3.710 \(\int \frac{(d x)^{27/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=420 \[ -\frac{69615 a^{5/4} d^{27/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{29/4}}-\frac{69615 a^{5/4} d^{27/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} b^{29/4}}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6} \]

[Out]

(-69615*a*d^13*Sqrt[d*x])/(4096*b^7) + (13923*d^11*(d*x)^(5/2))/(4096*b^6) - (d*(d*x)^(25/2))/(10*b*(a + b*x^2
)^5) - (5*d^3*(d*x)^(21/2))/(32*b^2*(a + b*x^2)^4) - (35*d^5*(d*x)^(17/2))/(128*b^3*(a + b*x^2)^3) - (595*d^7*
(d*x)^(13/2))/(1024*b^4*(a + b*x^2)^2) - (7735*d^9*(d*x)^(9/2))/(4096*b^5*(a + b*x^2)) - (69615*a^(5/4)*d^(27/
2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(29/4)) + (69615*a^(5/4)*d^(27/2
)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(29/4)) - (69615*a^(5/4)*d^(27/2)
*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^(29/4)) + (696
15*a^(5/4)*d^(27/2)*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^(29/4))

________________________________________________________________________________________

Rubi [A]  time = 0.529127, antiderivative size = 420, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ -\frac{69615 a^{5/4} d^{27/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{29/4}}-\frac{69615 a^{5/4} d^{27/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} b^{29/4}}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(-69615*a*d^13*Sqrt[d*x])/(4096*b^7) + (13923*d^11*(d*x)^(5/2))/(4096*b^6) - (d*(d*x)^(25/2))/(10*b*(a + b*x^2
)^5) - (5*d^3*(d*x)^(21/2))/(32*b^2*(a + b*x^2)^4) - (35*d^5*(d*x)^(17/2))/(128*b^3*(a + b*x^2)^3) - (595*d^7*
(d*x)^(13/2))/(1024*b^4*(a + b*x^2)^2) - (7735*d^9*(d*x)^(9/2))/(4096*b^5*(a + b*x^2)) - (69615*a^(5/4)*d^(27/
2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(29/4)) + (69615*a^(5/4)*d^(27/2
)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(29/4)) - (69615*a^(5/4)*d^(27/2)
*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^(29/4)) + (696
15*a^(5/4)*d^(27/2)*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^(29/4))

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)^{27/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{(d x)^{27/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}+\frac{1}{4} \left (5 b^4 d^2\right ) \int \frac{(d x)^{23/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}+\frac{1}{64} \left (105 b^2 d^4\right ) \int \frac{(d x)^{19/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}+\frac{1}{256} \left (595 d^6\right ) \int \frac{(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac{\left (7735 d^8\right ) \int \frac{(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (69615 d^{10}\right ) \int \frac{(d x)^{7/2}}{a b+b^2 x^2} \, dx}{8192 b^4}\\ &=\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{\left (69615 a d^{12}\right ) \int \frac{(d x)^{3/2}}{a b+b^2 x^2} \, dx}{8192 b^5}\\ &=-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (69615 a^2 d^{14}\right ) \int \frac{1}{\sqrt{d x} \left (a b+b^2 x^2\right )} \, dx}{8192 b^6}\\ &=-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (69615 a^2 d^{13}\right ) \operatorname{Subst}\left (\int \frac{1}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 b^6}\\ &=-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (69615 a^{3/2} d^{12}\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^6}+\frac{\left (69615 a^{3/2} d^{12}\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^6}\\ &=-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{\left (69615 a^{5/4} d^{27/2}\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^{29/4}}-\frac{\left (69615 a^{5/4} d^{27/2}\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^{29/4}}+\frac{\left (69615 a^{3/2} d^{14}\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^{15/2}}+\frac{\left (69615 a^{3/2} d^{14}\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^{15/2}}\\ &=-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{69615 a^{5/4} d^{27/2} \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^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \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^{29/4}}+\frac{\left (69615 a^{5/4} d^{27/2}\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^{29/4}}-\frac{\left (69615 a^{5/4} d^{27/2}\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^{29/4}}\\ &=-\frac{69615 a d^{13} \sqrt{d x}}{4096 b^7}+\frac{13923 d^{11} (d x)^{5/2}}{4096 b^6}-\frac{d (d x)^{25/2}}{10 b \left (a+b x^2\right )^5}-\frac{5 d^3 (d x)^{21/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac{35 d^5 (d x)^{17/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{595 d^7 (d x)^{13/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{7735 d^9 (d x)^{9/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{69615 a^{5/4} d^{27/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{29/4}}-\frac{69615 a^{5/4} d^{27/2} \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^{29/4}}+\frac{69615 a^{5/4} d^{27/2} \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^{29/4}}\\ \end{align*}

