\(\int \frac {x^{5/2} (a+b x^2)^2}{(c+d x^2)^3} \, dx\) [756]

Optimal result

Integrand size = 24, antiderivative size = 300 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {2 b^2 x^{3/2}}{3 d^3}-\frac {(b c-a d)^2 x^{3/2}}{4 d^3 \left (c+d x^2\right )^2}+\frac {(19 b c-3 a d) (b c-a d) x^{3/2}}{16 c d^3 \left (c+d x^2\right )}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}-\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}}+\frac {\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{32 \sqrt {2} c^{5/4} d^{15/4}} \] Output:

2/3*b^2*x^(3/2)/d^3-1/4*(-a*d+b*c)^2*x^(3/2)/d^3/(d*x^2+c)^2+1/16*(-3*a*d+ 
19*b*c)*(-a*d+b*c)*x^(3/2)/c/d^3/(d*x^2+c)+1/64*(-3*a^2*d^2-42*a*b*c*d+77* 
b^2*c^2)*arctan(1-2^(1/2)*d^(1/4)*x^(1/2)/c^(1/4))*2^(1/2)/c^(5/4)/d^(15/4 
)-1/64*(-3*a^2*d^2-42*a*b*c*d+77*b^2*c^2)*arctan(1+2^(1/2)*d^(1/4)*x^(1/2) 
/c^(1/4))*2^(1/2)/c^(5/4)/d^(15/4)+1/64*(-3*a^2*d^2-42*a*b*c*d+77*b^2*c^2) 
*arctanh(2^(1/2)*c^(1/4)*d^(1/4)*x^(1/2)/(c^(1/2)+d^(1/2)*x))*2^(1/2)/c^(5 
/4)/d^(15/4)
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.78 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{c} d^{3/4} x^{3/2} \left (3 a^2 d^2 \left (-c+3 d x^2\right )-6 a b c d \left (7 c+11 d x^2\right )+b^2 c \left (77 c^2+121 c d x^2+32 d^2 x^4\right )\right )}{\left (c+d x^2\right )^2}+3 \sqrt {2} \left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )+3 \sqrt {2} \left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{192 c^{5/4} d^{15/4}} \] Input:

Integrate[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]
 

Output:

((4*c^(1/4)*d^(3/4)*x^(3/2)*(3*a^2*d^2*(-c + 3*d*x^2) - 6*a*b*c*d*(7*c + 1 
1*d*x^2) + b^2*c*(77*c^2 + 121*c*d*x^2 + 32*d^2*x^4)))/(c + d*x^2)^2 + 3*S 
qrt[2]*(77*b^2*c^2 - 42*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])] + 3*Sqrt[2]*(77*b^2*c^2 - 42*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) 
])/(192*c^(5/4)*d^(15/4))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {366, 27, 362, 262, 266, 826, 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.

\(\displaystyle \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx\)

\(\Big \downarrow \) 366

\(\displaystyle \frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac {\int -\frac {x^{5/2} \left (8 a^2 d^2+8 b^2 c x^2 d-7 (b c-a d)^2\right )}{2 \left (d x^2+c\right )^2}dx}{4 c d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {x^{5/2} \left (8 a^2 d^2+8 b^2 c x^2 d-7 (b c-a d)^2\right )}{\left (d x^2+c\right )^2}dx}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 362

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \int \frac {x^{5/2}}{d x^2+c}dx}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {c \int \frac {\sqrt {x}}{d x^2+c}dx}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \int \frac {x}{d x^2+c}d\sqrt {x}}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\int \frac {\sqrt {d} x+\sqrt {c}}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\int \frac {\sqrt {c}-\sqrt {d} x}{d x^2+c}d\sqrt {x}}{2 \sqrt {d}}\right )}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{d} \left (x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}\right )}{\sqrt [4]{d} \left (x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{c}-2 \sqrt [4]{d} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{c} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}+\sqrt [4]{c}}{x+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{d}}+\frac {\sqrt {c}}{\sqrt {d}}}d\sqrt {x}}{2 \sqrt [4]{c} \sqrt {d}}}{2 \sqrt {d}}\right )}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \left (\frac {2 x^{3/2}}{3 d}-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} \sqrt [4]{d}}}{2 \sqrt {d}}\right )}{d}\right )}{4 c}-\frac {x^{7/2} (b c-a d) (a d+15 b c)}{2 c \left (c+d x^2\right )}}{8 c d^2}+\frac {x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2}\)

Input:

Int[(x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x]
 

Output:

((b*c - a*d)^2*x^(7/2))/(4*c*d^2*(c + d*x^2)^2) + (-1/2*((b*c - a*d)*(15*b 
*c + a*d)*x^(7/2))/(c*(c + d*x^2)) + ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2 
)*((2*x^(3/2))/(3*d) - (2*c*((-(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[d]) - (-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[d])))/d))/(4*c))/(8*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 262

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

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 362

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e 
*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1))   I 
nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N 
eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || 
  !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
 

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 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   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 = 0.56 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{3}}+\frac {\frac {2 \left (\frac {d \left (3 a^{2} d^{2}-22 a b c d +19 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c}+\left (-\frac {1}{32} a^{2} d^{2}-\frac {7}{16} a b c d +\frac {15}{32} b^{2} c^{2}\right ) x^{\frac {3}{2}}\right )}{\left (x^{2} d +c \right )^{2}}+\frac {\left (3 a^{2} d^{2}+42 a b c d -77 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{3}}\)	\(220\)
default	\(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{3}}+\frac {\frac {2 \left (\frac {d \left (3 a^{2} d^{2}-22 a b c d +19 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c}+\left (-\frac {1}{32} a^{2} d^{2}-\frac {7}{16} a b c d +\frac {15}{32} b^{2} c^{2}\right ) x^{\frac {3}{2}}\right )}{\left (x^{2} d +c \right )^{2}}+\frac {\left (3 a^{2} d^{2}+42 a b c d -77 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{3}}\)	\(220\)
risch	\(\frac {2 b^{2} x^{\frac {3}{2}}}{3 d^{3}}+\frac {\frac {\frac {d \left (3 a^{2} d^{2}-22 a b c d +19 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{16 c}+2 \left (-\frac {1}{32} a^{2} d^{2}-\frac {7}{16} a b c d +\frac {15}{32} b^{2} c^{2}\right ) x^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{2}}+\frac {\left (3 a^{2} d^{2}+42 a b c d -77 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{128 c d \left (\frac {c}{d}\right )^{\frac {1}{4}}}}{d^{3}}\)	\(220\)

Input:

int(x^(5/2)*(b*x^2+a)^2/(d*x^2+c)^3,x,method=_RETURNVERBOSE)
 

Output:

2/3*b^2*x^(3/2)/d^3+2/d^3*((1/32*d*(3*a^2*d^2-22*a*b*c*d+19*b^2*c^2)/c*x^( 
7/2)+(-1/32*a^2*d^2-7/16*a*b*c*d+15/32*b^2*c^2)*x^(3/2))/(d*x^2+c)^2+1/256 
*(3*a^2*d^2+42*a*b*c*d-77*b^2*c^2)/c/d/(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 = 0.12 (sec) , antiderivative size = 1539, normalized size of antiderivative = 5.13 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^(5/2)*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="fricas")
 

Output:

1/192*(3*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697 
544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 14 
57946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4 
536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*log(c^4*d^11*(-(35153041*b 
^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^ 
5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b 
^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(3/4) - (456533*b^ 
6*c^6 - 747054*a*b^5*c^5*d + 354123*a^2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^ 
3 - 13797*a^4*b^2*c^2*d^4 - 1134*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) - 3*(I 
*c*d^5*x^4 + 2*I*c^2*d^4*x^2 + I*c^3*d^3)*(-(35153041*b^8*c^8 - 76697544*a 
*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946 
*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a 
^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*log(I*c^4*d^11*(-(35153041*b^8* 
c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c 
^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2* 
c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(3/4) - (456533*b^6*c 
^6 - 747054*a*b^5*c^5*d + 354123*a^2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^3 - 
 13797*a^4*b^2*c^2*d^4 - 1134*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) - 3*(-I*c 
*d^5*x^4 - 2*I*c^2*d^4*x^2 - I*c^3*d^3)*(-(35153041*b^8*c^8 - 76697544*a*b 
^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 145794...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**(5/2)*(b*x**2+a)**2/(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.02 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{3}} + \frac {{\left (19 \, b^{2} c^{2} d - 22 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{\frac {7}{2}} + {\left (15 \, b^{2} c^{3} - 14 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{\frac {3}{2}}}{16 \, {\left (c d^{5} x^{4} + 2 \, c^{2} d^{4} x^{2} + c^{3} d^{3}\right )}} - \frac {{\left (77 \, b^{2} c^{2} - 42 \, a b c d - 3 \, a^{2} d^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{128 \, c d^{3}} \] Input:

integrate(x^(5/2)*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

2/3*b^2*x^(3/2)/d^3 + 1/16*((19*b^2*c^2*d - 22*a*b*c*d^2 + 3*a^2*d^3)*x^(7 
/2) + (15*b^2*c^3 - 14*a*b*c^2*d - a^2*c*d^2)*x^(3/2))/(c*d^5*x^4 + 2*c^2* 
d^4*x^2 + c^3*d^3) - 1/128*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*(2*sqrt(2 
)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sq 
rt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sq 
rt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d))) 
/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqr 
t(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/ 
4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/(c*d^3)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.42 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {2 \, b^{2} x^{\frac {3}{2}}}{3 \, d^{3}} + \frac {19 \, b^{2} c^{2} d x^{\frac {7}{2}} - 22 \, a b c d^{2} x^{\frac {7}{2}} + 3 \, a^{2} d^{3} x^{\frac {7}{2}} + 15 \, b^{2} c^{3} x^{\frac {3}{2}} - 14 \, a b c^{2} d x^{\frac {3}{2}} - a^{2} c d^{2} x^{\frac {3}{2}}}{16 \, {\left (d x^{2} + c\right )}^{2} c d^{3}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{2} d^{6}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{64 \, c^{2} d^{6}} + \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{2} d^{6}} - \frac {\sqrt {2} {\left (77 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{128 \, c^{2} d^{6}} \] Input:

integrate(x^(5/2)*(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")
 

