3.207 \(\int \frac {x^{23/2} (A+B x^2)}{(b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=343 \[ -\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} c^{17/4}}-\frac {9 \sqrt {x} (13 b B-5 A c)}{16 c^4}+\frac {9 x^{5/2} (13 b B-5 A c)}{80 b c^3}-\frac {x^{9/2} (13 b B-5 A c)}{16 b c^2 \left (b+c x^2\right )}-\frac {x^{13/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

[Out]

9/80*(-5*A*c+13*B*b)*x^(5/2)/b/c^3-1/4*(-A*c+B*b)*x^(13/2)/b/c/(c*x^2+b)^2-1/16*(-5*A*c+13*B*b)*x^(9/2)/b/c^2/
(c*x^2+b)-9/64*b^(1/4)*(-5*A*c+13*B*b)*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/c^(17/4)*2^(1/2)+9/64*b^(1/4)
*(-5*A*c+13*B*b)*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/b^(1/4))/c^(17/4)*2^(1/2)-9/128*b^(1/4)*(-5*A*c+13*B*b)*ln(b
^(1/2)+x*c^(1/2)-b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))/c^(17/4)*2^(1/2)+9/128*b^(1/4)*(-5*A*c+13*B*b)*ln(b^(1/2)+x*
c^(1/2)+b^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))/c^(17/4)*2^(1/2)-9/16*(-5*A*c+13*B*b)*x^(1/2)/c^4

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Rubi [A]  time = 0.28, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1584, 457, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {x^{9/2} (13 b B-5 A c)}{16 b c^2 \left (b+c x^2\right )}+\frac {9 x^{5/2} (13 b B-5 A c)}{80 b c^3}-\frac {9 \sqrt {x} (13 b B-5 A c)}{16 c^4}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt {2} c^{17/4}}-\frac {x^{13/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^(23/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(-9*(13*b*B - 5*A*c)*Sqrt[x])/(16*c^4) + (9*(13*b*B - 5*A*c)*x^(5/2))/(80*b*c^3) - ((b*B - A*c)*x^(13/2))/(4*b
*c*(b + c*x^2)^2) - ((13*b*B - 5*A*c)*x^(9/2))/(16*b*c^2*(b + c*x^2)) - (9*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 -
(Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*c^(17/4)) + (9*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 + (Sqrt[2]*c
^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*c^(17/4)) - (9*b^(1/4)*(13*b*B - 5*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^
(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*c^(17/4)) + (9*b^(1/4)*(13*b*B - 5*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*
c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*c^(17/4))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[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 288

