\(\int \frac {x^{11/2} (A+B x^2)}{b x^2+c x^4} \, dx\) [118]

Optimal result

Integrand size = 26, antiderivative size = 213 \[ \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {2 b (b B-A c) \sqrt {x}}{c^3}-\frac {2 (b B-A c) x^{5/2}}{5 c^2}+\frac {2 B x^{9/2}}{9 c}+\frac {b^{5/4} (b B-A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{13/4}}-\frac {b^{5/4} (b B-A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} c^{13/4}}-\frac {b^{5/4} (b B-A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt {2} c^{13/4}} \] Output:

2*b*(-A*c+B*b)*x^(1/2)/c^3-2/5*(-A*c+B*b)*x^(5/2)/c^2+2/9*B*x^(9/2)/c+1/2* 
b^(5/4)*(-A*c+B*b)*arctan(1-2^(1/2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/2)/c^(13 
/4)-1/2*b^(5/4)*(-A*c+B*b)*arctan(1+2^(1/2)*c^(1/4)*x^(1/2)/b^(1/4))*2^(1/ 
2)/c^(13/4)-1/2*b^(5/4)*(-A*c+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)/c^(13/4)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.81 \[ \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {2 \sqrt {x} \left (45 b^2 B-45 A b c-9 b B c x^2+9 A c^2 x^2+5 B c^2 x^4\right )}{45 c^3}+\frac {b^{5/4} (b B-A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} c^{13/4}}-\frac {b^{5/4} (b B-A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{\sqrt {2} c^{13/4}} \] Input:

Integrate[(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4),x]
 

Output:

(2*Sqrt[x]*(45*b^2*B - 45*A*b*c - 9*b*B*c*x^2 + 9*A*c^2*x^2 + 5*B*c^2*x^4) 
)/(45*c^3) + (b^(5/4)*(b*B - A*c)*ArcTan[(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2]*b^ 
(1/4)*c^(1/4)*Sqrt[x])])/(Sqrt[2]*c^(13/4)) - (b^(5/4)*(b*B - A*c)*ArcTanh 
[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + Sqrt[c]*x)])/(Sqrt[2]*c^(13/ 
4))
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.29, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {9, 363, 262, 262, 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.

\(\displaystyle \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \int \frac {x^{7/2} \left (A+B x^2\right )}{b+c x^2}dx\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \int \frac {x^{7/2}}{c x^2+b}dx}{c}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \int \frac {x^{3/2}}{c x^2+b}dx}{c}\right )}{c}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {b \int \frac {1}{\sqrt {x} \left (c x^2+b\right )}dx}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \int \frac {1}{c x^2+b}d\sqrt {x}}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {c} x+\sqrt {b}}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}\right )}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}}{2 \sqrt {b}}\right )}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \left (\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{c x^2+b}d\sqrt {x}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{b}}{x+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {b}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{c}\right )}{c}\right )}{c}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {2 B x^{9/2}}{9 c}-\frac {(b B-A c) \left (\frac {2 x^{5/2}}{5 c}-\frac {b \left (\frac {2 \sqrt {x}}{c}-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt [4]{c}}}{2 \sqrt {b}}\right )}{c}\right )}{c}\right )}{c}\)

Input:

Int[(x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4),x]
 

Output:

(2*B*x^(9/2))/(9*c) - ((b*B - A*c)*((2*x^(5/2))/(5*c) - (b*((2*Sqrt[x])/c 
- (2*b*((-(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] + Sqrt[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])))/c) 
)/c))/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 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 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 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 = 0.45 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {2 \left (-5 B \,c^{2} x^{4}-9 A \,c^{2} x^{2}+9 x^{2} B b c +45 A b c -45 B \,b^{2}\right ) \sqrt {x}}{45 c^{3}}+\frac {b \left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3}}\)	\(162\)
derivativedivides	\(-\frac {2 \left (-\frac {B \,c^{2} x^{\frac {9}{2}}}{9}-\frac {A \,c^{2} x^{\frac {5}{2}}}{5}+\frac {B b c \,x^{\frac {5}{2}}}{5}+A b c \sqrt {x}-B \,b^{2} \sqrt {x}\right )}{c^{3}}+\frac {b \left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3}}\)	\(164\)
default	\(-\frac {2 \left (-\frac {B \,c^{2} x^{\frac {9}{2}}}{9}-\frac {A \,c^{2} x^{\frac {5}{2}}}{5}+\frac {B b c \,x^{\frac {5}{2}}}{5}+A b c \sqrt {x}-B \,b^{2} \sqrt {x}\right )}{c^{3}}+\frac {b \left (A c -B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 c^{3}}\)	\(164\)

Input:

int(x^(11/2)*(B*x^2+A)/(c*x^4+b*x^2),x,method=_RETURNVERBOSE)
 

Output:

-2/45*(-5*B*c^2*x^4-9*A*c^2*x^2+9*B*b*c*x^2+45*A*b*c-45*B*b^2)*x^(1/2)/c^3 
+1/4*b*(A*c-B*b)/c^3*(b/c)^(1/4)*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 = 0.10 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.04 \[ \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {45 \, c^{3} \left (-\frac {B^{4} b^{9} - 4 \, A B^{3} b^{8} c + 6 \, A^{2} B^{2} b^{7} c^{2} - 4 \, A^{3} B b^{6} c^{3} + A^{4} b^{5} c^{4}}{c^{13}}\right )^{\frac {1}{4}} \log \left (c^{3} \left (-\frac {B^{4} b^{9} - 4 \, A B^{3} b^{8} c + 6 \, A^{2} B^{2} b^{7} c^{2} - 4 \, A^{3} B b^{6} c^{3} + A^{4} b^{5} c^{4}}{c^{13}}\right )^{\frac {1}{4}} - {\left (B b^{2} - A b c\right )} \sqrt {x}\right ) + 45 i \, c^{3} \left (-\frac {B^{4} b^{9} - 4 \, A B^{3} b^{8} c + 6 \, A^{2} B^{2} b^{7} c^{2} - 4 \, A^{3} B b^{6} c^{3} + A^{4} b^{5} c^{4}}{c^{13}}\right )^{\frac {1}{4}} \log \left (i \, c^{3} \left (-\frac {B^{4} b^{9} - 4 \, A B^{3} b^{8} c + 6 \, A^{2} B^{2} b^{7} c^{2} - 4 \, A^{3} B b^{6} c^{3} + A^{4} b^{5} c^{4}}{c^{13}}\right )^{\frac {1}{4}} - {\left (B b^{2} - A b c\right )} \sqrt {x}\right ) - 45 i \, c^{3} \left (-\frac {B^{4} b^{9} - 4 \, A B^{3} b^{8} c + 6 \, A^{2} B^{2} b^{7} c^{2} - 4 \, A^{3} B b^{6} c^{3} + A^{4} b^{5} c^{4}}{c^{13}}\right )^{\frac {1}{4}} \log \left (-i \, c^{3} \left (-\frac {B^{4} b^{9} - 4 \, A B^{3} b^{8} c + 6 \, A^{2} B^{2} b^{7} c^{2} - 4 \, A^{3} B b^{6} c^{3} + A^{4} b^{5} c^{4}}{c^{13}}\right )^{\frac {1}{4}} - {\left (B b^{2} - A b c\right )} \sqrt {x}\right ) - 45 \, c^{3} \left (-\frac {B^{4} b^{9} - 4 \, A B^{3} b^{8} c + 6 \, A^{2} B^{2} b^{7} c^{2} - 4 \, A^{3} B b^{6} c^{3} + A^{4} b^{5} c^{4}}{c^{13}}\right )^{\frac {1}{4}} \log \left (-c^{3} \left (-\frac {B^{4} b^{9} - 4 \, A B^{3} b^{8} c + 6 \, A^{2} B^{2} b^{7} c^{2} - 4 \, A^{3} B b^{6} c^{3} + A^{4} b^{5} c^{4}}{c^{13}}\right )^{\frac {1}{4}} - {\left (B b^{2} - A b c\right )} \sqrt {x}\right ) + 4 \, {\left (5 \, B c^{2} x^{4} + 45 \, B b^{2} - 45 \, A b c - 9 \, {\left (B b c - A c^{2}\right )} x^{2}\right )} \sqrt {x}}{90 \, c^{3}} \] Input:

integrate(x^(11/2)*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="fricas")
 

Output:

1/90*(45*c^3*(-(B^4*b^9 - 4*A*B^3*b^8*c + 6*A^2*B^2*b^7*c^2 - 4*A^3*B*b^6* 
c^3 + A^4*b^5*c^4)/c^13)^(1/4)*log(c^3*(-(B^4*b^9 - 4*A*B^3*b^8*c + 6*A^2* 
B^2*b^7*c^2 - 4*A^3*B*b^6*c^3 + A^4*b^5*c^4)/c^13)^(1/4) - (B*b^2 - A*b*c) 
*sqrt(x)) + 45*I*c^3*(-(B^4*b^9 - 4*A*B^3*b^8*c + 6*A^2*B^2*b^7*c^2 - 4*A^ 
3*B*b^6*c^3 + A^4*b^5*c^4)/c^13)^(1/4)*log(I*c^3*(-(B^4*b^9 - 4*A*B^3*b^8* 
c + 6*A^2*B^2*b^7*c^2 - 4*A^3*B*b^6*c^3 + A^4*b^5*c^4)/c^13)^(1/4) - (B*b^ 
2 - A*b*c)*sqrt(x)) - 45*I*c^3*(-(B^4*b^9 - 4*A*B^3*b^8*c + 6*A^2*B^2*b^7* 
c^2 - 4*A^3*B*b^6*c^3 + A^4*b^5*c^4)/c^13)^(1/4)*log(-I*c^3*(-(B^4*b^9 - 4 
*A*B^3*b^8*c + 6*A^2*B^2*b^7*c^2 - 4*A^3*B*b^6*c^3 + A^4*b^5*c^4)/c^13)^(1 
/4) - (B*b^2 - A*b*c)*sqrt(x)) - 45*c^3*(-(B^4*b^9 - 4*A*B^3*b^8*c + 6*A^2 
*B^2*b^7*c^2 - 4*A^3*B*b^6*c^3 + A^4*b^5*c^4)/c^13)^(1/4)*log(-c^3*(-(B^4* 
b^9 - 4*A*B^3*b^8*c + 6*A^2*B^2*b^7*c^2 - 4*A^3*B*b^6*c^3 + A^4*b^5*c^4)/c 
^13)^(1/4) - (B*b^2 - A*b*c)*sqrt(x)) + 4*(5*B*c^2*x^4 + 45*B*b^2 - 45*A*b 
*c - 9*(B*b*c - A*c^2)*x^2)*sqrt(x))/c^3
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\text {Timed out} \] Input:

integrate(x**(11/2)*(B*x**2+A)/(c*x**4+b*x**2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.22 \[ \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=-\frac {{\left (\frac {2 \, \sqrt {2} {\left (B b - A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (B b - A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (B b - A c\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (B b - A c\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}\right )} b^{2}}{4 \, c^{3}} + \frac {2 \, {\left (5 \, B c^{2} x^{\frac {9}{2}} - 9 \, {\left (B b c - A c^{2}\right )} x^{\frac {5}{2}} + 45 \, {\left (B b^{2} - A b c\right )} \sqrt {x}\right )}}{45 \, c^{3}} \] Input:

integrate(x^(11/2)*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="maxima")
 

Output:

-1/4*(2*sqrt(2)*(B*b - A*c)*arctan(1/2*sqrt(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)*(B*b - A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*s 
qrt(c)*sqrt(x))/sqrt(sqrt(b)*sqrt(c)))/(sqrt(b)*sqrt(sqrt(b)*sqrt(c))) + s 
qrt(2)*(B*b - A*c)*log(sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt( 
b))/(b^(3/4)*c^(1/4)) - sqrt(2)*(B*b - A*c)*log(-sqrt(2)*b^(1/4)*c^(1/4)*s 
qrt(x) + sqrt(c)*x + sqrt(b))/(b^(3/4)*c^(1/4)))*b^2/c^3 + 2/45*(5*B*c^2*x 
^(9/2) - 9*(B*b*c - A*c^2)*x^(5/2) + 45*(B*b^2 - A*b*c)*sqrt(x))/c^3
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.40 \[ \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=-\frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b^{2} - \left (b c^{3}\right )^{\frac {1}{4}} A b c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{4}} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b^{2} - \left (b c^{3}\right )^{\frac {1}{4}} A b c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{2 \, c^{4}} - \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b^{2} - \left (b c^{3}\right )^{\frac {1}{4}} A b c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{4}} + \frac {\sqrt {2} {\left (\left (b c^{3}\right )^{\frac {1}{4}} B b^{2} - \left (b c^{3}\right )^{\frac {1}{4}} A b c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{4 \, c^{4}} + \frac {2 \, {\left (5 \, B c^{8} x^{\frac {9}{2}} - 9 \, B b c^{7} x^{\frac {5}{2}} + 9 \, A c^{8} x^{\frac {5}{2}} + 45 \, B b^{2} c^{6} \sqrt {x} - 45 \, A b c^{7} \sqrt {x}\right )}}{45 \, c^{9}} \] Input:

integrate(x^(11/2)*(B*x^2+A)/(c*x^4+b*x^2),x, algorithm="giac")
 

Output:

-1/2*sqrt(2)*((b*c^3)^(1/4)*B*b^2 - (b*c^3)^(1/4)*A*b*c)*arctan(1/2*sqrt(2 
)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/c^4 - 1/2*sqrt(2)*((b*c^3 
)^(1/4)*B*b^2 - (b*c^3)^(1/4)*A*b*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1 
/4) - 2*sqrt(x))/(b/c)^(1/4))/c^4 - 1/4*sqrt(2)*((b*c^3)^(1/4)*B*b^2 - (b* 
c^3)^(1/4)*A*b*c)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^4 + 1 
/4*sqrt(2)*((b*c^3)^(1/4)*B*b^2 - (b*c^3)^(1/4)*A*b*c)*log(-sqrt(2)*sqrt(x 
)*(b/c)^(1/4) + x + sqrt(b/c))/c^4 + 2/45*(5*B*c^8*x^(9/2) - 9*B*b*c^7*x^( 
5/2) + 9*A*c^8*x^(5/2) + 45*B*b^2*c^6*sqrt(x) - 45*A*b*c^7*sqrt(x))/c^9
 

Mupad [B] (verification not implemented)

Time = 9.02 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.70 \[ \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx =\text {Too large to display} \] Input:

int((x^(11/2)*(A + B*x^2))/(b*x^2 + c*x^4),x)
 

Output:

x^(5/2)*((2*A)/(5*c) - (2*B*b)/(5*c^2)) + (2*B*x^(9/2))/(9*c) - ((-b)^(5/4 
)*atan((((-b)^(5/4)*((16*x^(1/2)*(B^2*b^6 + A^2*b^4*c^2 - 2*A*B*b^5*c))/c^ 
3 - ((-b)^(5/4)*(A*c - B*b)*(32*B*b^4 - 32*A*b^3*c))/(2*c^(13/4)))*(A*c - 
B*b)*1i)/(2*c^(13/4)) + ((-b)^(5/4)*((16*x^(1/2)*(B^2*b^6 + A^2*b^4*c^2 - 
2*A*B*b^5*c))/c^3 + ((-b)^(5/4)*(A*c - B*b)*(32*B*b^4 - 32*A*b^3*c))/(2*c^ 
(13/4)))*(A*c - B*b)*1i)/(2*c^(13/4)))/(((-b)^(5/4)*((16*x^(1/2)*(B^2*b^6 
+ A^2*b^4*c^2 - 2*A*B*b^5*c))/c^3 - ((-b)^(5/4)*(A*c - B*b)*(32*B*b^4 - 32 
*A*b^3*c))/(2*c^(13/4)))*(A*c - B*b))/(2*c^(13/4)) - ((-b)^(5/4)*((16*x^(1 
/2)*(B^2*b^6 + A^2*b^4*c^2 - 2*A*B*b^5*c))/c^3 + ((-b)^(5/4)*(A*c - B*b)*( 
32*B*b^4 - 32*A*b^3*c))/(2*c^(13/4)))*(A*c - B*b))/(2*c^(13/4))))*(A*c - B 
*b)*1i)/c^(13/4) - ((-b)^(5/4)*atan((((-b)^(5/4)*((16*x^(1/2)*(B^2*b^6 + A 
^2*b^4*c^2 - 2*A*B*b^5*c))/c^3 - ((-b)^(5/4)*(A*c - B*b)*(32*B*b^4 - 32*A* 
b^3*c)*1i)/(2*c^(13/4)))*(A*c - B*b))/(2*c^(13/4)) + ((-b)^(5/4)*((16*x^(1 
/2)*(B^2*b^6 + A^2*b^4*c^2 - 2*A*B*b^5*c))/c^3 + ((-b)^(5/4)*(A*c - B*b)*( 
32*B*b^4 - 32*A*b^3*c)*1i)/(2*c^(13/4)))*(A*c - B*b))/(2*c^(13/4)))/(((-b) 
^(5/4)*((16*x^(1/2)*(B^2*b^6 + A^2*b^4*c^2 - 2*A*B*b^5*c))/c^3 - ((-b)^(5/ 
4)*(A*c - B*b)*(32*B*b^4 - 32*A*b^3*c)*1i)/(2*c^(13/4)))*(A*c - B*b)*1i)/( 
2*c^(13/4)) - ((-b)^(5/4)*((16*x^(1/2)*(B^2*b^6 + A^2*b^4*c^2 - 2*A*B*b^5* 
c))/c^3 + ((-b)^(5/4)*(A*c - B*b)*(32*B*b^4 - 32*A*b^3*c)*1i)/(2*c^(13/4)) 
)*(A*c - B*b)*1i)/(2*c^(13/4))))*(A*c - B*b))/c^(13/4) - (b*x^(1/2)*((2...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.56 \[ \int \frac {x^{11/2} \left (A+B x^2\right )}{b x^2+c x^4} \, dx=\frac {-90 c^{\frac {7}{4}} b^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) a +90 c^{\frac {3}{4}} b^{\frac {13}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right )+90 c^{\frac {7}{4}} b^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right ) a -90 c^{\frac {3}{4}} b^{\frac {13}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {c}}{c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}}\right )-45 c^{\frac {7}{4}} b^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) a +45 c^{\frac {3}{4}} b^{\frac {13}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right )+45 c^{\frac {7}{4}} b^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right ) a -45 c^{\frac {3}{4}} b^{\frac {13}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, c^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}+\sqrt {b}+\sqrt {c}\, x \right )-360 \sqrt {x}\, a b \,c^{2}+72 \sqrt {x}\, a \,c^{3} x^{2}+360 \sqrt {x}\, b^{3} c -72 \sqrt {x}\, b^{2} c^{2} x^{2}+40 \sqrt {x}\, b \,c^{3} x^{4}}{180 c^{4}} \] Input:

int(x^(11/2)*(B*x^2+A)/(c*x^4+b*x^2),x)
 

Output:

( - 90*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 + 90*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 + 90*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sq 
rt(2) + 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*b*c - 90*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 - 45*c**(3/4)*b**(1/4)*sqrt(2)*log( - sqrt 
(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*a*b*c + 45*c**(3/4)*b 
**(1/4)*sqrt(2)*log( - sqrt(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt( 
c)*x)*b**3 + 45*c**(3/4)*b**(1/4)*sqrt(2)*log(sqrt(x)*c**(1/4)*b**(1/4)*sq 
rt(2) + sqrt(b) + sqrt(c)*x)*a*b*c - 45*c**(3/4)*b**(1/4)*sqrt(2)*log(sqrt 
(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*b**3 - 360*sqrt(x)*a* 
b*c**2 + 72*sqrt(x)*a*c**3*x**2 + 360*sqrt(x)*b**3*c - 72*sqrt(x)*b**2*c** 
2*x**2 + 40*sqrt(x)*b*c**3*x**4)/(180*c**4)