\(\int \frac {x^{3/2} (A+B x^2)}{(b x^2+c x^4)^3} \, dx\) [151]

Optimal result

Integrand size = 26, antiderivative size = 265 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {2 A}{7 b^3 x^{7/2}}-\frac {2 (b B-3 A c)}{3 b^4 x^{3/2}}-\frac {c (b B-A c) \sqrt {x}}{4 b^3 \left (b+c x^2\right )^2}-\frac {c (15 b B-23 A c) \sqrt {x}}{16 b^4 \left (b+c x^2\right )}+\frac {11 c^{3/4} (7 b B-15 A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}-\frac {11 c^{3/4} (7 b B-15 A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} b^{19/4}}-\frac {11 c^{3/4} (7 b B-15 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{32 \sqrt {2} b^{19/4}} \] Output:

-2/7*A/b^3/x^(7/2)-2/3*(-3*A*c+B*b)/b^4/x^(3/2)-1/4*c*(-A*c+B*b)*x^(1/2)/b 
^3/(c*x^2+b)^2-1/16*c*(-23*A*c+15*B*b)*x^(1/2)/b^4/(c*x^2+b)+11/64*c^(3/4) 
*(-15*A*c+7*B*b)*arctan(1-2^(1/2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/2)/b^(19/4 
)-11/64*c^(3/4)*(-15*A*c+7*B*b)*arctan(1+2^(1/2)*c^(1/4)*x^(1/2)/b^(1/4))* 
2^(1/2)/b^(19/4)-11/64*c^(3/4)*(-15*A*c+7*B*b)*arctanh(2^(1/2)*b^(1/4)*c^( 
1/4)*x^(1/2)/(b^(1/2)+c^(1/2)*x))*2^(1/2)/b^(19/4)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.79 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\frac {-\frac {4 b^{3/4} \left (7 b B x^2 \left (32 b^2+121 b c x^2+77 c^2 x^4\right )+3 A \left (32 b^3-160 b^2 c x^2-605 b c^2 x^4-385 c^3 x^6\right )\right )}{x^{7/2} \left (b+c x^2\right )^2}+231 \sqrt {2} c^{3/4} (7 b B-15 A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+231 \sqrt {2} c^{3/4} (-7 b B+15 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{1344 b^{19/4}} \] Input:

Integrate[(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
 

Output:

((-4*b^(3/4)*(7*b*B*x^2*(32*b^2 + 121*b*c*x^2 + 77*c^2*x^4) + 3*A*(32*b^3 
- 160*b^2*c*x^2 - 605*b*c^2*x^4 - 385*c^3*x^6)))/(x^(7/2)*(b + c*x^2)^2) + 
 231*Sqrt[2]*c^(3/4)*(7*b*B - 15*A*c)*ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2 
]*b^(1/4)*c^(1/4)*Sqrt[x])] + 231*Sqrt[2]*c^(3/4)*(-7*b*B + 15*A*c)*ArcTan 
h[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt[c]*x)])/(1344*b^(19/4) 
)
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.25, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {9, 362, 253, 264, 264, 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.

\(\displaystyle \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \int \frac {A+B x^2}{x^{9/2} \left (b+c x^2\right )^3}dx\)

\(\Big \downarrow \) 362

\(\displaystyle -\frac {(7 b B-15 A c) \int \frac {1}{x^{9/2} \left (c x^2+b\right )^2}dx}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 253

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \int \frac {1}{x^{9/2} \left (c x^2+b\right )}dx}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \int \frac {1}{x^{5/2} \left (c x^2+b\right )}dx}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {c \int \frac {1}{\sqrt {x} \left (c x^2+b\right )}dx}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \int \frac {1}{c x^2+b}d\sqrt {x}}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 755

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle -\frac {(7 b B-15 A c) \left (\frac {11 \left (-\frac {c \left (-\frac {2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{b}-\frac {2}{3 b x^{3/2}}\right )}{b}-\frac {2}{7 b x^{7/2}}\right )}{4 b}+\frac {1}{2 b x^{7/2} \left (b+c x^2\right )}\right )}{8 b c}-\frac {b B-A c}{4 b c x^{7/2} \left (b+c x^2\right )^2}\)

Input:

Int[(x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]
 

Output:

-1/4*(b*B - A*c)/(b*c*x^(7/2)*(b + c*x^2)^2) - ((7*b*B - 15*A*c)*(1/(2*b*x 
^(7/2)*(b + c*x^2)) + (11*(-2/(7*b*x^(7/2)) - (c*(-2/(3*b*x^(3/2)) - (2*c* 
((-(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4) 
)) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(Sqrt[2]*b^(1/4)*c^(1/4 
)))/(2*Sqrt[b]) + (-1/2*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sq 
rt[c]*x]/(Sqrt[2]*b^(1/4)*c^(1/4)) + Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4) 
*Sqrt[x] + Sqrt[c]*x]/(2*Sqrt[2]*b^(1/4)*c^(1/4)))/(2*Sqrt[b])))/b))/b))/( 
4*b)))/(8*b*c)
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

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 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && 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 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 = 0.43 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(-\frac {2 A}{7 b^{3} x^{\frac {7}{2}}}-\frac {2 \left (-3 A c +B b \right )}{3 b^{4} x^{\frac {3}{2}}}+\frac {2 c \left (\frac {\left (\frac {23}{32} A \,c^{2}-\frac {15}{32} B b c \right ) x^{\frac {5}{2}}+\frac {b \left (27 A c -19 B b \right ) \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {11 \left (15 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{b^{4}}\)	\(190\)
default	\(-\frac {2 A}{7 b^{3} x^{\frac {7}{2}}}-\frac {2 \left (-3 A c +B b \right )}{3 b^{4} x^{\frac {3}{2}}}+\frac {2 c \left (\frac {\left (\frac {23}{32} A \,c^{2}-\frac {15}{32} B b c \right ) x^{\frac {5}{2}}+\frac {b \left (27 A c -19 B b \right ) \sqrt {x}}{32}}{\left (c \,x^{2}+b \right )^{2}}+\frac {11 \left (15 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b}\right )}{b^{4}}\)	\(190\)
risch	\(-\frac {2 \left (-21 A c \,x^{2}+7 B b \,x^{2}+3 A b \right )}{21 b^{4} x^{\frac {7}{2}}}+\frac {c \left (\frac {2 \left (\frac {23}{32} A \,c^{2}-\frac {15}{32} B b c \right ) x^{\frac {5}{2}}+\frac {b \left (27 A c -19 B b \right ) \sqrt {x}}{16}}{\left (c \,x^{2}+b \right )^{2}}+\frac {11 \left (15 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b}\right )}{b^{4}}\)	\(192\)

Input:

int(x^(3/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x,method=_RETURNVERBOSE)
 

Output:

-2/7*A/b^3/x^(7/2)-2/3*(-3*A*c+B*b)/b^4/x^(3/2)+2/b^4*c*(((23/32*A*c^2-15/ 
32*B*b*c)*x^(5/2)+1/32*b*(27*A*c-19*B*b)*x^(1/2))/(c*x^2+b)^2+11/256*(15*A 
*c-7*B*b)*(b/c)^(1/4)/b*2^(1/2)*(ln((x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^( 
1/2))/(x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))+2*arctan(2^(1/2)/(b/c)^ 
(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 822, normalized size of antiderivative = 3.10 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^(3/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="fricas")
 

Output:

1/1344*(231*(b^4*c^2*x^8 + 2*b^5*c*x^6 + b^6*x^4)*(-(2401*B^4*b^4*c^3 - 20 
580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4* 
c^7)/b^19)^(1/4)*log(11*b^5*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66 
150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4) - 11* 
(7*B*b*c - 15*A*c^2)*sqrt(x)) - 231*(-I*b^4*c^2*x^8 - 2*I*b^5*c*x^6 - I*b^ 
6*x^4)*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 
 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4)*log(11*I*b^5*(-(2401*B^4*b 
^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 
 50625*A^4*c^7)/b^19)^(1/4) - 11*(7*B*b*c - 15*A*c^2)*sqrt(x)) - 231*(I*b^ 
4*c^2*x^8 + 2*I*b^5*c*x^6 + I*b^6*x^4)*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b 
^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^ 
(1/4)*log(-11*I*b^5*(-(2401*B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2* 
B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/b^19)^(1/4) - 11*(7*B*b*c 
 - 15*A*c^2)*sqrt(x)) - 231*(b^4*c^2*x^8 + 2*b^5*c*x^6 + b^6*x^4)*(-(2401* 
B^4*b^4*c^3 - 20580*A*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b* 
c^6 + 50625*A^4*c^7)/b^19)^(1/4)*log(-11*b^5*(-(2401*B^4*b^4*c^3 - 20580*A 
*B^3*b^3*c^4 + 66150*A^2*B^2*b^2*c^5 - 94500*A^3*B*b*c^6 + 50625*A^4*c^7)/ 
b^19)^(1/4) - 11*(7*B*b*c - 15*A*c^2)*sqrt(x)) - 4*(77*(7*B*b*c^2 - 15*A*c 
^3)*x^6 + 121*(7*B*b^2*c - 15*A*b*c^2)*x^4 + 96*A*b^3 + 32*(7*B*b^3 - 15*A 
*b^2*c)*x^2)*sqrt(x))/(b^4*c^2*x^8 + 2*b^5*c*x^6 + b^6*x^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**(3/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.21 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {77 \, {\left (7 \, B b c^{2} - 15 \, A c^{3}\right )} x^{6} + 121 \, {\left (7 \, B b^{2} c - 15 \, A b c^{2}\right )} x^{4} + 96 \, A b^{3} + 32 \, {\left (7 \, B b^{3} - 15 \, A b^{2} c\right )} x^{2}}{336 \, {\left (b^{4} c^{2} x^{\frac {15}{2}} + 2 \, b^{5} c x^{\frac {11}{2}} + b^{6} x^{\frac {7}{2}}\right )}} - \frac {11 \, {\left (\frac {2 \, \sqrt {2} {\left (7 \, B b c - 15 \, A c^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (7 \, B b c - 15 \, A c^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (7 \, B b c - 15 \, A c^{2}\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (7 \, B b c - 15 \, A c^{2}\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}\right )}}{128 \, b^{4}} \] Input:

integrate(x^(3/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="maxima")
 

Output:

-1/336*(77*(7*B*b*c^2 - 15*A*c^3)*x^6 + 121*(7*B*b^2*c - 15*A*b*c^2)*x^4 + 
 96*A*b^3 + 32*(7*B*b^3 - 15*A*b^2*c)*x^2)/(b^4*c^2*x^(15/2) + 2*b^5*c*x^( 
11/2) + b^6*x^(7/2)) - 11/128*(2*sqrt(2)*(7*B*b*c - 15*A*c^2)*arctan(1/2*s 
qrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)) 
)/(sqrt(b)*sqrt(sqrt(b)*sqrt(c))) + 2*sqrt(2)*(7*B*b*c - 15*A*c^2)*arctan( 
-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(b)*sq 
rt(c)))/(sqrt(b)*sqrt(sqrt(b)*sqrt(c))) + sqrt(2)*(7*B*b*c - 15*A*c^2)*log 
(sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(3/4)*c^(1/4)) 
- sqrt(2)*(7*B*b*c - 15*A*c^2)*log(-sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt 
(c)*x + sqrt(b))/(b^(3/4)*c^(1/4)))/b^4
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.19 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {11 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5}} - \frac {11 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, b^{5}} - \frac {11 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5}} + \frac {11 \, \sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 15 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, b^{5}} - \frac {15 \, B b c^{2} x^{\frac {5}{2}} - 23 \, A c^{3} x^{\frac {5}{2}} + 19 \, B b^{2} c \sqrt {x} - 27 \, A b c^{2} \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} b^{4}} - \frac {2 \, {\left (7 \, B b x^{2} - 21 \, A c x^{2} + 3 \, A b\right )}}{21 \, b^{4} x^{\frac {7}{2}}} \] Input:

integrate(x^(3/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="giac")
 

Output:

-11/64*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 15*(b*c^3)^(1/4)*A*c)*arctan(1/2*sqr 
t(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/b^5 - 11/64*sqrt(2)*(7 
*(b*c^3)^(1/4)*B*b - 15*(b*c^3)^(1/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b 
/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/b^5 - 11/128*sqrt(2)*(7*(b*c^3)^(1/4)* 
B*b - 15*(b*c^3)^(1/4)*A*c)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c 
))/b^5 + 11/128*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 15*(b*c^3)^(1/4)*A*c)*log(- 
sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^5 - 1/16*(15*B*b*c^2*x^(5/2 
) - 23*A*c^3*x^(5/2) + 19*B*b^2*c*sqrt(x) - 27*A*b*c^2*sqrt(x))/((c*x^2 + 
b)^2*b^4) - 2/21*(7*B*b*x^2 - 21*A*c*x^2 + 3*A*b)/(b^4*x^(7/2))
 