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

Rule 321

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 457

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {x^{23/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^{11/2} \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}+\frac {\left (\frac {13 b B}{2}-\frac {5 A c}{2}\right ) \int \frac {x^{11/2}}{\left (b+c x^2\right )^2} \, dx}{4 b c}\\ &=-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {(9 (13 b B-5 A c)) \int \frac {x^{7/2}}{b+c x^2} \, dx}{32 b c^2}\\ &=\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac {(9 (13 b B-5 A c)) \int \frac {x^{3/2}}{b+c x^2} \, dx}{32 c^3}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {(9 b (13 b B-5 A c)) \int \frac {1}{\sqrt {x} \left (b+c x^2\right )} \, dx}{32 c^4}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {(9 b (13 b B-5 A c)) \operatorname {Subst}\left (\int \frac {1}{b+c x^4} \, dx,x,\sqrt {x}\right )}{16 c^4}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {\left (9 \sqrt {b} (13 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}-\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 c^4}+\frac {\left (9 \sqrt {b} (13 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b}+\sqrt {c} x^2}{b+c x^4} \, dx,x,\sqrt {x}\right )}{32 c^4}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}+\frac {\left (9 \sqrt {b} (13 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{9/2}}+\frac {\left (9 \sqrt {b} (13 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{64 c^{9/2}}-\frac {\left (9 \sqrt [4]{b} (13 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {b}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{17/4}}-\frac {\left (9 \sqrt [4]{b} (13 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{b}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {b}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} c^{17/4}}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {\left (9 \sqrt [4]{b} (13 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}-\frac {\left (9 \sqrt [4]{b} (13 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}\\ &=-\frac {9 (13 b B-5 A c) \sqrt {x}}{16 c^4}+\frac {9 (13 b B-5 A c) x^{5/2}}{80 b c^3}-\frac {(b B-A c) x^{13/2}}{4 b c \left (b+c x^2\right )^2}-\frac {(13 b B-5 A c) x^{9/2}}{16 b c^2 \left (b+c x^2\right )}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )}{32 \sqrt {2} c^{17/4}}-\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}+\frac {9 \sqrt [4]{b} (13 b B-5 A c) \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{64 \sqrt {2} c^{17/4}}\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 435, normalized size = 1.27 \[ \frac {-\frac {160 A b^2 c^{5/4} \sqrt {x}}{\left (b+c x^2\right )^2}-90 \sqrt {2} \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )+90 \sqrt {2} \sqrt [4]{b} (13 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}+1\right )+\frac {680 A b c^{5/4} \sqrt {x}}{b+c x^2}+225 \sqrt {2} A \sqrt [4]{b} c \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )-225 \sqrt {2} A \sqrt [4]{b} c \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )+1280 A c^{5/4} \sqrt {x}-585 \sqrt {2} b^{5/4} B \log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )+585 \sqrt {2} b^{5/4} B \log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{c} \sqrt {x}+\sqrt {b}+\sqrt {c} x\right )+\frac {160 b^3 B \sqrt [4]{c} \sqrt {x}}{\left (b+c x^2\right )^2}-\frac {1000 b^2 B \sqrt [4]{c} \sqrt {x}}{b+c x^2}-3840 b B \sqrt [4]{c} \sqrt {x}+256 B c^{5/4} x^{5/2}}{640 c^{17/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(23/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(-3840*b*B*c^(1/4)*Sqrt[x] + 1280*A*c^(5/4)*Sqrt[x] + 256*B*c^(5/4)*x^(5/2) + (160*b^3*B*c^(1/4)*Sqrt[x])/(b +
c*x^2)^2 - (160*A*b^2*c^(5/4)*Sqrt[x])/(b + c*x^2)^2 - (1000*b^2*B*c^(1/4)*Sqrt[x])/(b + c*x^2) + (680*A*b*c^
(5/4)*Sqrt[x])/(b + c*x^2) - 90*Sqrt[2]*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]
+ 90*Sqrt[2]*b^(1/4)*(13*b*B - 5*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 585*Sqrt[2]*b^(5/4)*B*L
og[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 225*Sqrt[2]*A*b^(1/4)*c*Log[Sqrt[b] - Sqrt[2]*b^(1
/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 585*Sqrt[2]*b^(5/4)*B*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[
c]*x] - 225*Sqrt[2]*A*b^(1/4)*c*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(640*c^(17/4))

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fricas [B]  time = 1.01, size = 817, normalized size = 2.38 \[ -\frac {180 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {c^{8} \sqrt {-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}} + {\left (169 \, B^{2} b^{2} - 130 \, A B b c + 25 \, A^{2} c^{2}\right )} x} c^{13} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {3}{4}} + {\left (13 \, B b c^{13} - 5 \, A c^{14}\right )} \sqrt {x} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {3}{4}}}{28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}\right ) + 45 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} \log \left (9 \, c^{4} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} - 9 \, {\left (13 \, B b - 5 \, A c\right )} \sqrt {x}\right ) - 45 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} \log \left (-9 \, c^{4} \left (-\frac {28561 \, B^{4} b^{5} - 43940 \, A B^{3} b^{4} c + 25350 \, A^{2} B^{2} b^{3} c^{2} - 6500 \, A^{3} B b^{2} c^{3} + 625 \, A^{4} b c^{4}}{c^{17}}\right )^{\frac {1}{4}} - 9 \, {\left (13 \, B b - 5 \, A c\right )} \sqrt {x}\right ) - 4 \, {\left (32 \, B c^{3} x^{6} - 32 \, {\left (13 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} - 585 \, B b^{3} + 225 \, A b^{2} c - 81 \, {\left (13 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2}\right )} \sqrt {x}}{320 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="fricas")

[Out]

