\(\int \frac {(d x)^{23/2}}{(a^2+2 a b x^2+b^2 x^4)^{5/2}} \, dx\) [769]

Optimal result

Integrand size = 30, antiderivative size = 647 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

-1547/1024*d^7*(d*x)^(9/2)/b^4/((b*x^2+a)^2)^(1/2)-1/8*d*(d*x)^(21/2)/b/(b*x^2+a)^3/((b*x^2+a)^2)^(1/2)-7/32*d
^3*(d*x)^(17/2)/b^2/(b*x^2+a)^2/((b*x^2+a)^2)^(1/2)-119/256*d^5*(d*x)^(13/2)/b^3/(b*x^2+a)/((b*x^2+a)^2)^(1/2)
+13923/5120*d^9*(d*x)^(5/2)*(b*x^2+a)/b^5/((b*x^2+a)^2)^(1/2)-13923/4096*a^(5/4)*d^(23/2)*(b*x^2+a)*arctan(1-b
^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/b^(25/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)+13923/4096*a^(5/4)*d^(23/2)*(
b*x^2+a)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/b^(25/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)-13923/8192
*a^(5/4)*d^(23/2)*(b*x^2+a)*ln(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/b^(25/4)
*2^(1/2)/((b*x^2+a)^2)^(1/2)+13923/8192*a^(5/4)*d^(23/2)*(b*x^2+a)*ln(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)+a^(1/4
)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/b^(25/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)-13923/1024*a*d^11*(b*x^2+a)*(d*x)^(1/2)/b^
6/((b*x^2+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 294, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[In]

Int[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-1547*d^7*(d*x)^(9/2))/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(21/2))/(8*b*(a + b*x^2)^3*Sqrt[
a^2 + 2*a*b*x^2 + b^2*x^4]) - (7*d^3*(d*x)^(17/2))/(32*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (1
19*d^5*(d*x)^(13/2))/(256*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (13923*a*d^11*Sqrt[d*x]*(a + b*x^
2))/(1024*b^6*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*d^9*(d*x)^(5/2)*(a + b*x^2))/(5120*b^5*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4]) - (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d
])])/(2048*Sqrt[2]*b^(25/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*b^(25/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (139
23*a^(5/4)*d^(23/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/
(4096*Sqrt[2]*b^(25/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (13923*a^(5/4)*d^(23/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt
[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*b^(25/4)*Sqrt[a^2 + 2*a*b*x^2 + b^
2*x^4])

Rule 210

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 217

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 294

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 327

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 335

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 631

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 642

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 1126

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 1176

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 1179

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{23/2}}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 b^2 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^4} \, dx}{16 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (119 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{64 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (1547 d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{512 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx}{2048 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 a d^{10} \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{2048 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^2 d^{12} \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{2048 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^2 d^{11} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{3/2} d^{10} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2048 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{3/2} d^{10} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{2048 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 a^{5/4} d^{23/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{4096 \sqrt {2} b^{29/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 a^{5/4} d^{23/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{4096 \sqrt {2} b^{29/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{3/2} d^{12} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{4096 b^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{3/2} d^{12} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{4096 b^{15/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13923 a^{5/4} d^{23/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{29/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (13923 a^{5/4} d^{23/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{29/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a d^{11} \sqrt {d x} \left (a+b x^2\right )}{1024 b^6 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{2048 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{4096 \sqrt {2} b^{25/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.35 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {d^{11} \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (-69615 a^5-264537 a^4 b x^2-369733 a^3 b^2 x^4-220507 a^2 b^3 x^6-43008 a b^4 x^8+2048 b^5 x^{10}\right )+69615 \sqrt {2} a^{5/4} \left (a+b x^2\right )^4 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+69615 \sqrt {2} a^{5/4} \left (a+b x^2\right )^4 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{20480 b^{25/4} \sqrt {x} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \]

[In]

Integrate[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(d^11*Sqrt[d*x]*(4*b^(1/4)*Sqrt[x]*(-69615*a^5 - 264537*a^4*b*x^2 - 369733*a^3*b^2*x^4 - 220507*a^2*b^3*x^6 -
43008*a*b^4*x^8 + 2048*b^5*x^10) + 69615*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*
a^(1/4)*b^(1/4)*Sqrt[x])] + 69615*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqr
t[a] + Sqrt[b]*x)]))/(20480*b^(25/4)*Sqrt[x]*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.45

method	result	size
risch	\(-\frac {2 \left (-b \,x^{2}+25 a \right ) x \,d^{12} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 b^{6} \sqrt {d x}\, \left (b \,x^{2}+a \right )}+\frac {2 a^{2} d^{13} \left (\frac {-\frac {3683 a^{3} d^{6} \sqrt {d x}}{2048}-\frac {12357 a^{2} d^{4} b \left (d x \right )^{\frac {5}{2}}}{2048}-\frac {14145 a \,d^{2} b^{2} \left (d x \right )^{\frac {9}{2}}}{2048}-\frac {5599 b^{3} \left (d x \right )^{\frac {13}{2}}}{2048}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{4}}+\frac {13923 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16384 a \,d^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{6} \left (b \,x^{2}+a \right )}\)	\(288\)
default	\(\text {Expression too large to display}\)	\(1302\)

[In]

int((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*(-b*x^2+25*a)*x/b^6/(d*x)^(1/2)*d^12*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+2*a^2/b^6*d^13*((-3683/2048*a^3*d^6*(d
*x)^(1/2)-12357/2048*a^2*d^4*b*(d*x)^(5/2)-14145/2048*a*d^2*b^2*(d*x)^(9/2)-5599/2048*b^3*(d*x)^(13/2))/(b*d^2
*x^2+a*d^2)^4+13923/16384*(a*d^2/b)^(1/4)/a/d^2*2^(1/2)*(ln((d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)
^(1/2))/(d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2
)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)))*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.78 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} + 13923 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (-i \, b^{10} x^{8} - 4 i \, a b^{9} x^{6} - 6 i \, a^{2} b^{8} x^{4} - 4 i \, a^{3} b^{7} x^{2} - i \, a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} + 13923 i \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (i \, b^{10} x^{8} + 4 i \, a b^{9} x^{6} + 6 i \, a^{2} b^{8} x^{4} + 4 i \, a^{3} b^{7} x^{2} + i \, a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} - 13923 i \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) - 69615 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )} \log \left (13923 \, \sqrt {d x} a d^{11} - 13923 \, \left (-\frac {a^{5} d^{46}}{b^{25}}\right )^{\frac {1}{4}} b^{6}\right ) + 4 \, {\left (2048 \, b^{5} d^{11} x^{10} - 43008 \, a b^{4} d^{11} x^{8} - 220507 \, a^{2} b^{3} d^{11} x^{6} - 369733 \, a^{3} b^{2} d^{11} x^{4} - 264537 \, a^{4} b d^{11} x^{2} - 69615 \, a^{5} d^{11}\right )} \sqrt {d x}}{20480 \, {\left (b^{10} x^{8} + 4 \, a b^{9} x^{6} + 6 \, a^{2} b^{8} x^{4} + 4 \, a^{3} b^{7} x^{2} + a^{4} b^{6}\right )}} \]

[In]

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/20480*(69615*(-a^5*d^46/b^25)^(1/4)*(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4*b^6)*log(1
3923*sqrt(d*x)*a*d^11 + 13923*(-a^5*d^46/b^25)^(1/4)*b^6) - 69615*(-a^5*d^46/b^25)^(1/4)*(-I*b^10*x^8 - 4*I*a*
b^9*x^6 - 6*I*a^2*b^8*x^4 - 4*I*a^3*b^7*x^2 - I*a^4*b^6)*log(13923*sqrt(d*x)*a*d^11 + 13923*I*(-a^5*d^46/b^25)
^(1/4)*b^6) - 69615*(-a^5*d^46/b^25)^(1/4)*(I*b^10*x^8 + 4*I*a*b^9*x^6 + 6*I*a^2*b^8*x^4 + 4*I*a^3*b^7*x^2 + I
*a^4*b^6)*log(13923*sqrt(d*x)*a*d^11 - 13923*I*(-a^5*d^46/b^25)^(1/4)*b^6) - 69615*(-a^5*d^46/b^25)^(1/4)*(b^1
0*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4*b^6)*log(13923*sqrt(d*x)*a*d^11 - 13923*(-a^5*d^46/b
^25)^(1/4)*b^6) + 4*(2048*b^5*d^11*x^10 - 43008*a*b^4*d^11*x^8 - 220507*a^2*b^3*d^11*x^6 - 369733*a^3*b^2*d^11
*x^4 - 264537*a^4*b*d^11*x^2 - 69615*a^5*d^11)*sqrt(d*x))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*
x^2 + a^4*b^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((d*x)**(23/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {23}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")

[Out]

-4*a*d^(23/2)*integrate(x^(3/2)/(b^6*x^2 + a*b^5), x) + d^(23/2)*integrate(x^(7/2)/(b^5*x^2 + a*b^4), x) + 368
3/8192*(2*sqrt(2)*a^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b
)))/sqrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x)
)/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)) + sqrt(2)*a^(5/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)
*x + sqrt(a))/b^(1/4) - sqrt(2)*a^(5/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4))*d
^(23/2)/b^6 - 1/3072*(6925*a^2*b^3*d^(23/2)*x^(13/2) + 23395*a^3*b^2*d^(23/2)*x^(9/2) + 27135*a^4*b*d^(23/2)*x
^(5/2) + 11049*a^5*d^(23/2)*sqrt(x))/(b^10*x^8 + 4*a*b^9*x^6 + 6*a^2*b^8*x^4 + 4*a^3*b^7*x^2 + a^4*b^6) - 1/19
2*((617*a^2*b^4*d^(23/2)*x^5 + 1386*a^3*b^3*d^(23/2)*x^3 + 801*a^4*b^2*d^(23/2)*x)*x^(11/2) + 2*(519*a^3*b^3*d
^(23/2)*x^5 + 1182*a^4*b^2*d^(23/2)*x^3 + 695*a^5*b*d^(23/2)*x)*x^(7/2) + (453*a^4*b^2*d^(23/2)*x^5 + 1042*a^5
*b*d^(23/2)*x^3 + 621*a^6*d^(23/2)*x)*x^(3/2))/(a^3*b^8*x^6 + 3*a^4*b^7*x^4 + 3*a^5*b^6*x^2 + a^6*b^5 + (b^11*
x^6 + 3*a*b^10*x^4 + 3*a^2*b^9*x^2 + a^3*b^8)*x^6 + 3*(a*b^10*x^6 + 3*a^2*b^9*x^4 + 3*a^3*b^8*x^2 + a^4*b^7)*x
^4 + 3*(a^2*b^9*x^6 + 3*a^3*b^8*x^4 + 3*a^4*b^7*x^2 + a^5*b^6)*x^2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.64 \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\frac {1}{40960} \, d^{11} {\left (\frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{7} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{7} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {40 \, {\left (5599 \, \sqrt {d x} a^{2} b^{3} d^{8} x^{6} + 14145 \, \sqrt {d x} a^{3} b^{2} d^{8} x^{4} + 12357 \, \sqrt {d x} a^{4} b d^{8} x^{2} + 3683 \, \sqrt {d x} a^{5} d^{8}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{6} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {16384 \, {\left (\sqrt {d x} b^{20} d^{10} x^{2} - 25 \, \sqrt {d x} a b^{19} d^{10}\right )}}{b^{25} d^{10} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \]

[In]

integrate((d*x)^(23/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")

[Out]

1/40960*d^11*(139230*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a
*d^2/b)^(1/4))/(b^7*sgn(b*x^2 + a)) + 139230*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b
)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^7*sgn(b*x^2 + a)) + 69615*sqrt(2)*(a*b^3*d^2)^(1/4)*a*log(d*x + sqr
t(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^7*sgn(b*x^2 + a)) - 69615*sqrt(2)*(a*b^3*d^2)^(1/4)*a*log(d
*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^7*sgn(b*x^2 + a)) - 40*(5599*sqrt(d*x)*a^2*b^3*d^8*
x^6 + 14145*sqrt(d*x)*a^3*b^2*d^8*x^4 + 12357*sqrt(d*x)*a^4*b*d^8*x^2 + 3683*sqrt(d*x)*a^5*d^8)/((b*d^2*x^2 +
a*d^2)^4*b^6*sgn(b*x^2 + a)) + 16384*(sqrt(d*x)*b^20*d^10*x^2 - 25*sqrt(d*x)*a*b^19*d^10)/(b^25*d^10*sgn(b*x^2
+ a)))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d\,x\right )}^{23/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]

[In]

int((d*x)^(23/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)

[Out]

int((d*x)^(23/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)