\(\int \frac {A+B x^2}{\sqrt {x} (b x^2+c x^4)^2} \, dx\) [139]

Optimal result

Integrand size = 26, antiderivative size = 234 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^2} \, dx=-\frac {2 A}{7 b^2 x^{7/2}}-\frac {2 (b B-2 A c)}{3 b^3 x^{3/2}}-\frac {c (b B-A c) \sqrt {x}}{2 b^3 \left (b+c x^2\right )}+\frac {c^{3/4} (7 b B-11 A c) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{15/4}}-\frac {c^{3/4} (7 b B-11 A c) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{4 \sqrt {2} b^{15/4}}-\frac {c^{3/4} (7 b B-11 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{4 \sqrt {2} b^{15/4}} \] Output:

-2/7*A/b^2/x^(7/2)-2/3*(-2*A*c+B*b)/b^3/x^(3/2)-1/2*c*(-A*c+B*b)*x^(1/2)/b 
^3/(c*x^2+b)+1/8*c^(3/4)*(-11*A*c+7*B*b)*arctan(1-2^(1/2)*c^(1/4)*x^(1/2)/ 
b^(1/4))*2^(1/2)/b^(15/4)-1/8*c^(3/4)*(-11*A*c+7*B*b)*arctan(1+2^(1/2)*c^( 
1/4)*x^(1/2)/b^(1/4))*2^(1/2)/b^(15/4)-1/8*c^(3/4)*(-11*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^(15/4)
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.80 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^2} \, dx=\frac {-\frac {4 b^{3/4} \left (7 b B x^2 \left (4 b+7 c x^2\right )+A \left (12 b^2-44 b c x^2-77 c^2 x^4\right )\right )}{x^{7/2} \left (b+c x^2\right )}+21 \sqrt {2} c^{3/4} (7 b B-11 A c) \arctan \left (\frac {\sqrt {b}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}\right )+21 \sqrt {2} c^{3/4} (-7 b B+11 A c) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}}{\sqrt {b}+\sqrt {c} x}\right )}{168 b^{15/4}} \] Input:

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

Output:

((-4*b^(3/4)*(7*b*B*x^2*(4*b + 7*c*x^2) + A*(12*b^2 - 44*b*c*x^2 - 77*c^2* 
x^4)))/(x^(7/2)*(b + c*x^2)) + 21*Sqrt[2]*c^(3/4)*(7*b*B - 11*A*c)*ArcTan[ 
(Sqrt[b] - Sqrt[c]*x)/(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])] + 21*Sqrt[2]*c^(3 
/4)*(-7*b*B + 11*A*c)*ArcTanh[(Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[b] + 
 Sqrt[c]*x)])/(168*b^(15/4))
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 302, 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, 362, 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.

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

\(\Big \downarrow \) 9

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

\(\Big \downarrow \) 362

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 755

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

-1/2*(b*B - A*c)/(b*c*x^(7/2)*(b + c*x^2)) - ((7*b*B - 11*A*c)*(-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] + 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])))/b))/b))/(4*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 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 = 170, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {2 c \left (\frac {\left (\frac {A c}{4}-\frac {B b}{4}\right ) \sqrt {x}}{c \,x^{2}+b}+\frac {\left (11 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b}\right )}{b^{3}}-\frac {2 A}{7 b^{2} x^{\frac {7}{2}}}-\frac {2 \left (-2 A c +B b \right )}{3 b^{3} x^{\frac {3}{2}}}\)	\(170\)
default	\(\frac {2 c \left (\frac {\left (\frac {A c}{4}-\frac {B b}{4}\right ) \sqrt {x}}{c \,x^{2}+b}+\frac {\left (11 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b}\right )}{b^{3}}-\frac {2 A}{7 b^{2} x^{\frac {7}{2}}}-\frac {2 \left (-2 A c +B b \right )}{3 b^{3} x^{\frac {3}{2}}}\)	\(170\)
risch	\(-\frac {2 \left (-14 A c \,x^{2}+7 B b \,x^{2}+3 A b \right )}{21 b^{3} x^{\frac {7}{2}}}+\frac {c \left (\frac {2 \left (\frac {A c}{4}-\frac {B b}{4}\right ) \sqrt {x}}{c \,x^{2}+b}+\frac {\left (11 A c -7 B b \right ) \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {b}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b}\right )}{b^{3}}\)	\(172\)

Input:

int((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^2,x,method=_RETURNVERBOSE)
 

Output:

2/b^3*c*((1/4*A*c-1/4*B*b)*x^(1/2)/(c*x^2+b)+1/32*(11*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)))-2/7*A/b^2/x^(7/2)-2/3*(-2*A*c+B*b) 
/b^3/x^(3/2)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 742, normalized size of antiderivative = 3.17 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^2,x, algorithm="fricas")
 

Output:

1/168*(21*(b^3*c*x^6 + b^4*x^4)*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 
+ 35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^(1/4)*l 
og(b^4*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 35574*A^2*B^2*b^2*c^5 - 
 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^(1/4) - (7*B*b*c - 11*A*c^2)*sqr 
t(x)) - 21*(-I*b^3*c*x^6 - I*b^4*x^4)*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^ 
3*c^4 + 35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^( 
1/4)*log(I*b^4*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 35574*A^2*B^2*b 
^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^(1/4) - (7*B*b*c - 11*A* 
c^2)*sqrt(x)) - 21*(I*b^3*c*x^6 + I*b^4*x^4)*(-(2401*B^4*b^4*c^3 - 15092*A 
*B^3*b^3*c^4 + 35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/ 
b^15)^(1/4)*log(-I*b^4*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 35574*A 
^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^(1/4) - (7*B*b*c 
 - 11*A*c^2)*sqrt(x)) - 21*(b^3*c*x^6 + b^4*x^4)*(-(2401*B^4*b^4*c^3 - 150 
92*A*B^3*b^3*c^4 + 35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c 
^7)/b^15)^(1/4)*log(-b^4*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 35574 
*A^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^(1/4) - (7*B*b 
*c - 11*A*c^2)*sqrt(x)) - 4*(7*(7*B*b*c - 11*A*c^2)*x^4 + 12*A*b^2 + 4*(7* 
B*b^2 - 11*A*b*c)*x^2)*sqrt(x))/(b^3*c*x^6 + b^4*x^4)
 

Sympy [F(-1)]

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

integrate((B*x**2+A)/x**(1/2)/(c*x**4+b*x**2)**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.22 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^2} \, dx=-\frac {7 \, {\left (7 \, B b c - 11 \, A c^{2}\right )} x^{4} + 12 \, A b^{2} + 4 \, {\left (7 \, B b^{2} - 11 \, A b c\right )} x^{2}}{42 \, {\left (b^{3} c x^{\frac {11}{2}} + b^{4} x^{\frac {7}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (7 \, B b c - 11 \, A c^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (7 \, B b c - 11 \, A c^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (7 \, B b c - 11 \, A c^{2}\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (7 \, B b c - 11 \, A c^{2}\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}}{16 \, b^{3}} \] Input:

integrate((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^2,x, algorithm="maxima")
 

Output:

-1/42*(7*(7*B*b*c - 11*A*c^2)*x^4 + 12*A*b^2 + 4*(7*B*b^2 - 11*A*b*c)*x^2) 
/(b^3*c*x^(11/2) + b^4*x^(7/2)) - 1/16*(2*sqrt(2)*(7*B*b*c - 11*A*c^2)*arc 
tan(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)*(7*B*b*c - 11*A*c^2 
)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(s 
qrt(b)*sqrt(c)))/(sqrt(b)*sqrt(sqrt(b)*sqrt(c))) + sqrt(2)*(7*B*b*c - 11*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 - 11*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^3
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^2} \, dx=-\frac {\sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 11 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{8 \, b^{4}} - \frac {\sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 11 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{8 \, b^{4}} - \frac {\sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 11 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{16 \, b^{4}} + \frac {\sqrt {2} {\left (7 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 11 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{16 \, b^{4}} - \frac {B b c \sqrt {x} - A c^{2} \sqrt {x}}{2 \, {\left (c x^{2} + b\right )} b^{3}} - \frac {2 \, {\left (7 \, B b x^{2} - 14 \, A c x^{2} + 3 \, A b\right )}}{21 \, b^{3} x^{\frac {7}{2}}} \] Input:

integrate((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^2,x, algorithm="giac")
 

Output:

-1/8*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 11*(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^4 - 1/8*sqrt(2)*(7*(b* 
c^3)^(1/4)*B*b - 11*(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^4 - 1/16*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 
11*(b*c^3)^(1/4)*A*c)*log(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 
 + 1/16*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 11*(b*c^3)^(1/4)*A*c)*log(-sqrt(2)* 
sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 - 1/2*(B*b*c*sqrt(x) - A*c^2*sqrt 
(x))/((c*x^2 + b)*b^3) - 2/21*(7*B*b*x^2 - 14*A*c*x^2 + 3*A*b)/(b^3*x^(7/2 
))
 

