3.711 \(\int \frac{(d x)^{25/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=402 \[ -\frac{33649 a^{3/4} d^{25/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{27/4}}+\frac{33649 a^{3/4} d^{25/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{27/4}}+\frac{33649 a^{3/4} d^{25/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{27/4}}-\frac{33649 a^{3/4} d^{25/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} b^{27/4}}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}+\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6} \]

[Out]

(33649*d^11*(d*x)^(3/2))/(12288*b^6) - (d*(d*x)^(23/2))/(10*b*(a + b*x^2)^5) - (23*d^3*(d*x)^(19/2))/(160*b^2*
(a + b*x^2)^4) - (437*d^5*(d*x)^(15/2))/(1920*b^3*(a + b*x^2)^3) - (437*d^7*(d*x)^(11/2))/(1024*b^4*(a + b*x^2
)^2) - (4807*d^9*(d*x)^(7/2))/(4096*b^5*(a + b*x^2)) + (33649*a^(3/4)*d^(25/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(27/4)) - (33649*a^(3/4)*d^(25/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(27/4)) - (33649*a^(3/4)*d^(25/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(27/4)) + (33649*a^(3/4)*d^(25/2)*Log[Sqrt[a]*Sq
rt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(27/4))

________________________________________________________________________________________

Rubi [A]  time = 0.468539, 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, 297, 1162, 617, 204, 1165, 628} \[ -\frac{33649 a^{3/4} d^{25/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{27/4}}+\frac{33649 a^{3/4} d^{25/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} b^{27/4}}+\frac{33649 a^{3/4} d^{25/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{27/4}}-\frac{33649 a^{3/4} d^{25/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} b^{27/4}}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}+\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(25/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(33649*d^11*(d*x)^(3/2))/(12288*b^6) - (d*(d*x)^(23/2))/(10*b*(a + b*x^2)^5) - (23*d^3*(d*x)^(19/2))/(160*b^2*
(a + b*x^2)^4) - (437*d^5*(d*x)^(15/2))/(1920*b^3*(a + b*x^2)^3) - (437*d^7*(d*x)^(11/2))/(1024*b^4*(a + b*x^2
)^2) - (4807*d^9*(d*x)^(7/2))/(4096*b^5*(a + b*x^2)) + (33649*a^(3/4)*d^(25/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(27/4)) - (33649*a^(3/4)*d^(25/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*b^(27/4)) - (33649*a^(3/4)*d^(25/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(27/4)) + (33649*a^(3/4)*d^(25/2)*Log[Sqrt[a]*Sq
rt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*b^(27/4))

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 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

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])

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]

Rubi steps

\begin{align*} \int \frac{(d x)^{25/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{(d x)^{25/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}+\frac{1}{20} \left (23 b^4 d^2\right ) \int \frac{(d x)^{21/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{1}{320} \left (437 b^2 d^4\right ) \int \frac{(d x)^{17/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{1}{256} \left (437 d^6\right ) \int \frac{(d x)^{13/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac{\left (4807 d^8\right ) \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (33649 d^{10}\right ) \int \frac{(d x)^{5/2}}{a b+b^2 x^2} \, dx}{8192 b^4}\\ &=\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{\left (33649 a d^{12}\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{8192 b^5}\\ &=\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{\left (33649 a d^{11}\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 b^5}\\ &=\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (33649 a d^{11}\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^{11/2}}-\frac{\left (33649 a d^{11}\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^{11/2}}\\ &=\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{\left (33649 a^{3/4} d^{25/2}\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^{27/4}}-\frac{\left (33649 a^{3/4} d^{25/2}\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^{27/4}}-\frac{\left (33649 a d^{13}\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^7}-\frac{\left (33649 a d^{13}\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^7}\\ &=\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{33649 a^{3/4} d^{25/2} \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^{27/4}}+\frac{33649 a^{3/4} d^{25/2} \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^{27/4}}-\frac{\left (33649 a^{3/4} d^{25/2}\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^{27/4}}+\frac{\left (33649 a^{3/4} d^{25/2}\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^{27/4}}\\ &=\frac{33649 d^{11} (d x)^{3/2}}{12288 b^6}-\frac{d (d x)^{23/2}}{10 b \left (a+b x^2\right )^5}-\frac{23 d^3 (d x)^{19/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{437 d^5 (d x)^{15/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{437 d^7 (d x)^{11/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{4807 d^9 (d x)^{7/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{33649 a^{3/4} d^{25/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{27/4}}-\frac{33649 a^{3/4} d^{25/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} b^{27/4}}-\frac{33649 a^{3/4} d^{25/2} \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^{27/4}}+\frac{33649 a^{3/4} d^{25/2} \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^{27/4}}\\ \end{align*}

Mathematica [C]  time = 0.0351908, size = 109, normalized size = 0.27 \[ -\frac{2 d^{12} x \sqrt{d x} \left (-289731 a^2 b^3 x^6-482885 a^3 b^2 x^4-408595 a^4 b x^2-168245 a^5-76245 a b^4 x^8+168245 \left (a+b x^2\right )^5 \, _2F_1\left (\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right )-3315 b^5 x^{10}\right )}{9945 b^6 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(25/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(-2*d^12*x*Sqrt[d*x]*(-168245*a^5 - 408595*a^4*b*x^2 - 482885*a^3*b^2*x^4 - 289731*a^2*b^3*x^6 - 76245*a*b^4*x
^8 - 3315*b^5*x^10 + 168245*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]))/(9945*b^6*(a + b*x^2)
^5)

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Maple [A]  time = 0.074, size = 354, normalized size = 0.9 \begin{align*}{\frac{2\,{d}^{11}}{3\,{b}^{6}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{25457\,{d}^{21}{a}^{5}}{12288\,{b}^{6} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{3527\,{d}^{19}{a}^{4}}{384\,{b}^{5} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{95821\,{d}^{17}{a}^{3}}{6144\,{b}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{31149\,{d}^{15}{a}^{2}}{2560\,{b}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{15}{2}}}}+{\frac{15503\,{d}^{13}a}{4096\,{b}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{19}{2}}}}-{\frac{33649\,{d}^{13}a\sqrt{2}}{32768\,{b}^{7}}\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{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{33649\,{d}^{13}a\sqrt{2}}{16384\,{b}^{7}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{33649\,{d}^{13}a\sqrt{2}}{16384\,{b}^{7}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

