\(\int \frac {(a+b x^2)^2}{\sqrt {x} (c+d x^2)^2} \, dx\) [751]

Optimal result

Integrand size = 24, antiderivative size = 242 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {2 b^2 \sqrt {x}}{d^2}+\frac {(b c-a d)^2 \sqrt {x}}{2 c d^2 \left (c+d x^2\right )}+\frac {(b c-a d) (5 b c+3 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}}-\frac {(b c-a d) (5 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{4 \sqrt {2} c^{7/4} d^{9/4}} \] Output:

2*b^2*x^(1/2)/d^2+1/2*(-a*d+b*c)^2*x^(1/2)/c/d^2/(d*x^2+c)+1/8*(-a*d+b*c)* 
(3*a*d+5*b*c)*arctan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(7/4)/d^ 
(9/4)-1/8*(-a*d+b*c)*(3*a*d+5*b*c)*arctan(1+2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4 
))*2^(1/2)/c^(7/4)/d^(9/4)-1/8*(-a*d+b*c)*(3*a*d+5*b*c)*arctanh(2^(1/2)*c^ 
(1/4)*d^(1/4)*x^(1/2)/(c^(1/2)+d^(1/2)*x))*2^(1/2)/c^(7/4)/d^(9/4)
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {\frac {4 c^{3/4} \sqrt [4]{d} \sqrt {x} \left (-2 a b c d+a^2 d^2+b^2 c \left (5 c+4 d x^2\right )\right )}{c+d x^2}+\sqrt {2} \left (5 b^2 c^2-2 a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt {2} \left (5 b^2 c^2-2 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{8 c^{7/4} d^{9/4}} \] Input:

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

Output:

((4*c^(3/4)*d^(1/4)*Sqrt[x]*(-2*a*b*c*d + a^2*d^2 + b^2*c*(5*c + 4*d*x^2)) 
)/(c + d*x^2) + Sqrt[2]*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[c 
] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])] - Sqrt[2]*(5*b^2*c^2 - 2 
*a*b*c*d - 3*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + 
 Sqrt[d]*x)])/(8*c^(7/4)*d^(9/4))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {366, 27, 363, 266, 755, 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.

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

\(\Big \downarrow \) 366

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 363

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {(b c-a d) (3 a d+5 b c) \int \frac {1}{\sqrt {x} \left (d x^2+c\right )}dx-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \int \frac {1}{d x^2+c}d\sqrt {x}-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}\right )-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {c}}\right )-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \left (\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {c}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-8 b^2 c \sqrt {x}}{4 c d^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\sqrt {x} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}-\frac {2 (b c-a d) (3 a d+5 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {c}}\right )-8 b^2 c \sqrt {x}}{4 c d^2}\)

Input:

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

Output:

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

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 366

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, 
x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* 
b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1))   Int[(e*x)^m*(a + b*x^2)^(p 
 + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p 
, -1]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

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]
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.69

method	result	size
risch	\(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\left (2 a d -2 b c \right ) \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 c \left (x^{2} d +c \right )}+\frac {\left (3 a d +5 b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c^{2}}\right )}{d^{2}}\)	\(166\)
derivativedivides	\(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{2 c \left (x^{2} d +c \right )}+\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}}{d^{2}}\)	\(185\)
default	\(\frac {2 b^{2} \sqrt {x}}{d^{2}}+\frac {\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {x}}{2 c \left (x^{2} d +c \right )}+\frac {\left (3 a^{2} d^{2}+2 a b c d -5 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{16 c^{2}}}{d^{2}}\)	\(185\)

Input:

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

Output:

2*b^2*x^(1/2)/d^2+1/d^2*(2*a*d-2*b*c)*(1/4*(a*d-b*c)/c*x^(1/2)/(d*x^2+c)+1 
/32*(3*a*d+5*b*c)/c^2*(c/d)^(1/4)*2^(1/2)*(ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/ 
2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1 
/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 1220, normalized size of antiderivative = 5.04 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*((c*d^3*x^2 + c^2*d^2)*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6 
*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^ 
5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)*l 
og(c^2*d^2*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640* 
a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2* 
c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4) - (5*b^2*c^2 - 2* 
a*b*c*d - 3*a^2*d^2)*sqrt(x)) - (-I*c*d^3*x^2 - I*c^2*d^2)*(-(625*b^8*c^8 
- 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4* 
b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 
+ 81*a^8*d^8)/(c^7*d^9))^(1/4)*log(I*c^2*d^2*(-(625*b^8*c^8 - 1000*a*b^7*c 
^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 
984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/ 
(c^7*d^9))^(1/4) - (5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*sqrt(x)) - (I*c*d^3 
*x^2 + I*c^2*d^2)*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 
+ 1640*a^3*b^5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a 
^6*b^2*c^2*d^6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4)*log(-I*c^2 
*d^2*(-(625*b^8*c^8 - 1000*a*b^7*c^7*d - 900*a^2*b^6*c^6*d^2 + 1640*a^3*b^ 
5*c^5*d^3 + 646*a^4*b^4*c^4*d^4 - 984*a^5*b^3*c^3*d^5 - 324*a^6*b^2*c^2*d^ 
6 + 216*a^7*b*c*d^7 + 81*a^8*d^8)/(c^7*d^9))^(1/4) - (5*b^2*c^2 - 2*a*b*c* 
d - 3*a^2*d^2)*sqrt(x)) - (c*d^3*x^2 + c^2*d^2)*(-(625*b^8*c^8 - 1000*a...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1248 vs. \(2 (228) = 456\).