Output:

2/3*b^2*x^(3/2)/d^3 + 1/16*(19*b^2*c^2*d*x^(7/2) - 22*a*b*c*d^2*x^(7/2) + 
3*a^2*d^3*x^(7/2) + 15*b^2*c^3*x^(3/2) - 14*a*b*c^2*d*x^(3/2) - a^2*c*d^2* 
x^(3/2))/((d*x^2 + c)^2*c*d^3) - 1/64*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 
42*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sq 
rt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^2*d^6) - 1/64*sqrt(2)*(77*( 
c*d^3)^(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*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 
^6) + 1/128*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 
 3*(c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d)) 
/(c^2*d^6) - 1/128*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a* 
b*c*d - 3*(c*d^3)^(3/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sq 
rt(c/d))/(c^2*d^6)
 

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.66 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx=\frac {2\,b^2\,x^{3/2}}{3\,d^3}-\frac {x^{3/2}\,\left (\frac {a^2\,d^2}{16}+\frac {7\,a\,b\,c\,d}{8}-\frac {15\,b^2\,c^2}{16}\right )-\frac {x^{7/2}\,\left (3\,a^2\,d^3-22\,a\,b\,c\,d^2+19\,b^2\,c^2\,d\right )}{16\,c}}{c^2\,d^3+2\,c\,d^4\,x^2+d^5\,x^4}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}}{{\left (-c\right )}^{1/4}}\right )\,\left (3\,a^2\,d^2+42\,a\,b\,c\,d-77\,b^2\,c^2\right )}{32\,{\left (-c\right )}^{5/4}\,d^{15/4}}-\frac {\mathrm {atan}\left (\frac {d^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-c\right )}^{1/4}}\right )\,\left (3\,a^2\,d^2+42\,a\,b\,c\,d-77\,b^2\,c^2\right )\,1{}\mathrm {i}}{32\,{\left (-c\right )}^{5/4}\,d^{15/4}} \] Input:

int((x^(5/2)*(a + b*x^2)^2)/(c + d*x^2)^3,x)
 

Output:

(2*b^2*x^(3/2))/(3*d^3) - (x^(3/2)*((a^2*d^2)/16 - (15*b^2*c^2)/16 + (7*a* 
b*c*d)/8) - (x^(7/2)*(3*a^2*d^3 + 19*b^2*c^2*d - 22*a*b*c*d^2))/(16*c))/(c 
^2*d^3 + d^5*x^4 + 2*c*d^4*x^2) - (atan((d^(1/4)*x^(1/2))/(-c)^(1/4))*(3*a 
^2*d^2 - 77*b^2*c^2 + 42*a*b*c*d))/(32*(-c)^(5/4)*d^(15/4)) - (atan((d^(1/ 
4)*x^(1/2)*1i)/(-c)^(1/4))*(3*a^2*d^2 - 77*b^2*c^2 + 42*a*b*c*d)*1i)/(32*( 
-c)^(5/4)*d^(15/4))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1512, normalized size of antiderivative = 5.04 \[ \int \frac {x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^(5/2)*(b*x^2+a)^2/(d*x^2+c)^3,x)
 

Output:

( - 18*d**(1/4)*c**(3/4)*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**2*d**2 - 36*d**(1/4)*c**( 
3/4)*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**3*x**2 - 18*d**(1/4)*c**(3/4)*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**4*x**4 - 252*d**(1/4)*c**(3/4)*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**3*d - 504 
*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqr 
t(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*a*b*c**2*d**2*x**2 - 252*d**(1/4)*c**(3 
/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**3*x**4 + 462*d**(1/4)*c**(3/4)*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**4 + 924*d**(1/4)*c**(3/4)*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*d*x**2 + 462 
*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)*sqrt(2) - 2*sqrt(x)*sqr 
t(d))/(d**(1/4)*c**(1/4)*sqrt(2)))*b**2*c**2*d**2*x**4 + 18*d**(1/4)*c**(3 
/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**2*d**2 + 36*d**(1/4)*c**(3/4)*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**3*x**2 + 18*d**(1/4)*c**(3/4)*sqrt(2)*atan((d**(1/4)*c**(1/4)...