2/3*d^11*(d*x)^(3/2)/b^6+25457/12288*d^21*a^5/b^6/(b*d^2*x^2+a*d^2)^5*(d*x)^(3/2)+3527/384*d^19*a^4/b^5/(b*d^2
*x^2+a*d^2)^5*(d*x)^(7/2)+95821/6144*d^17*a^3/b^4/(b*d^2*x^2+a*d^2)^5*(d*x)^(11/2)+31149/2560*d^15*a^2/b^3/(b*
d^2*x^2+a*d^2)^5*(d*x)^(15/2)+15503/4096*d^13*a/b^2/(b*d^2*x^2+a*d^2)^5*(d*x)^(19/2)-33649/32768*d^13*a/b^7/(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)))-33649/16384*d^13*a/b^7/(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*
(d*x)^(1/2)+1)-33649/16384*d^13*a/b^7/(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)^(25/2)/(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 = 1.73702, size = 1258, normalized size = 3.13 \begin{align*} \frac{2018940 \, \left (-\frac{a^{3} d^{50}}{b^{27}}\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^{3} d^{50}}{b^{27}}\right )^{\frac{1}{4}} \sqrt{d x} a^{2} b^{7} d^{37} - \sqrt{a^{4} d^{75} x - \sqrt{-\frac{a^{3} d^{50}}{b^{27}}} a^{3} b^{13} d^{50}} \left (-\frac{a^{3} d^{50}}{b^{27}}\right )^{\frac{1}{4}} b^{7}}{a^{3} d^{50}}\right ) - 504735 \, \left (-\frac{a^{3} d^{50}}{b^{27}}\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 (38099255258449 \, \sqrt{d x} a^{2} d^{37} + 38099255258449 \, \left (-\frac{a^{3} d^{50}}{b^{27}}\right )^{\frac{3}{4}} b^{20}\right ) + 504735 \, \left (-\frac{a^{3} d^{50}}{b^{27}}\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 (38099255258449 \, \sqrt{d x} a^{2} d^{37} - 38099255258449 \, \left (-\frac{a^{3} d^{50}}{b^{27}}\right )^{\frac{3}{4}} b^{20}\right ) + 4 \,{\left (40960 \, b^{5} d^{12} x^{11} + 437345 \, a b^{4} d^{12} x^{9} + 1157176 \, a^{2} b^{3} d^{12} x^{7} + 1367810 \, a^{3} b^{2} d^{12} x^{5} + 769120 \, a^{4} b d^{12} x^{3} + 168245 \, a^{5} d^{12} x\right )} \sqrt{d x}}{245760 \,{\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)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/245760*(2018940*(-a^3*d^50/b^27)^(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^3*d^50/b^27)^(1/4)*sqrt(d*x)*a^2*b^7*d^37 - sqrt(a^4*d^75*x - sqrt(-a^3*d^50/b^
27)*a^3*b^13*d^50)*(-a^3*d^50/b^27)^(1/4)*b^7)/(a^3*d^50)) - 504735*(-a^3*d^50/b^27)^(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(38099255258449*sqrt(d*x)*a^2*d^37 + 38
099255258449*(-a^3*d^50/b^27)^(3/4)*b^20) + 504735*(-a^3*d^50/b^27)^(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(38099255258449*sqrt(d*x)*a^2*d^37 - 38099255258449*(-a^
3*d^50/b^27)^(3/4)*b^20) + 4*(40960*b^5*d^12*x^11 + 437345*a*b^4*d^12*x^9 + 1157176*a^2*b^3*d^12*x^7 + 1367810
*a^3*b^2*d^12*x^5 + 769120*a^4*b*d^12*x^3 + 168245*a^5*d^12*x)*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)

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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)**(25/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.30389, size = 463, normalized size = 1.15 \begin{align*} \frac{1}{491520} \, d^{11}{\left (\frac{327680 \, \sqrt{d x} d x}{b^{6}} - \frac{1009470 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \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^{9}} - \frac{1009470 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \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^{9}} + \frac{504735 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \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^{9}} - \frac{504735 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \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^{9}} + \frac{8 \,{\left (232545 \, \sqrt{d x} a b^{4} d^{11} x^{9} + 747576 \, \sqrt{d x} a^{2} b^{3} d^{11} x^{7} + 958210 \, \sqrt{d x} a^{3} b^{2} d^{11} x^{5} + 564320 \, \sqrt{d x} a^{4} b d^{11} x^{3} + 127285 \, \sqrt{d x} a^{5} d^{11} x\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)^(25/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/491520*d^11*(327680*sqrt(d*x)*d*x/b^6 - 1009470*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2
/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/b^9 - 1009470*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)
*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/b^9 + 504735*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d
^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/b^9 - 504735*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4
)*sqrt(d*x) + sqrt(a*d^2/b))/b^9 + 8*(232545*sqrt(d*x)*a*b^4*d^11*x^9 + 747576*sqrt(d*x)*a^2*b^3*d^11*x^7 + 95
8210*sqrt(d*x)*a^3*b^2*d^11*x^5 + 564320*sqrt(d*x)*a^4*b*d^11*x^3 + 127285*sqrt(d*x)*a^5*d^11*x)/((b*d^2*x^2 +
a*d^2)^5*b^6))