Time = 22.87 (sec) , antiderivative size = 1248, normalized size of antiderivative = 5.16 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((zoo*(-2*a**2/(7*x**(7/2)) - 4*a*b/(3*x**(3/2)) + 2*b**2*sqrt(x) 
), Eq(c, 0) & Eq(d, 0)), ((2*a**2*sqrt(x) + 4*a*b*x**(5/2)/5 + 2*b**2*x**( 
9/2)/9)/c**2, Eq(d, 0)), ((-2*a**2/(7*x**(7/2)) - 4*a*b/(3*x**(3/2)) + 2*b 
**2*sqrt(x))/d**2, Eq(c, 0)), (4*a**2*c*d**2*sqrt(x)/(8*c**3*d**2 + 8*c**2 
*d**3*x**2) - 3*a**2*c*d**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(8* 
c**3*d**2 + 8*c**2*d**3*x**2) + 3*a**2*c*d**2*(-c/d)**(1/4)*log(sqrt(x) + 
(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 6*a**2*c*d**2*(-c/d)**(1 
/4)*atan(sqrt(x)/(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 3*a**2* 
d**3*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2 
*d**3*x**2) + 3*a**2*d**3*x**2*(-c/d)**(1/4)*log(sqrt(x) + (-c/d)**(1/4))/ 
(8*c**3*d**2 + 8*c**2*d**3*x**2) + 6*a**2*d**3*x**2*(-c/d)**(1/4)*atan(sqr 
t(x)/(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 8*a*b*c**2*d*sqrt(x 
)/(8*c**3*d**2 + 8*c**2*d**3*x**2) - 2*a*b*c**2*d*(-c/d)**(1/4)*log(sqrt(x 
) - (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 2*a*b*c**2*d*(-c/d)* 
*(1/4)*log(sqrt(x) + (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 4*a 
*b*c**2*d*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2* 
d**3*x**2) - 2*a*b*c*d**2*x**2*(-c/d)**(1/4)*log(sqrt(x) - (-c/d)**(1/4))/ 
(8*c**3*d**2 + 8*c**2*d**3*x**2) + 2*a*b*c*d**2*x**2*(-c/d)**(1/4)*log(sqr 
t(x) + (-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x**2) + 4*a*b*c*d**2*x**2 
*(-c/d)**(1/4)*atan(sqrt(x)/(-c/d)**(1/4))/(8*c**3*d**2 + 8*c**2*d**3*x...
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {x}}{2 \, {\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac {2 \, b^{2} \sqrt {x}}{d^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (5 \, b^{2} c^{2} - 2 \, a b c d - 3 \, a^{2} d^{2}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{16 \, c d^{2}} \] Input:

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

Output:

1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)/(c*d^3*x^2 + c^2*d^2) + 2*b^2* 
sqrt(x)/d^2 - 1/16*(2*sqrt(2)*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*arctan(1 
/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt 
(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(5*b^2*c^2 - 2*a*b*c*d - 
 3*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt( 
x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(5*b^ 
2*c^2 - 2*a*b*c*d - 3*a^2*d^2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt( 
d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(5*b^2*c^2 - 2*a*b*c*d - 3*a^2 
*d^2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4) 
*d^(1/4)))/(c*d^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 388 vs. \(2 (184) = 368\).