Mathematica [A]  time = 0.186116, size = 432, normalized size = 1.03 \[ \frac{d^{13} \sqrt{d x} \left (-126156800 a^2 b^{17/4} x^{17/2}-306380800 a^3 b^{13/4} x^{13/2}-362086400 a^4 b^{9/4} x^{9/2}-217251840 a^5 b^{5/4} x^{5/2}[phone] a^2 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^4+5834400 a^3 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^3+4243200 a^4 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2+3394560 a^5 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )-3828825 \sqrt{2} a^{5/4} \left (a+b x^2\right )^5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+3828825 \sqrt{2} a^{5/4} \left (a+b x^2\right )^5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-7657650 \sqrt{2} a^{5/4} \left (a+b x^2\right )^5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+7657650 \sqrt{2} a^{5/4} \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-54312960 a^6 \sqrt [4]{b} \sqrt{x}-18022400 a b^{21/4} x^{21/2}+720896 b^{25/4} x^{25/2}\right )}{1802240 b^{29/4} \sqrt{x} \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(27/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^13*Sqrt[d*x]*(-54312960*a^6*b^(1/4)*Sqrt[x] - 217251840*a^5*b^(5/4)*x^(5/2) - 362086400*a^4*b^(9/4)*x^(9/2)
- 306380800*a^3*b^(13/4)*x^(13/2) - 126156800*a^2*b^(17/4)*x^(17/2) - 18022400*a*b^(21/4)*x^(21/2) + 720896*b
^(25/4)*x^(25/2) + 3394560*a^5*b^(1/4)*Sqrt[x]*(a + b*x^2) + 4243200*a^4*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 58344
00*a^3*b^(1/4)*Sqrt[x]*(a + b*x^2)^3 + 10210200*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^4 - 7657650*Sqrt[2]*a^(5/4)*(a
+ b*x^2)^5*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 7657650*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 3828825*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x] + 3828825*Sqrt[2]*a^(5/4)*(a + b*x^2)^5*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x]))/(1802240*b^(29/4)*Sqrt[x]*(a + b*x^2)^5)

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Maple [A]  time = 0.122, size = 370, normalized size = 0.9 \begin{align*}{\frac{2\,{d}^{11}}{5\,{b}^{6}} \left ( dx \right ) ^{{\frac{5}{2}}}}-12\,{\frac{a{d}^{13}\sqrt{dx}}{{b}^{7}}}-{\frac{20463\,{d}^{23}{a}^{6}}{4096\,{b}^{7} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}}\sqrt{dx}}-{\frac{56269\,{d}^{21}{a}^{5}}{2560\,{b}^{6} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{5}{2}}}}-{\frac{75471\,{d}^{19}{a}^{4}}{2048\,{b}^{5} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{9}{2}}}}-{\frac{3597\,{d}^{17}{a}^{3}}{128\,{b}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{13}{2}}}}-{\frac{34139\,{d}^{15}{a}^{2}}{4096\,{b}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{17}{2}}}}+{\frac{69615\,a{d}^{13}\sqrt{2}}{32768\,{b}^{7}}\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{69615\,a{d}^{13}\sqrt{2}}{16384\,{b}^{7}}\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{69615\,a{d}^{13}\sqrt{2}}{16384\,{b}^{7}}\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)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

2/5*d^11*(d*x)^(5/2)/b^6-12*a*d^13*(d*x)^(1/2)/b^7-20463/4096*d^23/b^7*a^6/(b*d^2*x^2+a*d^2)^5*(d*x)^(1/2)-562
69/2560*d^21/b^6*a^5/(b*d^2*x^2+a*d^2)^5*(d*x)^(5/2)-75471/2048*d^19/b^5*a^4/(b*d^2*x^2+a*d^2)^5*(d*x)^(9/2)-3
597/128*d^17/b^4*a^3/(b*d^2*x^2+a*d^2)^5*(d*x)^(13/2)-34139/4096*d^15/b^3*a^2/(b*d^2*x^2+a*d^2)^5*(d*x)^(17/2)
+69615/32768*d^13/b^7*a*(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)))+69615/16384*d^13/b^7*a*(a*d^2/b)^(1/4)*2^(1/2)*arcta
n(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+69615/16384*d^13/b^7*a*(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/
b)^(1/4)*(d*x)^(1/2)-1)