-1/320*(180*(c^6*x^4 + 2*b*c^5*x^2 + b^2*c^4)*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6
500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4)*arctan((sqrt(c^8*sqrt(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 253
50*A^2*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17) + (169*B^2*b^2 - 130*A*B*b*c + 25*A^2*c^2)*x)*c
^13*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(
3/4) + (13*B*b*c^13 - 5*A*c^14)*sqrt(x)*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^
3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(3/4))/(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^
3*B*b^2*c^3 + 625*A^4*b*c^4)) + 45*(c^6*x^4 + 2*b*c^5*x^2 + b^2*c^4)*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25
350*A^2*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4)*log(9*c^4*(-(28561*B^4*b^5 - 43940*A*B^3
*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4) - 9*(13*B*b - 5*A*c)*sqrt(x))
- 45*(c^6*x^4 + 2*b*c^5*x^2 + b^2*c^4)*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c^2 - 6500*A^
3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4)*log(-9*c^4*(-(28561*B^4*b^5 - 43940*A*B^3*b^4*c + 25350*A^2*B^2*b^3*c
^2 - 6500*A^3*B*b^2*c^3 + 625*A^4*b*c^4)/c^17)^(1/4) - 9*(13*B*b - 5*A*c)*sqrt(x)) - 4*(32*B*c^3*x^6 - 32*(13*
B*b*c^2 - 5*A*c^3)*x^4 - 585*B*b^3 + 225*A*b^2*c - 81*(13*B*b^2*c - 5*A*b*c^2)*x^2)*sqrt(x))/(c^6*x^4 + 2*b*c^
5*x^2 + b^2*c^4)

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giac [A]  time = 0.23, size = 321, normalized size = 0.94 \[ \frac {9 \, \sqrt {2} {\left (13 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, c^{5}} + \frac {9 \, \sqrt {2} {\left (13 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {b}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {b}{c}\right )^{\frac {1}{4}}}\right )}{64 \, c^{5}} + \frac {9 \, \sqrt {2} {\left (13 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, c^{5}} - \frac {9 \, \sqrt {2} {\left (13 \, \left (b c^{3}\right )^{\frac {1}{4}} B b - 5 \, \left (b c^{3}\right )^{\frac {1}{4}} A c\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {b}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {b}{c}}\right )}{128 \, c^{5}} - \frac {25 \, B b^{2} c x^{\frac {5}{2}} - 17 \, A b c^{2} x^{\frac {5}{2}} + 21 \, B b^{3} \sqrt {x} - 13 \, A b^{2} c \sqrt {x}}{16 \, {\left (c x^{2} + b\right )}^{2} c^{4}} + \frac {2 \, {\left (B c^{12} x^{\frac {5}{2}} - 15 \, B b c^{11} \sqrt {x} + 5 \, A c^{12} \sqrt {x}\right )}}{5 \, c^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="giac")

[Out]

9/64*sqrt(2)*(13*(b*c^3)^(1/4)*B*b - 5*(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))/c^5 + 9/64*sqrt(2)*(13*(b*c^3)^(1/4)*B*b - 5*(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))/c^5 + 9/128*sqrt(2)*(13*(b*c^3)^(1/4)*B*b - 5*(b*c^3)^(1/4)*A*c)*log(sqrt(2
)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^5 - 9/128*sqrt(2)*(13*(b*c^3)^(1/4)*B*b - 5*(b*c^3)^(1/4)*A*c)*log(-s
qrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^5 - 1/16*(25*B*b^2*c*x^(5/2) - 17*A*b*c^2*x^(5/2) + 21*B*b^3*sqr
t(x) - 13*A*b^2*c*sqrt(x))/((c*x^2 + b)^2*c^4) + 2/5*(B*c^12*x^(5/2) - 15*B*b*c^11*sqrt(x) + 5*A*c^12*sqrt(x))
/c^15

