\(\int \frac {(d x)^{21/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [713]

Optimal result

Integrand size = 28, antiderivative size = 385 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {4389 d^{21/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}+\frac {4389 d^{21/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}+\frac {4389 d^{21/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} \sqrt [4]{a} b^{23/4}} \]

[Out]

-1/10*d*(d*x)^(19/2)/b/(b*x^2+a)^5-19/160*d^3*(d*x)^(15/2)/b^2/(b*x^2+a)^4-19/128*d^5*(d*x)^(11/2)/b^3/(b*x^2+
a)^3-209/1024*d^7*(d*x)^(7/2)/b^4/(b*x^2+a)^2-1463/4096*d^9*(d*x)^(3/2)/b^5/(b*x^2+a)-4389/16384*d^(21/2)*arct
an(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(1/4)/b^(23/4)*2^(1/2)+4389/16384*d^(21/2)*arctan(1+b^(1/4
)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(1/4)/b^(23/4)*2^(1/2)+4389/32768*d^(21/2)*ln(a^(1/2)*d^(1/2)+x*b^(1/
2)*d^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/a^(1/4)/b^(23/4)*2^(1/2)-4389/32768*d^(21/2)*ln(a^(1/2)*d^(1/2
)+x*b^(1/2)*d^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/a^(1/4)/b^(23/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 294, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {4389 d^{21/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}+\frac {4389 d^{21/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} \sqrt [4]{a} b^{23/4}}+\frac {4389 d^{21/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} \sqrt [4]{a} b^{23/4}}-\frac {4389 d^{21/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} \sqrt [4]{a} b^{23/4}}-\frac {1463 d^9 (d x)^{3/2}}{4096 b^5 \left (a+b x^2\right )}-\frac {209 d^7 (d x)^{7/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {19 d^5 (d x)^{11/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {19 d^3 (d x)^{15/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{19/2}}{10 b \left (a+b x^2\right )^5} \]

[In]

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

[Out]

-1/10*(d*(d*x)^(19/2))/(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]])/(1638
4*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))

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 294

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 303

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 335

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 631

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 642

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 1176

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 1179

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]

Rubi steps \begin{align*} \text {integral}& = 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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A] (verified)

Time = 0.96 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.53 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^9 (d x)^{3/2} \left (-4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (7315 a^4+33440 a^3 b x^2+59470 a^2 b^2 x^4+50312 a b^3 x^6+19015 b^4 x^8\right )+21945 \sqrt {2} \left (a+b x^2\right )^5 \arctan \left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-21945 \sqrt {2} \left (a+b x^2\right )^5 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 \sqrt [4]{a} b^{23/4} x^{3/2} \left (a+b x^2\right )^5} \]

[In]

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

[Out]

(d^9*(d*x)^(3/2)*(-4*a^(1/4)*b^(3/4)*x^(3/2)*(7315*a^4 + 33440*a^3*b*x^2 + 59470*a^2*b^2*x^4 + 50312*a*b^3*x^6
+ 19015*b^4*x^8) + 21945*Sqrt[2]*(a + b*x^2)^5*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
)] - 21945*Sqrt[2]*(a + b*x^2)^5*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81920*a^(
1/4)*b^(23/4)*x^(3/2)*(a + b*x^2)^5)

Maple [A] (verified)

