a^{3/4} \left (b c^2+a d^2\right )^4 \sqrt {e}}-\frac {b^{3/4} \left (5 b c^2+6 \sqrt {a} \sqrt {b} d c-a d^2\right ) \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^4}{2 \sqrt {2} a^{3/4} \left (b c^2+a d^2\right )^4 \sqrt {e}}+\frac {b^{3/4} \left (5 b c^2+6 \sqrt {a} \sqrt {b} d c-a d^2\right ) \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^4}{2 \sqrt {2} a^{3/4} \left (b c^2+a d^2\right )^4 \sqrt {e}}-\frac {b^{3/4} \left (9 b c^2-4 \sqrt {a} \sqrt {b} d c-3 a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ) d^2}{4 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}+\frac {b^{3/4} \left (9 b c^2-4 \sqrt {a} \sqrt {b} d c-3 a d^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right ) d^2}{4 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}-\frac {b^{3/4} \left (9 b c^2+4 \sqrt {a} \sqrt {b} d c-3 a d^2\right ) \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^2}{8 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}+\frac {b^{3/4} \left (9 b c^2+4 \sqrt {a} \sqrt {b} d c-3 a d^2\right ) \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right ) d^2}{8 \sqrt {2} a^{7/4} \left (b c^2+a d^2\right )^3 \sqrt {e}}+\frac {b \sqrt {e x} \left (3 b c^2-4 b d x c-a d^2\right ) d^2}{2 a \left (b c^2+a d^2\right )^3 e \left (b x^2+a\right )}-\frac {b^{3/4} \left (21 b c^2-10 \sqrt {a} \sqrt {b} d c-21 a d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^2 \sqrt {e}}+\frac {b^{3/4} \left (21 b c^2-10 \sqrt {a} \sqrt {b} d c-21 a d^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}+1\right )}{32 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^2 \sqrt {e}}-\frac {b^{3/4} \left (21 b c^2+10 \sqrt {a} \sqrt {b} d c-21 a d^2\right ) \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right )}{64 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^2 \sqrt {e}}+\frac {b^{3/4} \left (21 b c^2+10 \sqrt {a} \sqrt {b} d c-21 a d^2\right ) \log \left (\sqrt {b} \sqrt {e} x+\sqrt {a} \sqrt {e}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {e x}\right )}{64 \sqrt {2} a^{11/4} \left (b c^2+a d^2\right )^2 \sqrt {e}}+\frac {b \sqrt {e x} \left (7 \left (b c^2-a d^2\right )-10 b c d x\right )}{16 a^2 \left (b c^2+a d^2\right )^2 e \left (b x^2+a\right )}+\frac {b \sqrt {e x} \left (b c^2-2 b d x c-a d^2\right )}{4 a \left (b c^2+a d^2\right )^2 e \left (b x^2+a\right )^2}\)

Input:

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

Output:

(d^6*Sqrt[e*x])/(c*(b*c^2 + a*d^2)^3*e*(c + d*x)) + (b*Sqrt[e*x]*(b*c^2 - 
a*d^2 - 2*b*c*d*x))/(4*a*(b*c^2 + a*d^2)^2*e*(a + b*x^2)^2) + (b*Sqrt[e*x] 
*(7*(b*c^2 - a*d^2) - 10*b*c*d*x))/(16*a^2*(b*c^2 + a*d^2)^2*e*(a + b*x^2) 
) + (b*d^2*Sqrt[e*x]*(3*b*c^2 - a*d^2 - 4*b*c*d*x))/(2*a*(b*c^2 + a*d^2)^3 
*e*(a + b*x^2)) + (12*b*Sqrt[c]*d^(11/2)*ArcTan[(Sqrt[d]*Sqrt[e*x])/(Sqrt[ 
c]*Sqrt[e])])/((b*c^2 + a*d^2)^4*Sqrt[e]) + (d^(11/2)*ArcTan[(Sqrt[d]*Sqrt 
[e*x])/(Sqrt[c]*Sqrt[e])])/(c^(3/2)*(b*c^2 + a*d^2)^3*Sqrt[e]) - (b^(3/4)* 
d^4*(5*b*c^2 - 6*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^(3/4)*(b*c^2 + a*d^2)^4*Sqrt[e]) 
 - (b^(3/4)*d^2*(9*b*c^2 - 4*Sqrt[a]*Sqrt[b]*c*d - 3*a*d^2)*ArcTan[1 - (Sq 
rt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(4*Sqrt[2]*a^(7/4)*(b*c^2 + a 
*d^2)^3*Sqrt[e]) - (b^(3/4)*(21*b*c^2 - 10*Sqrt[a]*Sqrt[b]*c*d - 21*a*d^2) 
*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(32*Sqrt[2]*a^ 
(11/4)*(b*c^2 + a*d^2)^2*Sqrt[e]) + (b^(3/4)*d^4*(5*b*c^2 - 6*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^(3/4)*(b*c^2 + a*d^2)^4*Sqrt[e]) + (b^(3/4)*d^2*(9*b*c^2 - 4* 
Sqrt[a]*Sqrt[b]*c*d - 3*a*d^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[e*x])/(a^( 
1/4)*Sqrt[e])])/(4*Sqrt[2]*a^(7/4)*(b*c^2 + a*d^2)^3*Sqrt[e]) + (b^(3/4)*( 
21*b*c^2 - 10*Sqrt[a]*Sqrt[b]*c*d - 21*a*d^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)* 
Sqrt[e*x])/(a^(1/4)*Sqrt[e])])/(32*Sqrt[2]*a^(11/4)*(b*c^2 + a*d^2)^2*S...
 