Time = 0.14 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\frac {2 \, b^{2} \sqrt {x}}{d^{2}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{2} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{8 \, c^{2} d^{3}} - \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{2} d^{3}} + \frac {\sqrt {2} {\left (5 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{16 \, c^{2} d^{3}} + \frac {b^{2} c^{2} \sqrt {x} - 2 \, a b c d \sqrt {x} + a^{2} d^{2} \sqrt {x}}{2 \, {\left (d x^{2} + c\right )} c d^{2}} \] Input:

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

Output:

2*b^2*sqrt(x)/d^2 - 1/8*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4) 
*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4 
) + 2*sqrt(x))/(c/d)^(1/4))/(c^2*d^3) - 1/8*sqrt(2)*(5*(c*d^3)^(1/4)*b^2*c 
^2 - 2*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2 
)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^2*d^3) - 1/16*sqrt(2)* 
(5*(c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a^2*d 
^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^3) + 1/16*sqrt 
(2)*(5*(c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d - 3*(c*d^3)^(1/4)*a 
^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^3) + 1/2* 
(b^2*c^2*sqrt(x) - 2*a*b*c*d*sqrt(x) + a^2*d^2*sqrt(x))/((d*x^2 + c)*c*d^2 
)
 

Mupad [B] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 1267, normalized size of antiderivative = 5.24 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*b^2*x^(1/2))/d^2 + (x^(1/2)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(2*c*(c*d^ 
2 + d^3*x^2)) + (atan(((((x^(1/2)*(9*a^4*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2 
*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) - ((a*d - b*c)*(3*a*d + 5 
*b*c)*(24*a^2*d^3 - 40*b^2*c^2*d + 16*a*b*c*d^2))/(8*(-c)^(7/4)*d^(9/4)))* 
(a*d - b*c)*(3*a*d + 5*b*c)*1i)/(8*(-c)^(7/4)*d^(9/4)) + (((x^(1/2)*(9*a^4 
*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3)) 
/(c^2*d) + ((a*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d^3 - 40*b^2*c^2*d + 16*a* 
b*c*d^2))/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b*c)*1i)/(8*(-c)^ 
(7/4)*d^(9/4)))/((((x^(1/2)*(9*a^4*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 
 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) - ((a*d - b*c)*(3*a*d + 5*b*c)* 
(24*a^2*d^3 - 40*b^2*c^2*d + 16*a*b*c*d^2))/(8*(-c)^(7/4)*d^(9/4)))*(a*d - 
 b*c)*(3*a*d + 5*b*c))/(8*(-c)^(7/4)*d^(9/4)) - (((x^(1/2)*(9*a^4*d^4 + 25 
*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a^3*b*c*d^3))/(c^2*d) 
+ ((a*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d^3 - 40*b^2*c^2*d + 16*a*b*c*d^2)) 
/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b*c))/(8*(-c)^(7/4)*d^(9/4 
))))*(a*d - b*c)*(3*a*d + 5*b*c)*1i)/(4*(-c)^(7/4)*d^(9/4)) + (atan(((((x^ 
(1/2)*(9*a^4*d^4 + 25*b^4*c^4 - 26*a^2*b^2*c^2*d^2 - 20*a*b^3*c^3*d + 12*a 
^3*b*c*d^3))/(c^2*d) - ((a*d - b*c)*(3*a*d + 5*b*c)*(24*a^2*d^3 - 40*b^2*c 
^2*d + 16*a*b*c*d^2)*1i)/(8*(-c)^(7/4)*d^(9/4)))*(a*d - b*c)*(3*a*d + 5*b* 
c))/(8*(-c)^(7/4)*d^(9/4)) + (((x^(1/2)*(9*a^4*d^4 + 25*b^4*c^4 - 26*a^...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 978, normalized size of antiderivative = 4.04 \[ \int \frac {\left (a+b x^2\right )^2}{\sqrt {x} \left (c+d x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x 
)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*c*d**2 - 6*d**(3/4)*c**(1/4)* 
sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/4)*c** 
(1/4)*sqrt(2)))*a**2*d**3*x**2 - 4*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4 
)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c 
**2*d - 4*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sq 
rt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c*d**2*x**2 + 10*d**(3/4)* 
c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d** 
(1/4)*c**(1/4)*sqrt(2)))*b**2*c**3 + 10*d**(3/4)*c**(1/4)*sqrt(2)*atan((d* 
*(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))* 
b**2*c**2*d*x**2 + 6*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqr 
t(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*c*d**2 + 6*d** 
(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(d) 
)/(d**(1/4)*c**(1/4)*sqrt(2)))*a**2*d**3*x**2 + 4*d**(3/4)*c**(1/4)*sqrt(2 
)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)* 
sqrt(2)))*a*b*c**2*d + 4*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4) 
*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c*d**2*x**2 
 - 10*d**(3/4)*c**(1/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x 
)*sqrt(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*b**2*c**3 - 10*d**(3/4)*c**(1/4)*s 
qrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(d))/(d**(1/4)*c...