3.712 \(\int \frac{(d x)^{23/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=402 \[ -\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{13923 \sqrt [4]{a} d^{23/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{25/4}}+\frac{13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} b^{25/4}}-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}+\frac{13923 d^{11} \sqrt{d x}}{4096 b^6} \]

[Out]

(13923*d^11*Sqrt[d*x])/(4096*b^6) - (d*(d*x)^(21/2))/(10*b*(a + b*x^2)^5) - (21*d^3*(d*x)^(17/2))/(160*b^2*(a
+ b*x^2)^4) - (119*d^5*(d*x)^(13/2))/(640*b^3*(a + b*x^2)^3) - (1547*d^7*(d*x)^(9/2))/(5120*b^4*(a + b*x^2)^2)
- (13923*d^9*(d*x)^(5/2))/(20480*b^5*(a + b*x^2)) + (13923*a^(1/4)*d^(23/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[
d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(25/4)) - (13923*a^(1/4)*d^(23/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(25/4)) + (13923*a^(1/4)*d^(23/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[
d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(25/4)) - (13923*a^(1/4)*d^(23/2)*Log[Sqrt[a]*Sqrt
[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(25/4))

________________________________________________________________________________________

Rubi [A]  time = 0.485538, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {28, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{13923 \sqrt [4]{a} d^{23/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{25/4}}+\frac{13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} b^{25/4}}-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}+\frac{13923 d^{11} \sqrt{d x}}{4096 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(13923*d^11*Sqrt[d*x])/(4096*b^6) - (d*(d*x)^(21/2))/(10*b*(a + b*x^2)^5) - (21*d^3*(d*x)^(17/2))/(160*b^2*(a
+ b*x^2)^4) - (119*d^5*(d*x)^(13/2))/(640*b^3*(a + b*x^2)^3) - (1547*d^7*(d*x)^(9/2))/(5120*b^4*(a + b*x^2)^2)
- (13923*d^9*(d*x)^(5/2))/(20480*b^5*(a + b*x^2)) + (13923*a^(1/4)*d^(23/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[
d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(25/4)) - (13923*a^(1/4)*d^(23/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(25/4)) + (13923*a^(1/4)*d^(23/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[
d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(25/4)) - (13923*a^(1/4)*d^(23/2)*Log[Sqrt[a]*Sqrt
[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(25/4))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 )^3} \, dx &=b^6 \int \frac{(d x)^{23/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}+\frac{1}{20} \left (21 b^4 d^2\right ) \int \frac{(d x)^{19/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{1}{320} \left (357 b^2 d^4\right ) \int \frac{(d x)^{15/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}+\frac{\left (1547 d^6\right ) \int \frac{(d x)^{11/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280}\\ &=-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}+\frac{\left (13923 d^8\right ) \int \frac{(d x)^{7/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{10240 b^2}\\ &=-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac{\left (13923 d^{10}\right ) \int \frac{(d x)^{3/2}}{a b+b^2 x^2} \, dx}{8192 b^4}\\ &=\frac{13923 d^{11} \sqrt{d x}}{4096 b^6}-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac{\left (13923 a d^{12}\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{8192 b^5}\\ &=\frac{13923 d^{11} \sqrt{d x}}{4096 b^6}-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac{\left (13923 a d^{11}\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 b^5}\\ &=\frac{13923 d^{11} \sqrt{d x}}{4096 b^6}-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}-\frac{\left (13923 \sqrt{a} d^{10}\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 )}{8192 b^5}-\frac{\left (13923 \sqrt{a} d^{10}\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 )}{8192 b^5}\\ &=\frac{13923 d^{11} \sqrt{d x}}{4096 b^6}-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac{\left (13923 \sqrt [4]{a} d^{23/2}\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 )}{16384 \sqrt{2} b^{25/4}}+\frac{\left (13923 \sqrt [4]{a} d^{23/2}\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 )}{16384 \sqrt{2} b^{25/4}}-\frac{\left (13923 \sqrt{a} d^{12}\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 )}{16384 b^{13/2}}-\frac{\left (13923 \sqrt{a} d^{12}\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 )}{16384 b^{13/2}}\\ &=\frac{13923 d^{11} \sqrt{d x}}{4096 b^6}-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac{13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} b^{25/4}}-\frac{\left (13923 \sqrt [4]{a} d^{23/2}\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 )}{8192 \sqrt{2} b^{25/4}}+\frac{\left (13923 \sqrt [4]{a} d^{23/2}\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 )}{8192 \sqrt{2} b^{25/4}}\\ &=\frac{13923 d^{11} \sqrt{d x}}{4096 b^6}-\frac{d (d x)^{21/2}}{10 b \left (a+b x^2\right )^5}-\frac{21 d^3 (d x)^{17/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{119 d^5 (d x)^{13/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac{1547 d^7 (d x)^{9/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac{13923 d^9 (d x)^{5/2}}{20480 b^5 \left (a+b x^2\right )}+\frac{13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt [4]{a} d^{23/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{25/4}}+\frac{13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} b^{25/4}}-\frac{13923 \sqrt [4]{a} d^{23/2} \log \left (\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}\right )}{16384 \sqrt{2} b^{25/4}}\\ \end{align*}

Mathematica [A]  time = 0.296907, size = 408, normalized size = 1.01 \[ \frac{d^{11} \sqrt{d x} \left (\frac{61276160 a^2 b^{13/4} x^{13/2}[phone] a^3 b^{9/4} x^{9/2}[phone] a^4 b^{5/4} x^{5/2}-1166880 a^2 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^3-848640 a^3 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2-678912 a^4 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )[phone] a^5 \sqrt [4]{b} \sqrt{x}[phone] a b^{17/4} x^{17/2}-2042040 a \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^4+765765 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-765765 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-1531530 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+3604480 b^{21/4} x^{21/2}}{\left (a+b x^2\right )^5}+1531530 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )\right )}{1802240 b^{25/4} \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(23/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^11*Sqrt[d*x]*(1531530*Sqrt[2]*a^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + (10862592*a^5*b^(1/4)
*Sqrt[x] + 43450368*a^4*b^(5/4)*x^(5/2) + 72417280*a^3*b^(9/4)*x^(9/2) + 61276160*a^2*b^(13/4)*x^(13/2) + 2523
1360*a*b^(17/4)*x^(17/2) + 3604480*b^(21/4)*x^(21/2) - 678912*a^4*b^(1/4)*Sqrt[x]*(a + b*x^2) - 848640*a^3*b^(
1/4)*Sqrt[x]*(a + b*x^2)^2 - 1166880*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 - 2042040*a*b^(1/4)*Sqrt[x]*(a + b*x^2)
^4 - 1531530*Sqrt[2]*a^(1/4)*(a + b*x^2)^5*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 765765*Sqrt[2]*a^(1
/4)*(a + b*x^2)^5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 765765*Sqrt[2]*a^(1/4)*(a + b*x
^2)^5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a + b*x^2)^5))/(1802240*b^(25/4)*Sqrt[x])

________________________________________________________________________________________

Maple [A]  time = 0.072, size = 351, normalized size = 0.9 \begin{align*} 2\,{\frac{{d}^{11}\sqrt{dx}}{{b}^{6}}}+{\frac{5731\,{d}^{21}{a}^{5}}{4096\,{b}^{6} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}}\sqrt{dx}}+{\frac{16169\,{d}^{19}{a}^{4}}{2560\,{b}^{5} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{22467\,{d}^{17}{a}^{3}}{2048\,{b}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{1129\,{d}^{15}{a}^{2}}{128\,{b}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{13}{2}}}}+{\frac{11743\,{d}^{13}a}{4096\,{b}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{17}{2}}}}-{\frac{13923\,{d}^{11}\sqrt{2}}{32768\,{b}^{6}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({ \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }-{\frac{13923\,{d}^{11}\sqrt{2}}{16384\,{b}^{6}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }-{\frac{13923\,{d}^{11}\sqrt{2}}{16384\,{b}^{6}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \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)^3,x)

[Out]

2*d^11*(d*x)^(1/2)/b^6+5731/4096*d^21/b^6*a^5/(b*d^2*x^2+a*d^2)^5*(d*x)^(1/2)+16169/2560*d^19/b^5*a^4/(b*d^2*x
^2+a*d^2)^5*(d*x)^(5/2)+22467/2048*d^17/b^4*a^3/(b*d^2*x^2+a*d^2)^5*(d*x)^(9/2)+1129/128*d^15/b^3*a^2/(b*d^2*x
^2+a*d^2)^5*(d*x)^(13/2)+11743/4096*d^13/b^2*a/(b*d^2*x^2+a*d^2)^5*(d*x)^(17/2)-13923/32768*d^11/b^6*(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)))-13923/16384*d^11/b^6*(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/
2)+1)-13923/16384*d^11/b^6*(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)