Defintions of rubi rules used

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 688, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(2 e^{7} \left (\frac {d^{6} \left (\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {e x}}{2 c \left (d e x +c e \right )}+\frac {\left (a \,d^{2}+13 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 c \sqrt {d e c}}\right )}{e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{4}}-\frac {b \left (\frac {\frac {c d \,b^{2} \left (21 a^{2} d^{4}+26 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) \left (e x \right )^{\frac {7}{2}}}{16 a^{2}}+\frac {b e \left (15 a^{3} d^{6}-9 a^{2} b \,c^{2} d^{4}-31 a \,b^{2} c^{4} d^{2}-7 b^{3} c^{6}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a^{2}}+\frac {b c d \,e^{2} \left (25 a^{2} d^{4}+34 b \,c^{2} d^{2} a +9 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{16 a}+\frac {e^{3} \left (19 a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-35 a \,b^{2} c^{4} d^{2}-11 b^{3} c^{6}\right ) \sqrt {e x}}{32 a}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {\left (77 a^{3} d^{6} e -187 a^{2} b \,c^{2} d^{4} e -93 a \,b^{2} c^{4} d^{2} e -21 b^{3} c^{6} 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 {\left (234 b c \,d^{5} a^{2}+52 a \,b^{2} c^{3} d^{3}+10 c^{5} d \,b^{3}\right ) \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}}}}{32 a^{2}}\right )}{e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{4}}\right )\)	\(688\)
default	\(2 e^{7} \left (\frac {d^{6} \left (\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {e x}}{2 c \left (d e x +c e \right )}+\frac {\left (a \,d^{2}+13 b \,c^{2}\right ) \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )}{2 c \sqrt {d e c}}\right )}{e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{4}}-\frac {b \left (\frac {\frac {c d \,b^{2} \left (21 a^{2} d^{4}+26 b \,c^{2} d^{2} a +5 b^{2} c^{4}\right ) \left (e x \right )^{\frac {7}{2}}}{16 a^{2}}+\frac {b e \left (15 a^{3} d^{6}-9 a^{2} b \,c^{2} d^{4}-31 a \,b^{2} c^{4} d^{2}-7 b^{3} c^{6}\right ) \left (e x \right )^{\frac {5}{2}}}{32 a^{2}}+\frac {b c d \,e^{2} \left (25 a^{2} d^{4}+34 b \,c^{2} d^{2} a +9 b^{2} c^{4}\right ) \left (e x \right )^{\frac {3}{2}}}{16 a}+\frac {e^{3} \left (19 a^{3} d^{6}-5 a^{2} b \,c^{2} d^{4}-35 a \,b^{2} c^{4} d^{2}-11 b^{3} c^{6}\right ) \sqrt {e x}}{32 a}}{\left (b \,e^{2} x^{2}+a \,e^{2}\right )^{2}}+\frac {\frac {\left (77 a^{3} d^{6} e -187 a^{2} b \,c^{2} d^{4} e -93 a \,b^{2} c^{4} d^{2} e -21 b^{3} c^{6} 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 {\left (234 b c \,d^{5} a^{2}+52 a \,b^{2} c^{3} d^{3}+10 c^{5} d \,b^{3}\right ) \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}}}}{32 a^{2}}\right )}{e^{7} \left (a \,d^{2}+b \,c^{2}\right )^{4}}\right )\)	\(688\)
pseudoelliptic	\(-\frac {77 \left (\left (d x +c \right ) e^{3} b \sqrt {2}\, \left (b \,x^{2}+a \right )^{2} \left (a^{3} d^{6}-\frac {17}{7} a^{2} b \,c^{2} d^{4}-\frac {93}{77} a \,b^{2} c^{4} d^{2}-\frac {3}{11} b^{3} c^{6}\right ) c \sqrt {d e 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}}+\frac {468 \left (\frac {2 \left (\frac {50 d \left (d x +c \right ) e^{2} \left (a \,d^{2}+\frac {9 b \,c^{2}}{25}\right ) b^{2} \left (a \,d^{2}+b \,c^{2}\right ) a \,c^{2} \sqrt {d e c}\, \left (e x \right )^{\frac {3}{2}}}{3}+5 \left (d x +c \right ) e \left (a^{2} d^{4}-\frac {8}{5} b \,c^{2} d^{2} a -\frac {7}{15} b^{2} c^{4}\right ) b^{2} \left (a \,d^{2}+b \,c^{2}\right ) c \sqrt {d e c}\, \left (e x \right )^{\frac {5}{2}}+14 d \left (d x +c \right ) b^{3} \left (a \,d^{2}+b \,c^{2}\right ) c^{2} \left (a \,d^{2}+\frac {5 b \,c^{2}}{21}\right ) \sqrt {d e c}\, \left (e x \right )^{\frac {7}{2}}-\frac {16 e^{3} \left (\left (\frac {11 b^{3} c^{6}}{16}+\frac {11 b^{3} c^{5} d x}{16}+\frac {3 a \,b^{2} c^{4} d^{2}}{2}+\frac {3 a \,b^{2} c^{3} d^{3} x}{2}-\frac {19 a^{2} b \,c^{2} d^{4}}{16}-\frac {19 a^{2} b c \,d^{5} x}{16}+a \,d^{6} \left (b \,x^{2}+a \right )^{2}\right ) \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {d e c}\, \sqrt {e x}+\left (a \,d^{2}+13 b \,c^{2}\right ) d^{6} \left (d x +c \right ) e \left (b \,x^{2}+a \right )^{2} a \arctan \left (\frac {d \sqrt {e x}}{\sqrt {d e c}}\right )\right ) a}{3}\right ) \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}}}{39}+d \left (d x +c \right ) e^{4} \left (a^{2} d^{4}+\frac {2}{9} b \,c^{2} d^{2} a +\frac {5}{117} b^{2} c^{4}\right ) b \sqrt {2}\, \left (b \,x^{2}+a \right )^{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 )+\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 )}{2}\right ) c^{2} \sqrt {d e c}\right ) a}{77}\right )}{128 \left (\frac {a \,e^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d e c}\, e^{4} a^{3} \left (a \,d^{2}+b \,c^{2}\right )^{4} \left (b \,x^{2}+a \right )^{2} c \left (d x +c \right )}\)	\(767\)

Input:

int(1/(e*x)^(1/2)/(d*x+c)^2/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2*e^7*(d^6/e^7/(a*d^2+b*c^2)^4*(1/2*(a*d^2+b*c^2)/c*(e*x)^(1/2)/(d*e*x+c*e 
)+1/2*(a*d^2+13*b*c^2)/c/(d*e*c)^(1/2)*arctan(d*(e*x)^(1/2)/(d*e*c)^(1/2)) 
)-b/e^7/(a*d^2+b*c^2)^4*((1/16*c*d*b^2*(21*a^2*d^4+26*a*b*c^2*d^2+5*b^2*c^ 
4)/a^2*(e*x)^(7/2)+1/32*b*e*(15*a^3*d^6-9*a^2*b*c^2*d^4-31*a*b^2*c^4*d^2-7 
*b^3*c^6)/a^2*(e*x)^(5/2)+1/16*b*c*d*e^2*(25*a^2*d^4+34*a*b*c^2*d^2+9*b^2* 
c^4)/a*(e*x)^(3/2)+1/32*e^3*(19*a^3*d^6-5*a^2*b*c^2*d^4-35*a*b^2*c^4*d^2-1 
1*b^3*c^6)/a*(e*x)^(1/2))/(b*e^2*x^2+a*e^2)^2+1/32/a^2*(1/8*(77*a^3*d^6*e- 
187*a^2*b*c^2*d^4*e-93*a*b^2*c^4*d^2*e-21*b^3*c^6*e)*(a*e^2/b)^(1/4)/a/e^2 
*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*(234*a^2*b*c*d^5+52*a*b^2*c^3*d^3+10*b^3*c^5*d)/b/(a*e^2/b)^(1/4) 
*2^(1/2)*(ln((e*x-(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2))/(e* 
x+(a*e^2/b)^(1/4)*(e*x)^(1/2)*2^(1/2)+(a*e^2/b)^(1/2)))+2*arctan(2^(1/2)/( 
a*e^2/b)^(1/4)*(e*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*e^2/b)^(1/4)*(e*x)^(1/2) 
-1)))))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)^(1/2)/(d*x+c)^2/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x)**(1/2)/(d*x+c)**2/(b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x)^(1/2)/(d*x+c)^2/(b*x^2+a)^3,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. 1529 vs. \(2 (620) = 1240\).