Time = 20.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(2 d^{11} \left (\frac {-\frac {1463 d^{8} a^{4} \left (d x \right )^{\frac {3}{2}}}{8192 b^{5}}-\frac {209 d^{6} a^{3} \left (d x \right )^{\frac {7}{2}}}{256 b^{4}}-\frac {5947 d^{4} a^{2} \left (d x \right )^{\frac {11}{2}}}{4096 b^{3}}-\frac {6289 d^{2} a \left (d x \right )^{\frac {15}{2}}}{5120 b^{2}}-\frac {3803 \left (d x \right )^{\frac {19}{2}}}{8192 b}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {4389 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b^{6} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\)	\(235\)
default	\(2 d^{11} \left (\frac {-\frac {1463 d^{8} a^{4} \left (d x \right )^{\frac {3}{2}}}{8192 b^{5}}-\frac {209 d^{6} a^{3} \left (d x \right )^{\frac {7}{2}}}{256 b^{4}}-\frac {5947 d^{4} a^{2} \left (d x \right )^{\frac {11}{2}}}{4096 b^{3}}-\frac {6289 d^{2} a \left (d x \right )^{\frac {15}{2}}}{5120 b^{2}}-\frac {3803 \left (d x \right )^{\frac {19}{2}}}{8192 b}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {4389 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 b^{6} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\)	\(235\)
pseudoelliptic	\(-\frac {1463 \left (8 \left (\frac {3803}{1463} b^{4} x^{8}+\frac {2648}{385} a \,b^{3} x^{6}+\frac {626}{77} a^{2} b^{2} x^{4}+\frac {32}{7} a^{3} b \,x^{2}+a^{4}\right ) x b \sqrt {d x}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}-3 \sqrt {2}\, d \left (b \,x^{2}+a \right )^{5} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right )\right ) d^{10}}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} b^{6} \left (b \,x^{2}+a \right )^{5}}\)	\(246\)

[In]

int((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*d^11*((-1463/8192/b^5*d^8*a^4*(d*x)^(3/2)-209/256/b^4*d^6*a^3*(d*x)^(7/2)-5947/4096/b^3*d^4*a^2*(d*x)^(11/2)
-6289/5120/b^2*d^2*a*(d*x)^(15/2)-3803/8192/b*(d*x)^(19/2))/(b*d^2*x^2+a*d^2)^5+4389/65536/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)))+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/
2)-1)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.41 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {21945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} \log \left (84546715869 \, \sqrt {d x} d^{31} + 84546715869 \, \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {3}{4}} a b^{17}\right ) - 21945 \, {\left (i \, b^{10} x^{10} + 5 i \, a b^{9} x^{8} + 10 i \, a^{2} b^{8} x^{6} + 10 i \, a^{3} b^{7} x^{4} + 5 i \, a^{4} b^{6} x^{2} + i \, a^{5} b^{5}\right )} \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} \log \left (84546715869 \, \sqrt {d x} d^{31} + 84546715869 i \, \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {3}{4}} a b^{17}\right ) - 21945 \, {\left (-i \, b^{10} x^{10} - 5 i \, a b^{9} x^{8} - 10 i \, a^{2} b^{8} x^{6} - 10 i \, a^{3} b^{7} x^{4} - 5 i \, a^{4} b^{6} x^{2} - i \, a^{5} b^{5}\right )} \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} \log \left (84546715869 \, \sqrt {d x} d^{31} - 84546715869 i \, \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {3}{4}} a b^{17}\right ) - 21945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {1}{4}} \log \left (84546715869 \, \sqrt {d x} d^{31} - 84546715869 \, \left (-\frac {d^{42}}{a b^{23}}\right )^{\frac {3}{4}} a b^{17}\right ) - 4 \, {\left (19015 \, b^{4} d^{10} x^{9} + 50312 \, a b^{3} d^{10} x^{7} + 59470 \, a^{2} b^{2} d^{10} x^{5} + 33440 \, a^{3} b d^{10} x^{3} + 7315 \, a^{4} d^{10} x\right )} \sqrt {d x}}{81920 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \]

[In]

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

[Out]

1/81920*(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*(I*b^10*x^10
+ 5*I*a*b^9*x^8 + 10*I*a^2*b^8*x^6 + 10*I*a^3*b^7*x^4 + 5*I*a^4*b^6*x^2 + I*a^5*b^5)*(-d^42/(a*b^23))^(1/4)*l
og(84546715869*sqrt(d*x)*d^31 + 84546715869*I*(-d^42/(a*b^23))^(3/4)*a*b^17) - 21945*(-I*b^10*x^10 - 5*I*a*b^9
*x^8 - 10*I*a^2*b^8*x^6 - 10*I*a^3*b^7*x^4 - 5*I*a^4*b^6*x^2 - I*a^5*b^5)*(-d^42/(a*b^23))^(1/4)*log(845467158
69*sqrt(d*x)*d^31 - 84546715869*I*(-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 - 84546
715869*(-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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.98 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {21945 \, d^{12} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{b^{5}} - \frac {8 \, {\left (19015 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{12} + 50312 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{14} + 59470 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{16} + 33440 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{18} + 7315 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{20}\right )}}{b^{10} d^{10} x^{10} + 5 \, a b^{9} d^{10} x^{8} + 10 \, a^{2} b^{8} d^{10} x^{6} + 10 \, a^{3} b^{7} d^{10} x^{4} + 5 \, a^{4} b^{6} d^{10} x^{2} + a^{5} b^{5} d^{10}}}{163840 \, d} \]

