\(\int \frac {(d x)^{11/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [718]

Optimal result

Integrand size = 28, antiderivative size = 391 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {63 d^{11/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}-\frac {63 d^{11/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{11/4} b^{13/4}} \]

[Out]

-1/10*d*(d*x)^(9/2)/b/(b*x^2+a)^5-9/160*d^3*(d*x)^(5/2)/b^2/(b*x^2+a)^4-63/16384*d^(11/2)*arctan(1-b^(1/4)*2^(
1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(11/4)/b^(13/4)*2^(1/2)+63/16384*d^(11/2)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(
1/2)/a^(1/4)/d^(1/2))/a^(11/4)/b^(13/4)*2^(1/2)-63/32768*d^(11/2)*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^(11/4)/b^(13/4)*2^(1/2)+63/32768*d^(11/2)*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^(11/4)/b^(13/4)*2^(1/2)-3/128*d^5*(d*x)^(1/2)/b^3/(b*x^2+a)^3+3/1024*d
^5*(d*x)^(1/2)/a/b^3/(b*x^2+a)^2+21/4096*d^5*(d*x)^(1/2)/a^2/b^3/(b*x^2+a)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 294, 296, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {63 d^{11/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}-\frac {63 d^{11/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{11/4} b^{13/4}}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5} \]

[In]

Int[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/10*(d*(d*x)^(9/2))/(b*(a + b*x^2)^5) - (9*d^3*(d*x)^(5/2))/(160*b^2*(a + b*x^2)^4) - (3*d^5*Sqrt[d*x])/(128
*b^3*(a + b*x^2)^3) + (3*d^5*Sqrt[d*x])/(1024*a*b^3*(a + b*x^2)^2) + (21*d^5*Sqrt[d*x])/(4096*a^2*b^3*(a + b*x
^2)) - (63*d^(11/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(11/4)*b^(13/4)
) + (63*d^(11/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(11/4)*b^(13/4)) -
(63*d^(11/2)*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^(
11/4)*b^(13/4)) + (63*d^(11/2)*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^(11/4)*b^(13/4))

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 217

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 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 296

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] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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)^{11/2}}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (9 b^4 d^2\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^5} \, dx \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{64} \left (9 b^2 d^4\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {1}{256} \left (3 d^6\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {\left (21 d^6\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a b} \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (63 d^6\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^2 b^2} \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (63 d^5\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^2 b^2} \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (63 d^4\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 a^{5/2} b^2}+\frac {\left (63 d^4\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 a^{5/2} b^2} \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {\left (63 d^{11/2}\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} a^{11/4} b^{13/4}}-\frac {\left (63 d^{11/2}\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} a^{11/4} b^{13/4}}+\frac {\left (63 d^6\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 a^{5/2} b^{7/2}}+\frac {\left (63 d^6\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 a^{5/2} b^{7/2}} \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {63 d^{11/2} \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^{11/4} b^{13/4}}+\frac {63 d^{11/2} \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^{11/4} b^{13/4}}+\frac {\left (63 d^{11/2}\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} a^{11/4} b^{13/4}}-\frac {\left (63 d^{11/2}\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} a^{11/4} b^{13/4}} \\ & = -\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {63 d^{11/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}-\frac {63 d^{11/2} \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^{11/4} b^{13/4}}+\frac {63 d^{11/2} \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^{11/4} b^{13/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.48 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {d^5 \sqrt {d x} \left (\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-315 a^4-1512 a^3 b x^2-2870 a^2 b^2 x^4+480 a b^3 x^6+105 b^4 x^8\right )}{\left (a+b x^2\right )^5}-315 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+315 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{11/4} b^{13/4} \sqrt {x}} \]

[In]

Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^5*Sqrt[d*x]*((4*a^(3/4)*b^(1/4)*Sqrt[x]*(-315*a^4 - 1512*a^3*b*x^2 - 2870*a^2*b^2*x^4 + 480*a*b^3*x^6 + 105
*b^4*x^8))/(a + b*x^2)^5 - 315*Sqrt[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 315*S
qrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81920*a^(11/4)*b^(13/4)*Sqrt[x])

Maple [A] (verified)