Time = 0.25 (sec) , antiderivative size = 1529, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x)^(1/2)/(d*x+c)^2/(b*x^2+a)^3,x, algorithm="giac")
 

Output:

sqrt(e*x)*d^6/((b^3*c^7 + 3*a*b^2*c^5*d^2 + 3*a^2*b*c^3*d^4 + a^3*c*d^6)*( 
d*e*x + c*e)) + 1/32*(21*(a*b^3*e^2)^(1/4)*b^4*c^6*e + 93*(a*b^3*e^2)^(1/4 
)*a*b^3*c^4*d^2*e + 187*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^4*e - 77*(a*b^3*e^ 
2)^(1/4)*a^3*b*d^6*e - 10*(a*b^3*e^2)^(3/4)*b^2*c^5*d - 52*(a*b^3*e^2)^(3/ 
4)*a*b*c^3*d^3 - 234*(a*b^3*e^2)^(3/4)*a^2*c*d^5)*arctan(1/2*sqrt(2)*(sqrt 
(2)*(a*e^2/b)^(1/4) + 2*sqrt(e*x))/(a*e^2/b)^(1/4))/(sqrt(2)*a^3*b^5*c^8*e 
^2 + 4*sqrt(2)*a^4*b^4*c^6*d^2*e^2 + 6*sqrt(2)*a^5*b^3*c^4*d^4*e^2 + 4*sqr 
t(2)*a^6*b^2*c^2*d^6*e^2 + sqrt(2)*a^7*b*d^8*e^2) + 1/32*(21*(a*b^3*e^2)^( 
1/4)*b^4*c^6*e + 93*(a*b^3*e^2)^(1/4)*a*b^3*c^4*d^2*e + 187*(a*b^3*e^2)^(1 
/4)*a^2*b^2*c^2*d^4*e - 77*(a*b^3*e^2)^(1/4)*a^3*b*d^6*e - 10*(a*b^3*e^2)^ 
(3/4)*b^2*c^5*d - 52*(a*b^3*e^2)^(3/4)*a*b*c^3*d^3 - 234*(a*b^3*e^2)^(3/4) 
*a^2*c*d^5)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*e^2/b)^(1/4) - 2*sqrt(e*x))/(a 
*e^2/b)^(1/4))/(sqrt(2)*a^3*b^5*c^8*e^2 + 4*sqrt(2)*a^4*b^4*c^6*d^2*e^2 + 
6*sqrt(2)*a^5*b^3*c^4*d^4*e^2 + 4*sqrt(2)*a^6*b^2*c^2*d^6*e^2 + sqrt(2)*a^ 
7*b*d^8*e^2) + 1/64*(21*(a*b^3*e^2)^(1/4)*b^4*c^6*e + 93*(a*b^3*e^2)^(1/4) 
*a*b^3*c^4*d^2*e + 187*(a*b^3*e^2)^(1/4)*a^2*b^2*c^2*d^4*e - 77*(a*b^3*e^2 
)^(1/4)*a^3*b*d^6*e + 10*(a*b^3*e^2)^(3/4)*b^2*c^5*d + 52*(a*b^3*e^2)^(3/4 
)*a*b*c^3*d^3 + 234*(a*b^3*e^2)^(3/4)*a^2*c*d^5)*log(e*x + sqrt(2)*(a*e^2/ 
b)^(1/4)*sqrt(e*x) + sqrt(a*e^2/b))/(sqrt(2)*a^3*b^5*c^8*e^2 + 4*sqrt(2)*a 
^4*b^4*c^6*d^2*e^2 + 6*sqrt(2)*a^5*b^3*c^4*d^4*e^2 + 4*sqrt(2)*a^6*b^2*...
 