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maple [A]  time = 0.08, size = 381, normalized size = 1.11 \[ \frac {17 A b \,x^{\frac {5}{2}}}{16 \left (c \,x^{2}+b \right )^{2} c^{2}}-\frac {25 B \,b^{2} x^{\frac {5}{2}}}{16 \left (c \,x^{2}+b \right )^{2} c^{3}}+\frac {13 A \,b^{2} \sqrt {x}}{16 \left (c \,x^{2}+b \right )^{2} c^{3}}-\frac {21 B \,b^{3} \sqrt {x}}{16 \left (c \,x^{2}+b \right )^{2} c^{4}}+\frac {2 B \,x^{\frac {5}{2}}}{5 c^{3}}-\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 c^{3}}-\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 c^{3}}-\frac {45 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, A \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{128 c^{3}}+\frac {117 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}-1\right )}{64 c^{4}}+\frac {117 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {b}{c}\right )^{\frac {1}{4}}}+1\right )}{64 c^{4}}+\frac {117 \left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, B b \ln \left (\frac {x +\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}{x -\left (\frac {b}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {b}{c}}}\right )}{128 c^{4}}+\frac {2 A \sqrt {x}}{c^{3}}-\frac {6 B b \sqrt {x}}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(23/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

2/5/c^3*B*x^(5/2)+2/c^3*A*x^(1/2)-6/c^4*b*B*x^(1/2)+17/16*b/c^2/(c*x^2+b)^2*x^(5/2)*A-25/16*b^2/c^3/(c*x^2+b)^
2*x^(5/2)*B+13/16*b^2/c^3/(c*x^2+b)^2*A*x^(1/2)-21/16*b^3/c^4/(c*x^2+b)^2*B*x^(1/2)-45/64/c^3*(b/c)^(1/4)*2^(1
/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-45/64/c^3*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)
-1)-45/128/c^3*(b/c)^(1/4)*2^(1/2)*A*ln((x+(b/c)^(1/4)*2^(1/2)*x^(1/2)+(b/c)^(1/2))/(x-(b/c)^(1/4)*2^(1/2)*x^(
1/2)+(b/c)^(1/2)))+117/64*b/c^4*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+117/64*b/c^4*(b/c)
^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+117/128*b/c^4*(b/c)^(1/4)*2^(1/2)*B*ln((x+(b/c)^(1/4)*2
^(1/2)*x^(1/2)+(b/c)^(1/2))/(x-(b/c)^(1/4)*2^(1/2)*x^(1/2)+(b/c)^(1/2)))

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maxima [A]  time = 3.08, size = 306, normalized size = 0.89 \[ -\frac {{\left (25 \, B b^{2} c - 17 \, A b c^{2}\right )} x^{\frac {5}{2}} + {\left (21 \, B b^{3} - 13 \, A b^{2} c\right )} \sqrt {x}}{16 \, {\left (c^{6} x^{4} + 2 \, b c^{5} x^{2} + b^{2} c^{4}\right )}} + \frac {9 \, {\left (\frac {2 \, \sqrt {2} {\left (13 \, B b - 5 \, A c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {2 \, \sqrt {2} {\left (13 \, B b - 5 \, A c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {b} \sqrt {c}}}\right )}{\sqrt {b} \sqrt {\sqrt {b} \sqrt {c}}} + \frac {\sqrt {2} {\left (13 \, B b - 5 \, A c\right )} \log \left (\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (13 \, B b - 5 \, A c\right )} \log \left (-\sqrt {2} b^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {b}\right )}{b^{\frac {3}{4}} c^{\frac {1}{4}}}\right )} b}{128 \, c^{4}} + \frac {2 \, {\left (B c x^{\frac {5}{2}} - 5 \, {\left (3 \, B b - A c\right )} \sqrt {x}\right )}}{5 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(23/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x, algorithm="maxima")

[Out]

-1/16*((25*B*b^2*c - 17*A*b*c^2)*x^(5/2) + (21*B*b^3 - 13*A*b^2*c)*sqrt(x))/(c^6*x^4 + 2*b*c^5*x^2 + b^2*c^4)
+ 9/128*(2*sqrt(2)*(13*B*b - 5*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)*(13*B*b - 5*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))) + sqrt(2)*(13*B*b - 5*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)*(13*B*b - 5*A*c)*log
(-sqrt(2)*b^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(b))/(b^(3/4)*c^(1/4)))*b/c^4 + 2/5*(B*c*x^(5/2) - 5*(3*B*
b - A*c)*sqrt(x))/c^4