Time = 19.90 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.60

method	result	size
derivativedivides	\(2 d^{11} \left (\frac {-\frac {63 d^{4} a^{2} \sqrt {d x}}{8192 b^{3}}-\frac {189 d^{2} a \left (d x \right )^{\frac {5}{2}}}{5120 b^{2}}-\frac {287 \left (d x \right )^{\frac {9}{2}}}{4096 b}+\frac {3 \left (d x \right )^{\frac {13}{2}}}{256 a \,d^{2}}+\frac {21 b \left (d x \right )^{\frac {17}{2}}}{8192 a^{2} d^{4}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {63 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{3} d^{6} b^{3}}\right )\)	\(236\)
default	\(2 d^{11} \left (\frac {-\frac {63 d^{4} a^{2} \sqrt {d x}}{8192 b^{3}}-\frac {189 d^{2} a \left (d x \right )^{\frac {5}{2}}}{5120 b^{2}}-\frac {287 \left (d x \right )^{\frac {9}{2}}}{4096 b}+\frac {3 \left (d x \right )^{\frac {13}{2}}}{256 a \,d^{2}}+\frac {21 b \left (d x \right )^{\frac {17}{2}}}{8192 a^{2} d^{4}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {63 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{3} d^{6} b^{3}}\right )\)	\(236\)
pseudoelliptic	\(\frac {d^{5} \left (\left (840 a \,x^{8} b^{4}+3840 a^{2} x^{6} b^{3}-22960 a^{3} x^{4} b^{2}-12096 x^{2} a^{4} b -2520 a^{5}\right ) \sqrt {d x}+315 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{163840 a^{3} b^{3} \left (b \,x^{2}+a \right )^{5}}\)	\(240\)

[In]

int((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*d^11*((-63/8192/b^3*d^4*a^2*(d*x)^(1/2)-189/5120/b^2*d^2*a*(d*x)^(5/2)-287/4096/b*(d*x)^(9/2)+3/256/a/d^2*(d
*x)^(13/2)+21/8192/a^2/d^4*b*(d*x)^(17/2))/(b*d^2*x^2+a*d^2)^5+63/65536/a^3/d^6/b^3*(a*d^2/b)^(1/4)*2^(1/2)*(l
n((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 = 0.26 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.47 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {315 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (63 \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) - 315 \, {\left (-i \, a^{2} b^{8} x^{10} - 5 i \, a^{3} b^{7} x^{8} - 10 i \, a^{4} b^{6} x^{6} - 10 i \, a^{5} b^{5} x^{4} - 5 i \, a^{6} b^{4} x^{2} - i \, a^{7} b^{3}\right )} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (63 i \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) - 315 \, {\left (i \, a^{2} b^{8} x^{10} + 5 i \, a^{3} b^{7} x^{8} + 10 i \, a^{4} b^{6} x^{6} + 10 i \, a^{5} b^{5} x^{4} + 5 i \, a^{6} b^{4} x^{2} + i \, a^{7} b^{3}\right )} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (-63 i \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) - 315 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (-63 \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) + 4 \, {\left (105 \, b^{4} d^{5} x^{8} + 480 \, a b^{3} d^{5} x^{6} - 2870 \, a^{2} b^{2} d^{5} x^{4} - 1512 \, a^{3} b d^{5} x^{2} - 315 \, a^{4} d^{5}\right )} \sqrt {d x}}{81920 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )}} \]

[In]

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(315*(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^22
/(a^11*b^13))^(1/4)*log(63*a^3*b^3*(-d^22/(a^11*b^13))^(1/4) + 63*sqrt(d*x)*d^5) - 315*(-I*a^2*b^8*x^10 - 5*I*
a^3*b^7*x^8 - 10*I*a^4*b^6*x^6 - 10*I*a^5*b^5*x^4 - 5*I*a^6*b^4*x^2 - I*a^7*b^3)*(-d^22/(a^11*b^13))^(1/4)*log
(63*I*a^3*b^3*(-d^22/(a^11*b^13))^(1/4) + 63*sqrt(d*x)*d^5) - 315*(I*a^2*b^8*x^10 + 5*I*a^3*b^7*x^8 + 10*I*a^4
*b^6*x^6 + 10*I*a^5*b^5*x^4 + 5*I*a^6*b^4*x^2 + I*a^7*b^3)*(-d^22/(a^11*b^13))^(1/4)*log(-63*I*a^3*b^3*(-d^22/
(a^11*b^13))^(1/4) + 63*sqrt(d*x)*d^5) - 315*(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^22/(a^11*b^13))^(1/4)*log(-63*a^3*b^3*(-d^22/(a^11*b^13))^(1/4) + 63*sqrt(d*x)*d
^5) + 4*(105*b^4*d^5*x^8 + 480*a*b^3*d^5*x^6 - 2870*a^2*b^2*d^5*x^4 - 1512*a^3*b*d^5*x^2 - 315*a^4*d^5)*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)

Sympy [F]

\[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\left (d x\right )^{\frac {11}{2}}}{\left (a + b x^{2}\right )^{6}}\, dx \]

