\(\int \frac {(d x)^{5/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [721]

Optimal result

Integrand size = 28, antiderivative size = 389 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}-\frac {117 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{17/4} b^{7/4}}-\frac {117 d^{5/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{17/4} b^{7/4}} \]

[Out]

-1/10*d*(d*x)^(3/2)/b/(b*x^2+a)^5+3/160*d*(d*x)^(3/2)/a/b/(b*x^2+a)^4+13/640*d*(d*x)^(3/2)/a^2/b/(b*x^2+a)^3+1
17/5120*d*(d*x)^(3/2)/a^3/b/(b*x^2+a)^2+117/4096*d*(d*x)^(3/2)/a^4/b/(b*x^2+a)-117/16384*d^(5/2)*arctan(1-b^(1
/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(17/4)/b^(7/4)*2^(1/2)+117/16384*d^(5/2)*arctan(1+b^(1/4)*2^(1/2)*(
d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(17/4)/b^(7/4)*2^(1/2)+117/32768*d^(5/2)*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^(17/4)/b^(7/4)*2^(1/2)-117/32768*d^(5/2)*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^(17/4)/b^(7/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 389, 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, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {117 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{17/4} b^{7/4}}-\frac {117 d^{5/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5} \]

[In]

Int[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/10*(d*(d*x)^(3/2))/(b*(a + b*x^2)^5) + (3*d*(d*x)^(3/2))/(160*a*b*(a + b*x^2)^4) + (13*d*(d*x)^(3/2))/(640*
a^2*b*(a + b*x^2)^3) + (117*d*(d*x)^(3/2))/(5120*a^3*b*(a + b*x^2)^2) + (117*d*(d*x)^(3/2))/(4096*a^4*b*(a + b
*x^2)) - (117*d^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(17/4)*b^(7/4
)) + (117*d^(5/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(17/4)*b^(7/4)) +
(117*d^(5/2)*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^(
17/4)*b^(7/4)) - (117*d^(5/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(1
6384*Sqrt[2]*a^(17/4)*b^(7/4))

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 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 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)^{5/2}}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (3 b^4 d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^5} \, dx \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {\left (39 b^3 d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^4} \, dx}{320 a} \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {\left (117 b^2 d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280 a^2} \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {\left (117 b d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a^3} \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {\left (117 d^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a^4} \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {(117 d) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^4} \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}-\frac {(117 d) \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^4 \sqrt {b}}+\frac {(117 d) \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^4 \sqrt {b}} \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {\left (117 d^{5/2}\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^{17/4} b^{7/4}}+\frac {\left (117 d^{5/2}\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^{17/4} b^{7/4}}+\frac {\left (117 d^3\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^4 b^2}+\frac {\left (117 d^3\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^4 b^2} \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {117 d^{5/2} \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^{17/4} b^{7/4}}-\frac {117 d^{5/2} \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^{17/4} b^{7/4}}+\frac {\left (117 d^{5/2}\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^{17/4} b^{7/4}}-\frac {\left (117 d^{5/2}\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^{17/4} b^{7/4}} \\ & = -\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}-\frac {117 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \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^{17/4} b^{7/4}}-\frac {117 d^{5/2} \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^{17/4} b^{7/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.47 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{5/2} \left (\frac {4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (-195 a^4+4960 a^3 b x^2+5330 a^2 b^2 x^4+2808 a b^3 x^6+585 b^4 x^8\right )}{\left (a+b x^2\right )^5}-585 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-585 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{17/4} b^{7/4} x^{5/2}} \]

[In]

Integrate[(d*x)^(5/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((d*x)^(5/2)*((4*a^(1/4)*b^(3/4)*x^(3/2)*(-195*a^4 + 4960*a^3*b*x^2 + 5330*a^2*b^2*x^4 + 2808*a*b^3*x^6 + 585*
b^4*x^8))/(a + b*x^2)^5 - 585*Sqrt[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 585*Sq
rt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81920*a^(17/4)*b^(7/4)*x^(5/2))

Maple [A] (verified)