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mupad [B]  time = 0.25, size = 865, normalized size = 2.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(23/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x)

[Out]

(x^(5/2)*((17*A*b*c^2)/16 - (25*B*b^2*c)/16) - x^(1/2)*((21*B*b^3)/16 - (13*A*b^2*c)/16))/(b^2*c^4 + c^6*x^4 +
2*b*c^5*x^2) + x^(1/2)*((2*A)/c^3 - (6*B*b)/c^4) + (2*B*x^(5/2))/(5*c^3) + ((-b)^(1/4)*atan((((-b)^(1/4)*((81
*x^(1/2)*(169*B^2*b^4 + 25*A^2*b^2*c^2 - 130*A*B*b^3*c))/(64*c^5) - (81*(-b)^(1/4)*(5*A*c - 13*B*b)*(13*B*b^3
- 5*A*b^2*c))/(64*c^(21/4)))*(5*A*c - 13*B*b)*9i)/(64*c^(17/4)) + ((-b)^(1/4)*((81*x^(1/2)*(169*B^2*b^4 + 25*A
^2*b^2*c^2 - 130*A*B*b^3*c))/(64*c^5) + (81*(-b)^(1/4)*(5*A*c - 13*B*b)*(13*B*b^3 - 5*A*b^2*c))/(64*c^(21/4)))
*(5*A*c - 13*B*b)*9i)/(64*c^(17/4)))/((9*(-b)^(1/4)*((81*x^(1/2)*(169*B^2*b^4 + 25*A^2*b^2*c^2 - 130*A*B*b^3*c
))/(64*c^5) - (81*(-b)^(1/4)*(5*A*c - 13*B*b)*(13*B*b^3 - 5*A*b^2*c))/(64*c^(21/4)))*(5*A*c - 13*B*b))/(64*c^(
17/4)) - (9*(-b)^(1/4)*((81*x^(1/2)*(169*B^2*b^4 + 25*A^2*b^2*c^2 - 130*A*B*b^3*c))/(64*c^5) + (81*(-b)^(1/4)*
(5*A*c - 13*B*b)*(13*B*b^3 - 5*A*b^2*c))/(64*c^(21/4)))*(5*A*c - 13*B*b))/(64*c^(17/4))))*(5*A*c - 13*B*b)*9i)
/(32*c^(17/4)) + (9*(-b)^(1/4)*atan(((9*(-b)^(1/4)*((81*x^(1/2)*(169*B^2*b^4 + 25*A^2*b^2*c^2 - 130*A*B*b^3*c)
)/(64*c^5) - ((-b)^(1/4)*(5*A*c - 13*B*b)*(13*B*b^3 - 5*A*b^2*c)*81i)/(64*c^(21/4)))*(5*A*c - 13*B*b))/(64*c^(
17/4)) + (9*(-b)^(1/4)*((81*x^(1/2)*(169*B^2*b^4 + 25*A^2*b^2*c^2 - 130*A*B*b^3*c))/(64*c^5) + ((-b)^(1/4)*(5*
A*c - 13*B*b)*(13*B*b^3 - 5*A*b^2*c)*81i)/(64*c^(21/4)))*(5*A*c - 13*B*b))/(64*c^(17/4)))/(((-b)^(1/4)*((81*x^
(1/2)*(169*B^2*b^4 + 25*A^2*b^2*c^2 - 130*A*B*b^3*c))/(64*c^5) - ((-b)^(1/4)*(5*A*c - 13*B*b)*(13*B*b^3 - 5*A*
b^2*c)*81i)/(64*c^(21/4)))*(5*A*c - 13*B*b)*9i)/(64*c^(17/4)) - ((-b)^(1/4)*((81*x^(1/2)*(169*B^2*b^4 + 25*A^2
*b^2*c^2 - 130*A*B*b^3*c))/(64*c^5) + ((-b)^(1/4)*(5*A*c - 13*B*b)*(13*B*b^3 - 5*A*b^2*c)*81i)/(64*c^(21/4)))*
(5*A*c - 13*B*b)*9i)/(64*c^(17/4))))*(5*A*c - 13*B*b))/(32*c^(17/4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(23/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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