Mupad [B] (verification not implemented)

Time = 9.45 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.54 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int((A + B*x^2)/(x^(1/2)*(b*x^2 + c*x^4)^2),x)
 

Output:

((2*x^2*(11*A*c - 7*B*b))/(21*b^2) - (2*A)/(7*b) + (c*x^4*(11*A*c - 7*B*b) 
)/(6*b^3))/(b*x^(7/2) + c*x^(11/2)) + ((-c)^(3/4)*atan((((-c)^(3/4)*(11*A* 
c - 7*B*b)*(x^(1/2)*(3872*A^2*b^9*c^7 + 1568*B^2*b^11*c^5 - 4928*A*B*b^10* 
c^6) - ((-c)^(3/4)*(11*A*c - 7*B*b)*(2816*A*b^13*c^5 - 1792*B*b^14*c^4)*1i 
)/(8*b^(15/4))))/(8*b^(15/4)) + ((-c)^(3/4)*(11*A*c - 7*B*b)*(x^(1/2)*(387 
2*A^2*b^9*c^7 + 1568*B^2*b^11*c^5 - 4928*A*B*b^10*c^6) + ((-c)^(3/4)*(11*A 
*c - 7*B*b)*(2816*A*b^13*c^5 - 1792*B*b^14*c^4)*1i)/(8*b^(15/4))))/(8*b^(1 
5/4)))/(((-c)^(3/4)*(11*A*c - 7*B*b)*(x^(1/2)*(3872*A^2*b^9*c^7 + 1568*B^2 
*b^11*c^5 - 4928*A*B*b^10*c^6) - ((-c)^(3/4)*(11*A*c - 7*B*b)*(2816*A*b^13 
*c^5 - 1792*B*b^14*c^4)*1i)/(8*b^(15/4)))*1i)/(8*b^(15/4)) - ((-c)^(3/4)*( 
11*A*c - 7*B*b)*(x^(1/2)*(3872*A^2*b^9*c^7 + 1568*B^2*b^11*c^5 - 4928*A*B* 
b^10*c^6) + ((-c)^(3/4)*(11*A*c - 7*B*b)*(2816*A*b^13*c^5 - 1792*B*b^14*c^ 
4)*1i)/(8*b^(15/4)))*1i)/(8*b^(15/4))))*(11*A*c - 7*B*b))/(4*b^(15/4)) - ( 
(-c)^(3/4)*atan((A^3*c^8*x^(1/2)*1331i - B^3*b^3*c^5*x^(1/2)*343i - A^2*B* 
b*c^7*x^(1/2)*2541i + A*B^2*b^2*c^6*x^(1/2)*1617i)/(b^(1/4)*(-c)^(19/4)*(c 
*(c*(1331*A^3*c - 2541*A^2*B*b) + 1617*A*B^2*b^2) - 343*B^3*b^3)))*(11*A*c 
 - 7*B*b)*1i)/(4*b^(15/4))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 693, normalized size of antiderivative = 2.96 \[ \int \frac {A+B x^2}{\sqrt {x} \left (b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:

int((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^2,x)
 

Output:

( - 462*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*c*x**3 - 462*sqrt(x) 
*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(x)*sqr 
t(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*a*c**2*x**5 + 294*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*x**3 + 294*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*ata 
n((c**(1/4)*b**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(c))/(c**(1/4)*b**(1/4)*sqrt( 
2)))*b**2*c*x**5 + 462*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*c*x** 
3 + 462*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**2*x**5 - 294*sqrt(x 
)*c**(3/4)*b**(1/4)*sqrt(2)*atan((c**(1/4)*b**(1/4)*sqrt(2) + 2*sqrt(x)*sq 
rt(c))/(c**(1/4)*b**(1/4)*sqrt(2)))*b**3*x**3 - 294*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*x**5 - 231*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*lo 
g( - sqrt(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*a*b*c*x**3 - 
 231*sqrt(x)*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*c**2*x**5 + 147*sqrt(x)*c**(3/4)*b**(1/4)*s 
qrt(2)*log( - sqrt(x)*c**(1/4)*b**(1/4)*sqrt(2) + sqrt(b) + sqrt(c)*x)*b** 
3*x**3 + 147*sqrt(x)*c**(3/4)*b**(1/4)*sqrt(2)*log( - sqrt(x)*c**(1/4)*...