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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.508811, antiderivative size = 647, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1112, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{13923 a d^{11} \sqrt{d x} \left (a+b x^2\right )}{1024 b^6 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 d^9 (d x)^{5/2} \left (a+b x^2\right )}{5120 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1547 d^7 (d x)^{9/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{119 d^5 (d x)^{13/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7 d^3 (d x)^{17/2}}{32 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{13923 a^{5/4} d^{23/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} b^{25/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{21/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[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 1112

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 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 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 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 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 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 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])

Rubi steps

\begin{align*} \int \frac{(d x)^{23/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.318006, size = 401, normalized size = 0.62 \[ \frac{(d x)^{23/2} \left (a+b x^2\right ) \left (-21446656 a^2 b^{13/4} x^{13/2}-39829504 a^3 b^{9/4} x^{9/2}-32587776 a^4 b^{5/4} x^{5/2}+2042040 a^2 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^3+1166880 a^3 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2+848640 a^4 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )-765765 \sqrt{2} a^{5/4} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+765765 \sqrt{2} a^{5/4} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-1531530 \sqrt{2} a^{5/4} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+1531530 \sqrt{2} a^{5/4} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-10183680 a^5 \sqrt [4]{b} \sqrt{x}-3784704 a b^{17/4} x^{17/2}+180224 b^{21/4} x^{21/2}\right )}{450560 b^{25/4} x^{23/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d*x)^(23/2)*(a + b*x^2)*(-10183680*a^5*b^(1/4)*Sqrt[x] - 32587776*a^4*b^(5/4)*x^(5/2) - 39829504*a^3*b^(9/4)
*x^(9/2) - 21446656*a^2*b^(13/4)*x^(13/2) - 3784704*a*b^(17/4)*x^(17/2) + 180224*b^(21/4)*x^(21/2) + 848640*a^
4*b^(1/4)*Sqrt[x]*(a + b*x^2) + 1166880*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 2042040*a^2*b^(1/4)*Sqrt[x]*(a + b
*x^2)^3 - 1531530*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 1531530*Sqrt[2
]*a^(5/4)*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 765765*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 765765*Sqrt[2]*a^(5/4)*(a + b*x^2)^4*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(450560*b^(25/4)*x^(23/2)*((a + b*x^2)^2)^(5/2))

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Maple [B]  time = 0.236, size = 1287, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40960*(139230*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^5*d^6+
139230*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^5*d^6-1638400*(
d*x)^(1/2)*x^2*a^4*b*d^6+65536*(d*x)^(5/2)*x^6*a*b^4*d^4-409600*(d*x)^(1/2)*x^8*a*b^4*d^6+98304*(d*x)^(5/2)*x^
4*a^2*b^3*d^4-1638400*(d*x)^(1/2)*x^6*a^2*b^3*d^6+65536*(d*x)^(5/2)*x^2*a^3*b^2*d^4-2457600*(d*x)^(1/2)*x^4*a^
3*b^2*d^6+69615*(a*d^2/b)^(1/4)*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)))*a^5*d^6-223960*(d*x)^(13/2)*a^2*b^3+16384*(d*x)^(5/2)*x^8*b^
5*d^4-565800*(d*x)^(9/2)*a^3*b^2*d^2-477896*(d*x)^(5/2)*a^4*b*d^4-556920*(d*x)^(1/2)*a^5*d^6+69615*(a*d^2/b)^(
1/4)*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)))*x^8*a*b^4*d^6+556920*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4
))/(a*d^2/b)^(1/4))*x^6*a^2*b^3*d^6+417690*(a*d^2/b)^(1/4)*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)))*x^4*a^3*b^2*d^6+835380*(a*d^2/b)^
(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*a^3*b^2*d^6+835380*(a*d^2/b)^(
1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*a^3*b^2*d^6+278460*(a*d^2/b)^(1
/4)*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)))*x^2*a^4*b*d^6+556920*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4)
)/(a*d^2/b)^(1/4))*x^2*a^4*b*d^6+556920*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(
a*d^2/b)^(1/4))*x^2*a^4*b*d^6+139230*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d
^2/b)^(1/4))*x^8*a*b^4*d^6+139230*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)-(a*d^2/b)^(1/4))/(a*d^2/
b)^(1/4))*x^8*a*b^4*d^6+278460*(a*d^2/b)^(1/4)*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)))*x^6*a^2*b^3*d^6+556920*(a*d^2/b)^(1/4)*2^(1/2
)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^6*a^2*b^3*d^6)*d^5*(b*x^2+a)/b^6/((b*x^2+a)^
2)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72312, size = 1067, normalized size = 1.65 \begin{align*} \frac{278460 \, \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 )} \arctan \left (-\frac{\left (-\frac{a^{5} d^{46}}{b^{25}}\right )^{\frac{3}{4}} \sqrt{d x} a b^{19} d^{11} - \left (-\frac{a^{5} d^{46}}{b^{25}}\right )^{\frac{3}{4}} \sqrt{a^{2} d^{23} x + \sqrt{-\frac{a^{5} d^{46}}{b^{25}}} b^{12}} b^{19}}{a^{5} d^{46}}\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 ) - 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*(278460*(-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)*arct
an(-((-a^5*d^46/b^25)^(3/4)*sqrt(d*x)*a*b^19*d^11 - (-a^5*d^46/b^25)^(3/4)*sqrt(a^2*d^23*x + sqrt(-a^5*d^46/b^
25)*b^12)*b^19)/(a^5*d^46)) + 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(13923*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)*
(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(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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.36281, size = 622, normalized size = 0.96 \begin{align*} \frac{1}{40960} \, d^{10}{\left (\frac{139230 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \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 d^{4} x^{2} + a d^{4}\right )} + \frac{139230 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \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 d^{4} x^{2} + a d^{4}\right )} + \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \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 d^{4} x^{2} + a d^{4}\right )} - \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} a d \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 d^{4} x^{2} + a d^{4}\right )} - \frac{40 \,{\left (5599 \, \sqrt{d x} a^{2} b^{3} d^{9} x^{6} + 14145 \, \sqrt{d x} a^{3} b^{2} d^{9} x^{4} + 12357 \, \sqrt{d x} a^{4} b d^{9} x^{2} + 3683 \, \sqrt{d x} a^{5} d^{9}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{6} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{16384 \,{\left (\sqrt{d x} b^{20} d^{6} x^{2} - 25 \, \sqrt{d x} a b^{19} d^{6}\right )}}{b^{25} d^{5} \mathrm{sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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^10*(139230*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*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*d^4*x^2 + a*d^4)) + 139230*sqrt(2)*(a*b^3*d^2)^(1/4)*a*d*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*d^4*x^2 + a*d^4)) + 69615*sqrt(2)*(a*b^3*d^2)^(
1/4)*a*d*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^7*sgn(b*d^4*x^2 + a*d^4)) - 69615*sqr
t(2)*(a*b^3*d^2)^(1/4)*a*d*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^7*sgn(b*d^4*x^2 + a
*d^4)) - 40*(5599*sqrt(d*x)*a^2*b^3*d^9*x^6 + 14145*sqrt(d*x)*a^3*b^2*d^9*x^4 + 12357*sqrt(d*x)*a^4*b*d^9*x^2
+ 3683*sqrt(d*x)*a^5*d^9)/((b*d^2*x^2 + a*d^2)^4*b^6*sgn(b*d^4*x^2 + a*d^4)) + 16384*(sqrt(d*x)*b^20*d^6*x^2 -
25*sqrt(d*x)*a*b^19*d^6)/(b^25*d^5*sgn(b*d^4*x^2 + a*d^4)))