[In]

integrate((d*x)**(11/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Integral((d*x)**(11/2)/(a + b*x**2)**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.01 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (105 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{8} + 480 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{10} - 2870 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{12} - 1512 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{14} - 315 \, \sqrt {d x} a^{4} d^{16}\right )}}{a^{2} b^{8} d^{10} x^{10} + 5 \, a^{3} b^{7} d^{10} x^{8} + 10 \, a^{4} b^{6} d^{10} x^{6} + 10 \, a^{5} b^{5} d^{10} x^{4} + 5 \, a^{6} b^{4} d^{10} x^{2} + a^{7} b^{3} d^{10}} + \frac {315 \, {\left (\frac {\sqrt {2} d^{8} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{8} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{7} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{7} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{2} b^{3}}}{163840 \, d} \]

[In]

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/163840*(8*(105*(d*x)^(17/2)*b^4*d^8 + 480*(d*x)^(13/2)*a*b^3*d^10 - 2870*(d*x)^(9/2)*a^2*b^2*d^12 - 1512*(d*
x)^(5/2)*a^3*b*d^14 - 315*sqrt(d*x)*a^4*d^16)/(a^2*b^8*d^10*x^10 + 5*a^3*b^7*d^10*x^8 + 10*a^4*b^6*d^10*x^6 +
10*a^5*b^5*d^10*x^4 + 5*a^6*b^4*d^10*x^2 + a^7*b^3*d^10) + 315*(sqrt(2)*d^8*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^
(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) - sqrt(2)*d^8*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(
1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) + 2*sqrt(2)*d^7*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(a)) + 2*sqrt(2)*
d^7*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(s
qrt(a)*sqrt(b)*d)*sqrt(a)))/(a^2*b^3))/d

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.87 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{163840} \, d^{5} {\left (\frac {630 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{4}} + \frac {630 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{4}} + \frac {315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{4}} - \frac {315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{4}} + \frac {8 \, {\left (105 \, \sqrt {d x} b^{4} d^{10} x^{8} + 480 \, \sqrt {d x} a b^{3} d^{10} x^{6} - 2870 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} - 1512 \, \sqrt {d x} a^{3} b d^{10} x^{2} - 315 \, \sqrt {d x} a^{4} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{2} b^{3}}\right )} \]

[In]

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^5*(630*sqrt(2)*(a*b^3*d^2)^(1/4)*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^4) + 630*sqrt(2)*(a*b^3*d^2)^(1/4)*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^4) + 315*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqr
t(a*d^2/b))/(a^3*b^4) - 315*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2
/b))/(a^3*b^4) + 8*(105*sqrt(d*x)*b^4*d^10*x^8 + 480*sqrt(d*x)*a*b^3*d^10*x^6 - 2870*sqrt(d*x)*a^2*b^2*d^10*x^
4 - 1512*sqrt(d*x)*a^3*b*d^10*x^2 - 315*sqrt(d*x)*a^4*d^10)/((b*d^2*x^2 + a*d^2)^5*a^2*b^3))

Mupad [B] (verification not implemented)

Time = 14.13 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.53 \[ \int \frac {(d x)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {\frac {287\,d^{11}\,{\left (d\,x\right )}^{9/2}}{2048\,b}-\frac {3\,d^9\,{\left (d\,x\right )}^{13/2}}{128\,a}+\frac {63\,a^2\,d^{15}\,\sqrt {d\,x}}{4096\,b^3}+\frac {189\,a\,d^{13}\,{\left (d\,x\right )}^{5/2}}{2560\,b^2}-\frac {21\,b\,d^7\,{\left (d\,x\right )}^{17/2}}{4096\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {63\,d^{11/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{11/4}\,b^{13/4}}-\frac {63\,d^{11/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{11/4}\,b^{13/4}} \]

[In]

int((d*x)^(11/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

- ((287*d^11*(d*x)^(9/2))/(2048*b) - (3*d^9*(d*x)^(13/2))/(128*a) + (63*a^2*d^15*(d*x)^(1/2))/(4096*b^3) + (18
9*a*d^13*(d*x)^(5/2))/(2560*b^2) - (21*b*d^7*(d*x)^(17/2))/(4096*a^2))/(a^5*d^10 + b^5*d^10*x^10 + 5*a^4*b*d^1
0*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) - (63*d^(11/2)*atan((b^(1/4)*(d*x)^(1/2)
)/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(11/4)*b^(13/4)) - (63*d^(11/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^
(1/2))))/(8192*(-a)^(11/4)*b^(13/4))