3.717 \(\int \frac{(d x)^{13/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=391 \[ \frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{77 d^{13/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{9/4} b^{15/4}}-\frac{77 d^{13/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{9/4} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{9/4} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{9/4} b^{15/4}}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5} \]

[Out]

-(d*(d*x)^(11/2))/(10*b*(a + b*x^2)^5) - (11*d^3*(d*x)^(7/2))/(160*b^2*(a + b*x^2)^4) - (77*d^5*(d*x)^(3/2))/(
1920*b^3*(a + b*x^2)^3) + (77*d^5*(d*x)^(3/2))/(5120*a*b^3*(a + b*x^2)^2) + (77*d^5*(d*x)^(3/2))/(4096*a^2*b^3
*(a + b*x^2)) - (77*d^(13/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(9/4)*
b^(15/4)) + (77*d^(13/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(9/4)*b^(1
5/4)) + (77*d^(13/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt
[2]*a^(9/4)*b^(15/4)) - (77*d^(13/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*
x]])/(16384*Sqrt[2]*a^(9/4)*b^(15/4))

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Rubi [A]  time = 0.479026, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {28, 288, 290, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{77 d^{13/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{9/4} b^{15/4}}-\frac{77 d^{13/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{9/4} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{9/4} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{9/4} b^{15/4}}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(11/2))/(10*b*(a + b*x^2)^5) - (11*d^3*(d*x)^(7/2))/(160*b^2*(a + b*x^2)^4) - (77*d^5*(d*x)^(3/2))/(
1920*b^3*(a + b*x^2)^3) + (77*d^5*(d*x)^(3/2))/(5120*a*b^3*(a + b*x^2)^2) + (77*d^5*(d*x)^(3/2))/(4096*a^2*b^3
*(a + b*x^2)) - (77*d^(13/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(9/4)*
b^(15/4)) + (77*d^(13/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(9/4)*b^(1
5/4)) + (77*d^(13/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt
[2]*a^(9/4)*b^(15/4)) - (77*d^(13/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*
x]])/(16384*Sqrt[2]*a^(9/4)*b^(15/4))

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 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && 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)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{(d x)^{13/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}+\frac{1}{20} \left (11 b^4 d^2\right ) \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{1}{320} \left (77 b^2 d^4\right ) \int \frac{(d x)^{5/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{\left (77 d^6\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280}\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac{\left (77 d^6\right ) \int \frac{\sqrt{d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a b}\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{\left (77 d^6\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{8192 a^2 b^2}\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{\left (77 d^5\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 a^2 b^2}\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac{\left (77 d^5\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 a^2 b^{5/2}}+\frac{\left (77 d^5\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 a^2 b^{5/2}}\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{\left (77 d^{13/2}\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} a^{9/4} b^{15/4}}+\frac{\left (77 d^{13/2}\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} a^{9/4} b^{15/4}}+\frac{\left (77 d^7\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 a^2 b^4}+\frac{\left (77 d^7\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 a^2 b^4}\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac{77 d^{13/2} \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} a^{9/4} b^{15/4}}-\frac{77 d^{13/2} \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} a^{9/4} b^{15/4}}+\frac{\left (77 d^{13/2}\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} a^{9/4} b^{15/4}}-\frac{\left (77 d^{13/2}\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} a^{9/4} b^{15/4}}\\ &=-\frac{d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac{11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac{77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac{77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{9/4} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{9/4} b^{15/4}}+\frac{77 d^{13/2} \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} a^{9/4} b^{15/4}}-\frac{77 d^{13/2} \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} a^{9/4} b^{15/4}}\\ \end{align*}

Mathematica [C]  time = 0.030944, size = 85, normalized size = 0.22 \[ \frac{2 d^6 x \sqrt{d x} \left (77 \left (a+b x^2\right )^5 \, _2F_1\left (\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right )-a^3 \left (77 a^2+187 a b x^2+221 b^2 x^4\right )\right )}{1989 a^3 b^3 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*d^6*x*Sqrt[d*x]*(-(a^3*(77*a^2 + 187*a*b*x^2 + 221*b^2*x^4)) + 77*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7
/4, -((b*x^2)/a)]))/(1989*a^3*b^3*(a + b*x^2)^5)