________________________________________________________________________________________

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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.69568, size = 1141, normalized size = 2.84 \begin{align*} -\frac{278460 \, \left (-\frac{a d^{46}}{b^{25}}\right )^{\frac{1}{4}}{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \arctan \left (-\frac{\left (-\frac{a d^{46}}{b^{25}}\right )^{\frac{3}{4}} \sqrt{d x} b^{19} d^{11} - \sqrt{d^{23} x + \sqrt{-\frac{a d^{46}}{b^{25}}} b^{12}} \left (-\frac{a d^{46}}{b^{25}}\right )^{\frac{3}{4}} b^{19}}{a d^{46}}\right ) + 69615 \, \left (-\frac{a d^{46}}{b^{25}}\right )^{\frac{1}{4}}{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (13923 \, \sqrt{d x} d^{11} + 13923 \, \left (-\frac{a d^{46}}{b^{25}}\right )^{\frac{1}{4}} b^{6}\right ) - 69615 \, \left (-\frac{a d^{46}}{b^{25}}\right )^{\frac{1}{4}}{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )} \log \left (13923 \, \sqrt{d x} d^{11} - 13923 \, \left (-\frac{a d^{46}}{b^{25}}\right )^{\frac{1}{4}} b^{6}\right ) - 4 \,{\left (40960 \, b^{5} d^{11} x^{10} + 263515 \, a b^{4} d^{11} x^{8} + 590240 \, a^{2} b^{3} d^{11} x^{6} + 634270 \, a^{3} b^{2} d^{11} x^{4} + 334152 \, a^{4} b d^{11} x^{2} + 69615 \, a^{5} d^{11}\right )} \sqrt{d x}}{81920 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} 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)^3,x, algorithm="fricas")

[Out]

-1/81920*(278460*(-a*d^46/b^25)^(1/4)*(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*
x^2 + a^5*b^6)*arctan(-((-a*d^46/b^25)^(3/4)*sqrt(d*x)*b^19*d^11 - sqrt(d^23*x + sqrt(-a*d^46/b^25)*b^12)*(-a*
d^46/b^25)^(3/4)*b^19)/(a*d^46)) + 69615*(-a*d^46/b^25)^(1/4)*(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*
a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)*log(13923*sqrt(d*x)*d^11 + 13923*(-a*d^46/b^25)^(1/4)*b^6) - 69615*(-a*
d^46/b^25)^(1/4)*(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)*log(13
923*sqrt(d*x)*d^11 - 13923*(-a*d^46/b^25)^(1/4)*b^6) - 4*(40960*b^5*d^11*x^10 + 263515*a*b^4*d^11*x^8 + 590240
*a^2*b^3*d^11*x^6 + 634270*a^3*b^2*d^11*x^4 + 334152*a^4*b*d^11*x^2 + 69615*a^5*d^11)*sqrt(d*x))/(b^11*x^10 +
5*a*b^10*x^8 + 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)

________________________________________________________________________________________

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)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.33164, size = 466, normalized size = 1.16 \begin{align*} -\frac{1}{163840} \, d^{10}{\left (\frac{139230 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} 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}} + \frac{139230 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} 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}} + \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} 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}} - \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} 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}} - \frac{327680 \, \sqrt{d x} d}{b^{6}} - \frac{8 \,{\left (58715 \, \sqrt{d x} a b^{4} d^{11} x^{8} + 180640 \, \sqrt{d x} a^{2} b^{3} d^{11} x^{6} + 224670 \, \sqrt{d x} a^{3} b^{2} d^{11} x^{4} + 129352 \, \sqrt{d x} a^{4} b d^{11} x^{2} + 28655 \, \sqrt{d x} a^{5} d^{11}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} 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)^3,x, algorithm="giac")

[Out]

-1/163840*d^10*(139230*sqrt(2)*(a*b^3*d^2)^(1/4)*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 + 139230*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqr
t(d*x))/(a*d^2/b)^(1/4))/b^7 + 69615*sqrt(2)*(a*b^3*d^2)^(1/4)*d*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) +
sqrt(a*d^2/b))/b^7 - 69615*sqrt(2)*(a*b^3*d^2)^(1/4)*d*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d
^2/b))/b^7 - 327680*sqrt(d*x)*d/b^6 - 8*(58715*sqrt(d*x)*a*b^4*d^11*x^8 + 180640*sqrt(d*x)*a^2*b^3*d^11*x^6 +
224670*sqrt(d*x)*a^3*b^2*d^11*x^4 + 129352*sqrt(d*x)*a^4*b*d^11*x^2 + 28655*sqrt(d*x)*a^5*d^11)/((b*d^2*x^2 +
a*d^2)^5*b^6))