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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)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69083, size = 1229, normalized size = 2.93 \begin{align*} \frac{1392300 \, \left (-\frac{a^{5} d^{54}}{b^{29}}\right )^{\frac{1}{4}}{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )} \arctan \left (-\frac{\left (-\frac{a^{5} d^{54}}{b^{29}}\right )^{\frac{3}{4}} \sqrt{d x} a b^{22} d^{13} - \left (-\frac{a^{5} d^{54}}{b^{29}}\right )^{\frac{3}{4}} \sqrt{a^{2} d^{27} x + \sqrt{-\frac{a^{5} d^{54}}{b^{29}}} b^{14}} b^{22}}{a^{5} d^{54}}\right ) + 348075 \, \left (-\frac{a^{5} d^{54}}{b^{29}}\right )^{\frac{1}{4}}{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )} \log \left (69615 \, \sqrt{d x} a d^{13} + 69615 \, \left (-\frac{a^{5} d^{54}}{b^{29}}\right )^{\frac{1}{4}} b^{7}\right ) - 348075 \, \left (-\frac{a^{5} d^{54}}{b^{29}}\right )^{\frac{1}{4}}{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )} \log \left (69615 \, \sqrt{d x} a d^{13} - 69615 \, \left (-\frac{a^{5} d^{54}}{b^{29}}\right )^{\frac{1}{4}} b^{7}\right ) + 4 \,{\left (8192 \, b^{6} d^{13} x^{12} - 204800 \, a b^{5} d^{13} x^{10} - 1317575 \, a^{2} b^{4} d^{13} x^{8} - 2951200 \, a^{3} b^{3} d^{13} x^{6} - 3171350 \, a^{4} b^{2} d^{13} x^{4} - 1670760 \, a^{5} b d^{13} x^{2} - 348075 \, a^{6} d^{13}\right )} \sqrt{d x}}{81920 \,{\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(1392300*(-a^5*d^54/b^29)^(1/4)*(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b
^8*x^2 + a^5*b^7)*arctan(-((-a^5*d^54/b^29)^(3/4)*sqrt(d*x)*a*b^22*d^13 - (-a^5*d^54/b^29)^(3/4)*sqrt(a^2*d^27
*x + sqrt(-a^5*d^54/b^29)*b^14)*b^22)/(a^5*d^54)) + 348075*(-a^5*d^54/b^29)^(1/4)*(b^12*x^10 + 5*a*b^11*x^8 +
10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7)*log(69615*sqrt(d*x)*a*d^13 + 69615*(-a^5*d^54/b^29
)^(1/4)*b^7) - 348075*(-a^5*d^54/b^29)^(1/4)*(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*
a^4*b^8*x^2 + a^5*b^7)*log(69615*sqrt(d*x)*a*d^13 - 69615*(-a^5*d^54/b^29)^(1/4)*b^7) + 4*(8192*b^6*d^13*x^12
- 204800*a*b^5*d^13*x^10 - 1317575*a^2*b^4*d^13*x^8 - 2951200*a^3*b^3*d^13*x^6 - 3171350*a^4*b^2*d^13*x^4 - 16
70760*a^5*b*d^13*x^2 - 348075*a^6*d^13)*sqrt(d*x))/(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^
4 + 5*a^4*b^8*x^2 + a^5*b^7)

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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)**(27/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.27497, size = 510, normalized size = 1.21 \begin{align*} \frac{1}{163840} \, d^{12}{\left (\frac{696150 \, \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^{8}} + \frac{696150 \, \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^{8}} + \frac{348075 \, \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^{8}} - \frac{348075 \, \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^{8}} - \frac{8 \,{\left (170695 \, \sqrt{d x} a^{2} b^{4} d^{11} x^{8} + 575520 \, \sqrt{d x} a^{3} b^{3} d^{11} x^{6} + 754710 \, \sqrt{d x} a^{4} b^{2} d^{11} x^{4} + 450152 \, \sqrt{d x} a^{5} b d^{11} x^{2} + 102315 \, \sqrt{d x} a^{6} d^{11}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{7}} + \frac{65536 \,{\left (\sqrt{d x} b^{24} d^{6} x^{2} - 30 \, \sqrt{d x} a b^{23} d^{6}\right )}}{b^{30} d^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(27/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^12*(696150*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^8 + 696150*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^8 + 348075*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^8 - 348075*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^8 - 8*(170695*sqrt(d*x)*a^2*b^4*d^11*x^8 + 575520*sqrt(d*x)*a^3*b^3*d^11*x^6 + 754710*sqrt(d
*x)*a^4*b^2*d^11*x^4 + 450152*sqrt(d*x)*a^5*b*d^11*x^2 + 102315*sqrt(d*x)*a^6*d^11)/((b*d^2*x^2 + a*d^2)^5*b^7
) + 65536*(sqrt(d*x)*b^24*d^6*x^2 - 30*sqrt(d*x)*a*b^23*d^6)/(b^30*d^5))