Mupad [B] (verification not implemented)

Time = 12.40 (sec) , antiderivative size = 8820, normalized size of antiderivative = 11.82 \[ \int \frac {1}{\sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int(1/((e*x)^(1/2)*(a + b*x^2)^3*(c + d*x)^2),x)
 

Output:

symsum(log((2528253*b^14*c^15*d^10*e^8 + 151476601*a^2*b^12*c^11*d^14*e^8 
+ 357116355*a^3*b^11*c^9*d^16*e^8 + 287856551*a^4*b^10*c^7*d^18*e^8 - 7964 
1683*a^5*b^9*c^5*d^20*e^8 + 350658723*a^6*b^8*c^3*d^22*e^8 + 30957759*a*b^ 
13*c^13*d^12*e^8 + 27563921*a^7*b^7*c*d^24*e^8)/(65536*(a^8*b^12*c^26 + a^ 
20*c^2*d^24 + 12*a^19*b*c^4*d^22 + 12*a^9*b^11*c^24*d^2 + 66*a^10*b^10*c^2 
2*d^4 + 220*a^11*b^9*c^20*d^6 + 495*a^12*b^8*c^18*d^8 + 792*a^13*b^7*c^16* 
d^10 + 924*a^14*b^6*c^14*d^12 + 792*a^15*b^5*c^12*d^14 + 495*a^16*b^4*c^10 
*d^16 + 220*a^17*b^3*c^8*d^18 + 66*a^18*b^2*c^6*d^20)) - root(1073741824*a 
^26*b*c^5*d^30*e^3*g^6 + 8053063680*a^25*b^2*c^7*d^28*e^3*g^6 + 8053063680 
*a^13*b^14*c^31*d^4*e^3*g^6 + 37580963840*a^24*b^3*c^9*d^26*e^3*g^6 + 7677 
25404160*a^20*b^7*c^17*d^18*e^3*g^6 + 767725404160*a^18*b^9*c^21*d^14*e^3* 
g^6 + 37580963840*a^14*b^13*c^29*d^6*e^3*g^6 + 122138132480*a^23*b^4*c^11* 
d^24*e^3*g^6 + 122138132480*a^15*b^12*c^27*d^8*e^3*g^6 + 293131517952*a^22 
*b^5*c^13*d^22*e^3*g^6 + 293131517952*a^16*b^11*c^25*d^10*e^3*g^6 + 107374 
1824*a^12*b^15*c^33*d^2*e^3*g^6 + 537407782912*a^21*b^6*c^15*d^20*e^3*g^6 
+ 537407782912*a^17*b^10*c^23*d^12*e^3*g^6 + 863691079680*a^19*b^8*c^19*d^ 
16*e^3*g^6 + 67108864*a^27*c^3*d^32*e^3*g^6 + 67108864*a^11*b^16*c^35*e^3* 
g^6 + 42677436416*a^18*b^3*c^6*d^25*e^2*g^4 - 12586319872*a^9*b^12*c^24*d^ 
7*e^2*g^4 - 214224601088*a^12*b^9*c^18*d^13*e^2*g^4 + 7975600128*a^19*b^2* 
c^4*d^27*e^2*g^4 + 115199442944*a^17*b^4*c^8*d^23*e^2*g^4 + 16656551116...
 

Reduce [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 8127, normalized size of antiderivative = 10.89 \[ \int \frac {1}{\sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(1/(e*x)^(1/2)/(d*x+c)^2/(b*x^2+a)^3,x)
 

Output:

(sqrt(e)*(468*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**4*b*c**4*d**5 + 468*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**4*b*c**3*d**6*x + 104*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**3*b**2*c**6*d**3 + 104*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**3*b**2*c**5*d**4*x + 936*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**3*b**2* 
c**4*d**5*x**2 + 936*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr 
t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3*b**2*c**3*d**6 
*x**3 + 20*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*s 
qrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**3*c**8*d + 20*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**2*b**3*c**7*d**2*x + 208*b**(1/4)*a**(3/4)*s 
qrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**( 
1/4)*sqrt(2)))*a**2*b**3*c**6*d**3*x**2 + 208*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)))*a**2*b**3*c**5*d**4*x**3 + 468*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)))*...