Mupad [B] (verification not implemented)

Time = 9.28 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.36 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int((x^(3/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x)
 

Output:

((2*x^2*(15*A*c - 7*B*b))/(21*b^2) - (2*A)/(7*b) + (11*c^2*x^6*(15*A*c - 7 
*B*b))/(48*b^4) + (121*c*x^4*(15*A*c - 7*B*b))/(336*b^3))/(b^2*x^(7/2) + c 
^2*x^(15/2) + 2*b*c*x^(11/2)) + (11*(-c)^(3/4)*atan(((11*(-c)^(3/4)*(15*A* 
c - 7*B*b)*(x^(1/2)*(446054400*A^2*b^12*c^7 + 97140736*B^2*b^14*c^5 - 4163 
17440*A*B*b^13*c^6) - ((-c)^(3/4)*(15*A*c - 7*B*b)*(173015040*A*b^17*c^5 - 
 80740352*B*b^18*c^4)*11i)/(64*b^(19/4))))/(64*b^(19/4)) + (11*(-c)^(3/4)* 
(15*A*c - 7*B*b)*(x^(1/2)*(446054400*A^2*b^12*c^7 + 97140736*B^2*b^14*c^5 
- 416317440*A*B*b^13*c^6) + ((-c)^(3/4)*(15*A*c - 7*B*b)*(173015040*A*b^17 
*c^5 - 80740352*B*b^18*c^4)*11i)/(64*b^(19/4))))/(64*b^(19/4)))/(((-c)^(3/ 
4)*(15*A*c - 7*B*b)*(x^(1/2)*(446054400*A^2*b^12*c^7 + 97140736*B^2*b^14*c 
^5 - 416317440*A*B*b^13*c^6) - ((-c)^(3/4)*(15*A*c - 7*B*b)*(173015040*A*b 
^17*c^5 - 80740352*B*b^18*c^4)*11i)/(64*b^(19/4)))*11i)/(64*b^(19/4)) - (( 
-c)^(3/4)*(15*A*c - 7*B*b)*(x^(1/2)*(446054400*A^2*b^12*c^7 + 97140736*B^2 
*b^14*c^5 - 416317440*A*B*b^13*c^6) + ((-c)^(3/4)*(15*A*c - 7*B*b)*(173015 
040*A*b^17*c^5 - 80740352*B*b^18*c^4)*11i)/(64*b^(19/4)))*11i)/(64*b^(19/4 
))))*(15*A*c - 7*B*b))/(32*b^(19/4)) - ((-c)^(3/4)*atan((A^3*c^8*x^(1/2)*3 
375i - B^3*b^3*c^5*x^(1/2)*343i - A^2*B*b*c^7*x^(1/2)*4725i + A*B^2*b^2*c^ 
6*x^(1/2)*2205i)/(b^(1/4)*(-c)^(19/4)*(c*(c*(3375*A^3*c - 4725*A^2*B*b) + 
2205*A*B^2*b^2) - 343*B^3*b^3)))*(15*A*c - 7*B*b)*11i)/(32*b^(19/4))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1041, normalized size of antiderivative = 3.93 \[ \int \frac {x^{3/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^(3/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x)
 

Output:

( - 6930*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) 
 - 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*b**2*c*x**3 - 13860*s 
qrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt( 
x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*b*c**2*x**5 - 6930*sqrt(x)*c**( 
3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c)) 
/(c**(1/4)*b**(1/4)*sqrt(2)))*a*c**3*x**7 + 3234*sqrt(x)*c**(3/4)*b**(1/4) 
*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c))/(c**(1/4)*b* 
*(1/4)*sqrt(2)))*b**4*x**3 + 6468*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan(( 
c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)) 
)*b**3*c*x**5 + 3234*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**( 
1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*b**2*c**2*x 
**7 + 6930*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt( 
2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*b**2*c*x**3 + 13860 
*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) + 2*sqr 
t(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*b*c**2*x**5 + 6930*sqrt(x)*c* 
*(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c 
))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*c**3*x**7 - 3234*sqrt(x)*c**(3/4)*b**(1/ 
4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)* 
b**(1/4)*sqrt(2)))*b**4*x**3 - 6468*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*atan 
((c**(1/4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqr...