3.713 \(\int \frac{(d x)^{21/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=385 \[ -\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{4389 d^{21/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}+\frac{4389 d^{21/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5} \]

[Out]

-(d*(d*x)^(19/2))/(10*b*(a + b*x^2)^5) - (19*d^3*(d*x)^(15/2))/(160*b^2*(a + b*x^2)^4) - (19*d^5*(d*x)^(11/2))
/(128*b^3*(a + b*x^2)^3) - (209*d^7*(d*x)^(7/2))/(1024*b^4*(a + b*x^2)^2) - (1463*d^9*(d*x)^(3/2))/(4096*b^5*(
a + b*x^2)) - (4389*d^(21/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(1/4)*
b^(23/4)) + (4389*d^(21/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(1/4)*b^
(23/4)) + (4389*d^(21/2)*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^(1/4)*b^(23/4)) - (4389*d^(21/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[d*x]])/(16384*Sqrt[2]*a^(1/4)*b^(23/4))

________________________________________________________________________________________

Rubi [A]  time = 0.448027, antiderivative size = 385, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321, Rules used = {28, 288, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{4389 d^{21/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}+\frac{4389 d^{21/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(19/2))/(10*b*(a + b*x^2)^5) - (19*d^3*(d*x)^(15/2))/(160*b^2*(a + b*x^2)^4) - (19*d^5*(d*x)^(11/2))
/(128*b^3*(a + b*x^2)^3) - (209*d^7*(d*x)^(7/2))/(1024*b^4*(a + b*x^2)^2) - (1463*d^9*(d*x)^(3/2))/(4096*b^5*(
a + b*x^2)) - (4389*d^(21/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(1/4)*
b^(23/4)) + (4389*d^(21/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(1/4)*b^
(23/4)) + (4389*d^(21/2)*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^(1/4)*b^(23/4)) - (4389*d^(21/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*S
qrt[d*x]])/(16384*Sqrt[2]*a^(1/4)*b^(23/4))

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 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)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{(d x)^{21/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}+\frac{1}{20} \left (19 b^4 d^2\right ) \int \frac{(d x)^{17/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac{1}{64} \left (57 b^2 d^4\right ) \int \frac{(d x)^{13/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}+\frac{1}{256} \left (209 d^6\right ) \int \frac{(d x)^{9/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac{\left (1463 d^8\right ) \int \frac{(d x)^{5/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (4389 d^{10}\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{8192 b^4}\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (4389 d^9\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^4}\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{\left (4389 d^9\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^{9/2}}+\frac{\left (4389 d^9\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^{9/2}}\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{\left (4389 d^{21/2}\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} \sqrt [4]{a} b^{23/4}}+\frac{\left (4389 d^{21/2}\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} \sqrt [4]{a} b^{23/4}}+\frac{\left (4389 d^{11}\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^6}+\frac{\left (4389 d^{11}\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^6}\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}+\frac{4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}+\frac{\left (4389 d^{21/2}\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} \sqrt [4]{a} b^{23/4}}-\frac{\left (4389 d^{21/2}\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} \sqrt [4]{a} b^{23/4}}\\ &=-\frac{d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac{19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac{19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac{209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac{1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac{4389 d^{21/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}+\frac{4389 d^{21/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} \sqrt [4]{a} b^{23/4}}+\frac{4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}-\frac{4389 d^{21/2} \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} \sqrt [4]{a} b^{23/4}}\\ \end{align*}

Mathematica [C]  time = 0.0368045, size = 104, normalized size = 0.27 \[ \frac{2 d^9 (d x)^{3/2} \left (7315 \left (a+b x^2\right )^5 \, _2F_1\left (\frac{3}{4},6;\frac{7}{4};-\frac{b x^2}{a}\right )-a \left (20995 a^2 b^2 x^4+17765 a^3 b x^2+7315 a^4+12597 a b^3 x^6+3315 b^4 x^8\right )\right )}{3315 a b^5 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*d^9*(d*x)^(3/2)*(-(a*(7315*a^4 + 17765*a^3*b*x^2 + 20995*a^2*b^2*x^4 + 12597*a*b^3*x^6 + 3315*b^4*x^8)) + 7
315*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]))/(3315*a*b^5*(a + b*x^2)^5)

________________________________________________________________________________________

Maple [A]  time = 0.072, size = 335, normalized size = 0.9 \begin{align*} -{\frac{1463\,{d}^{19}{a}^{4}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{5}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{209\,{d}^{17}{a}^{3}}{128\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{4}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{5947\,{d}^{15}{a}^{2}}{2048\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{6289\,{d}^{13}a}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{b}^{2}} \left ( dx \right ) ^{{\frac{15}{2}}}}-{\frac{3803\,{d}^{11}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}b} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{4389\,{d}^{11}\sqrt{2}}{32768\,{b}^{6}}\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{4389\,{d}^{11}\sqrt{2}}{16384\,{b}^{6}}\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{4389\,{d}^{11}\sqrt{2}}{16384\,{b}^{6}}\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)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-1463/4096*d^19/(b*d^2*x^2+a*d^2)^5/b^5*a^4*(d*x)^(3/2)-209/128*d^17/(b*d^2*x^2+a*d^2)^5/b^4*a^3*(d*x)^(7/2)-5
947/2048*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(11/2)-6289/2560*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(15/2)-3
803/4096*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(19/2)+4389/32768*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)))+4389/16
384*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)+4389/16384*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)

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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)^(21/2)/(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 = 1.74617, size = 1158, normalized size = 3.01 \begin{align*} -\frac{87780 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{1}{4}} \sqrt{d x} b^{6} d^{31} - \sqrt{d^{63} x - \sqrt{-\frac{d^{42}}{a b^{23}}} a b^{11} d^{42}} \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{1}{4}} b^{6}}{d^{42}}\right ) - 21945 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{1}{4}} \log \left (84546715869 \, \sqrt{d x} d^{31} + 84546715869 \, \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{3}{4}} a b^{17}\right ) + 21945 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{1}{4}} \log \left (84546715869 \, \sqrt{d x} d^{31} - 84546715869 \, \left (-\frac{d^{42}}{a b^{23}}\right )^{\frac{3}{4}} a b^{17}\right ) + 4 \,{\left (19015 \, b^{4} d^{10} x^{9} + 50312 \, a b^{3} d^{10} x^{7} + 59470 \, a^{2} b^{2} d^{10} x^{5} + 33440 \, a^{3} b d^{10} x^{3} + 7315 \, a^{4} d^{10} x\right )} \sqrt{d x}}{81920 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/81920*(87780*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(-d^42/(
a*b^23))^(1/4)*arctan(-((-d^42/(a*b^23))^(1/4)*sqrt(d*x)*b^6*d^31 - sqrt(d^63*x - sqrt(-d^42/(a*b^23))*a*b^11*
d^42)*(-d^42/(a*b^23))^(1/4)*b^6)/d^42) - 21945*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5
*a^4*b^6*x^2 + a^5*b^5)*(-d^42/(a*b^23))^(1/4)*log(84546715869*sqrt(d*x)*d^31 + 84546715869*(-d^42/(a*b^23))^(
3/4)*a*b^17) + 21945*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(-d
^42/(a*b^23))^(1/4)*log(84546715869*sqrt(d*x)*d^31 - 84546715869*(-d^42/(a*b^23))^(3/4)*a*b^17) + 4*(19015*b^4
*d^10*x^9 + 50312*a*b^3*d^10*x^7 + 59470*a^2*b^2*d^10*x^5 + 33440*a^3*b*d^10*x^3 + 7315*a^4*d^10*x)*sqrt(d*x))
/(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)

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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)**(21/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.17143, size = 459, normalized size = 1.19 \begin{align*} \frac{1}{163840} \, d^{9}{\left (\frac{43890 \, \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 b^{8}} + \frac{43890 \, \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 b^{8}} - \frac{21945 \, \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 b^{8}} + \frac{21945 \, \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 b^{8}} - \frac{8 \,{\left (19015 \, \sqrt{d x} b^{4} d^{11} x^{9} + 50312 \, \sqrt{d x} a b^{3} d^{11} x^{7} + 59470 \, \sqrt{d x} a^{2} b^{2} d^{11} x^{5} + 33440 \, \sqrt{d x} a^{3} b d^{11} x^{3} + 7315 \, \sqrt{d x} a^{4} d^{11} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^9*(43890*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*b^8) + 43890*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*b^8) - 21945*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + s
qrt(a*d^2/b))/(a*b^8) + 21945*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*b^8) - 8*(19015*sqrt(d*x)*b^4*d^11*x^9 + 50312*sqrt(d*x)*a*b^3*d^11*x^7 + 59470*sqrt(d*x)*a^2*b^2*d^
11*x^5 + 33440*sqrt(d*x)*a^3*b*d^11*x^3 + 7315*sqrt(d*x)*a^4*d^11*x)/((b*d^2*x^2 + a*d^2)^5*b^5))