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Maple [A]  time = 0.073, size = 339, normalized size = 0.9 \begin{align*} -{\frac{77\,{d}^{15}{a}^{2}}{12288\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{3}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{11\,{d}^{13}a}{384\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{313\,{d}^{11}}{6144\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{231\,{d}^{9}}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a} \left ( dx \right ) ^{{\frac{15}{2}}}}+{\frac{77\,{d}^{7}b}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{77\,{d}^{7}\sqrt{2}}{32768\,{a}^{2}{b}^{4}}\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{77\,{d}^{7}\sqrt{2}}{16384\,{a}^{2}{b}^{4}}\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{77\,{d}^{7}\sqrt{2}}{16384\,{a}^{2}{b}^{4}}\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)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-77/12288*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(3/2)-11/384*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(7/2)-313/6
144*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(11/2)+231/2560*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(15/2)+77/4096*d^7/(b*d^2
*x^2+a*d^2)^5/a^2*b*(d*x)^(19/2)+77/32768*d^7/a^2/b^4/(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)))+77/16384*d^7/a^2/b^4/(
a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+77/16384*d^7/a^2/b^4/(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)^(13/2)/(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 = 1.49084, size = 1181, normalized size = 3.02 \begin{align*} -\frac{4620 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac{d^{26}}{a^{9} b^{15}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (-\frac{d^{26}}{a^{9} b^{15}}\right )^{\frac{1}{4}} \sqrt{d x} a^{2} b^{4} d^{19} - \sqrt{d^{39} x - \sqrt{-\frac{d^{26}}{a^{9} b^{15}}} a^{5} b^{7} d^{26}} \left (-\frac{d^{26}}{a^{9} b^{15}}\right )^{\frac{1}{4}} a^{2} b^{4}}{d^{26}}\right ) - 1155 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac{d^{26}}{a^{9} b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} + 456533 \, \left (-\frac{d^{26}}{a^{9} b^{15}}\right )^{\frac{3}{4}} a^{7} b^{11}\right ) + 1155 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac{d^{26}}{a^{9} b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} - 456533 \, \left (-\frac{d^{26}}{a^{9} b^{15}}\right )^{\frac{3}{4}} a^{7} b^{11}\right ) - 4 \,{\left (1155 \, b^{4} d^{6} x^{9} + 5544 \, a b^{3} d^{6} x^{7} - 3130 \, a^{2} b^{2} d^{6} x^{5} - 1760 \, a^{3} b d^{6} x^{3} - 385 \, a^{4} d^{6} x\right )} \sqrt{d x}}{245760 \,{\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/245760*(4620*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)*(-d
^26/(a^9*b^15))^(1/4)*arctan(-((-d^26/(a^9*b^15))^(1/4)*sqrt(d*x)*a^2*b^4*d^19 - sqrt(d^39*x - sqrt(-d^26/(a^9
*b^15))*a^5*b^7*d^26)*(-d^26/(a^9*b^15))^(1/4)*a^2*b^4)/d^26) - 1155*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^
6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)*(-d^26/(a^9*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 + 456533*
(-d^26/(a^9*b^15))^(3/4)*a^7*b^11) + 1155*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*
a^6*b^4*x^2 + a^7*b^3)*(-d^26/(a^9*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 - 456533*(-d^26/(a^9*b^15))^(3/4)*a^
7*b^11) - 4*(1155*b^4*d^6*x^9 + 5544*a*b^3*d^6*x^7 - 3130*a^2*b^2*d^6*x^5 - 1760*a^3*b*d^6*x^3 - 385*a^4*d^6*x
)*sqrt(d*x))/(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)

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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)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.19677, size = 463, normalized size = 1.18 \begin{align*} \frac{1}{491520} \, d^{5}{\left (\frac{2310 \, \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 )}{a^{3} b^{6}} + \frac{2310 \, \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 )}{a^{3} b^{6}} - \frac{1155 \, \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 )}{a^{3} b^{6}} + \frac{1155 \, \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 )}{a^{3} b^{6}} + \frac{8 \,{\left (1155 \, \sqrt{d x} b^{4} d^{11} x^{9} + 5544 \, \sqrt{d x} a b^{3} d^{11} x^{7} - 3130 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{5} - 1760 \, \sqrt{d x} a^{3} b d^{11} x^{3} - 385 \, \sqrt{d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{2} b^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/491520*d^5*(2310*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))/(a^3*b^6) + 2310*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))/(a^3*b^6) - 1155*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))/(a^3*b^6) + 1155*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))/(a^3*b^6) + 8*(1155*sqrt(d*x)*b^4*d^11*x^9 + 5544*sqrt(d*x)*a*b^3*d^11*x^7 - 3130*sqrt(d*x)*a^2*b^2*d
^11*x^5 - 1760*sqrt(d*x)*a^3*b*d^11*x^3 - 385*sqrt(d*x)*a^4*d^11*x)/((b*d^2*x^2 + a*d^2)^5*a^2*b^3))