[In]

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

[Out]

1/163840*(21945*d^12*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(
sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b
^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) - sqrt(2)*log(sqrt(b)
*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x
- sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)))/b^5 - 8*(19015*(d*x)^(19/2)*b^
4*d^12 + 50312*(d*x)^(15/2)*a*b^3*d^14 + 59470*(d*x)^(11/2)*a^2*b^2*d^16 + 33440*(d*x)^(7/2)*a^3*b*d^18 + 7315
*(d*x)^(3/2)*a^4*d^20)/(b^10*d^10*x^10 + 5*a*b^9*d^10*x^8 + 10*a^2*b^8*d^10*x^6 + 10*a^3*b^7*d^10*x^4 + 5*a^4*
b^6*d^10*x^2 + a^5*b^5*d^10))/d

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.91 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{163840} \, d^{10} {\left (\frac {43890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{8} d} + \frac {43890 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{8} d} - \frac {21945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{8} d} + \frac {21945 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{8} d} - \frac {8 \, {\left (19015 \, \sqrt {d x} b^{4} d^{10} x^{9} + 50312 \, \sqrt {d x} a b^{3} d^{10} x^{7} + 59470 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{5} + 33440 \, \sqrt {d x} a^{3} b d^{10} x^{3} + 7315 \, \sqrt {d x} a^{4} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{5}}\right )} \]

[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^10*(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*d) + 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*d) - 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*d) + 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*d) - 8*(19015*sqrt(d*x)*b^4*d^10*x^9 + 50312*sqrt(d*x)*a*b^3*d^10*x^7 + 59470*sqrt(d*x)*a
^2*b^2*d^10*x^5 + 33440*sqrt(d*x)*a^3*b*d^10*x^3 + 7315*sqrt(d*x)*a^4*d^10*x)/((b*d^2*x^2 + a*d^2)^5*b^5))

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.55 \[ \int \frac {(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {4389\,d^{21/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{1/4}\,b^{23/4}}-\frac {\frac {3803\,d^{11}\,{\left (d\,x\right )}^{19/2}}{4096\,b}+\frac {5947\,a^2\,d^{15}\,{\left (d\,x\right )}^{11/2}}{2048\,b^3}+\frac {209\,a^3\,d^{17}\,{\left (d\,x\right )}^{7/2}}{128\,b^4}+\frac {1463\,a^4\,d^{19}\,{\left (d\,x\right )}^{3/2}}{4096\,b^5}+\frac {6289\,a\,d^{13}\,{\left (d\,x\right )}^{15/2}}{2560\,b^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {4389\,d^{21/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{1/4}\,b^{23/4}} \]

[In]

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

[Out]

(4389*d^(21/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(1/4)*b^(23/4)) - ((3803*d^11*(d*x
)^(19/2))/(4096*b) + (5947*a^2*d^15*(d*x)^(11/2))/(2048*b^3) + (209*a^3*d^17*(d*x)^(7/2))/(128*b^4) + (1463*a^
4*d^19*(d*x)^(3/2))/(4096*b^5) + (6289*a*d^13*(d*x)^(15/2))/(2560*b^2))/(a^5*d^10 + b^5*d^10*x^10 + 5*a^4*b*d^
10*x^2 + 5*a*b^4*d^10*x^8 + 10*a^3*b^2*d^10*x^4 + 10*a^2*b^3*d^10*x^6) - (4389*d^(21/2)*atanh((b^(1/4)*(d*x)^(
1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(1/4)*b^(23/4))