Time = 20.14 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(2 d^{11} \left (\frac {-\frac {39 \left (d x \right )^{\frac {3}{2}}}{8192 b}+\frac {31 \left (d x \right )^{\frac {7}{2}}}{256 a \,d^{2}}+\frac {533 b \left (d x \right )^{\frac {11}{2}}}{4096 a^{2} d^{4}}+\frac {351 b^{2} \left (d x \right )^{\frac {15}{2}}}{5120 a^{3} d^{6}}+\frac {117 b^{3} \left (d x \right )^{\frac {19}{2}}}{8192 a^{4} d^{8}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{4} d^{8} b^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\)	\(238\)
default	\(2 d^{11} \left (\frac {-\frac {39 \left (d x \right )^{\frac {3}{2}}}{8192 b}+\frac {31 \left (d x \right )^{\frac {7}{2}}}{256 a \,d^{2}}+\frac {533 b \left (d x \right )^{\frac {11}{2}}}{4096 a^{2} d^{4}}+\frac {351 b^{2} \left (d x \right )^{\frac {15}{2}}}{5120 a^{3} d^{6}}+\frac {117 b^{3} \left (d x \right )^{\frac {19}{2}}}{8192 a^{4} d^{8}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {117 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{4} d^{8} b^{2} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\)	\(238\)
pseudoelliptic	\(\frac {39 \left (-8 \left (-3 b^{4} x^{8}-\frac {72}{5} a \,b^{3} x^{6}-\frac {82}{3} a^{2} b^{2} x^{4}-\frac {992}{39} a^{3} b \,x^{2}+a^{4}\right ) x b \sqrt {d x}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}+3 \sqrt {2}\, d \left (b \,x^{2}+a \right )^{5} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )\right )\right ) d^{2}}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{4} b^{2} \left (b \,x^{2}+a \right )^{5}}\)	\(249\)

[In]

int((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*d^11*((-39/8192/b*(d*x)^(3/2)+31/256/a/d^2*(d*x)^(7/2)+533/4096/a^2/d^4*b*(d*x)^(11/2)+351/5120/a^3/d^6*b^2*
(d*x)^(15/2)+117/8192/a^4/d^8*b^3*(d*x)^(19/2))/(b*d^2*x^2+a*d^2)^5+117/65536/a^4/d^8/b^2/(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 = 567, normalized size of antiderivative = 1.46 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {585 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (1601613 \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) - 585 \, {\left (i \, a^{4} b^{6} x^{10} + 5 i \, a^{5} b^{5} x^{8} + 10 i \, a^{6} b^{4} x^{6} + 10 i \, a^{7} b^{3} x^{4} + 5 i \, a^{8} b^{2} x^{2} + i \, a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (1601613 i \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) - 585 \, {\left (-i \, a^{4} b^{6} x^{10} - 5 i \, a^{5} b^{5} x^{8} - 10 i \, a^{6} b^{4} x^{6} - 10 i \, a^{7} b^{3} x^{4} - 5 i \, a^{8} b^{2} x^{2} - i \, a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (-1601613 i \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) - 585 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (-1601613 \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) + 4 \, {\left (585 \, b^{4} d^{2} x^{9} + 2808 \, a b^{3} d^{2} x^{7} + 5330 \, a^{2} b^{2} d^{2} x^{5} + 4960 \, a^{3} b d^{2} x^{3} - 195 \, a^{4} d^{2} x\right )} \sqrt {d x}}{81920 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )}} \]

[In]

integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(585*(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)*(-d^10/(
a^17*b^7))^(1/4)*log(1601613*a^13*b^5*(-d^10/(a^17*b^7))^(3/4) + 1601613*sqrt(d*x)*d^7) - 585*(I*a^4*b^6*x^10
+ 5*I*a^5*b^5*x^8 + 10*I*a^6*b^4*x^6 + 10*I*a^7*b^3*x^4 + 5*I*a^8*b^2*x^2 + I*a^9*b)*(-d^10/(a^17*b^7))^(1/4)*
log(1601613*I*a^13*b^5*(-d^10/(a^17*b^7))^(3/4) + 1601613*sqrt(d*x)*d^7) - 585*(-I*a^4*b^6*x^10 - 5*I*a^5*b^5*
x^8 - 10*I*a^6*b^4*x^6 - 10*I*a^7*b^3*x^4 - 5*I*a^8*b^2*x^2 - I*a^9*b)*(-d^10/(a^17*b^7))^(1/4)*log(-1601613*I
*a^13*b^5*(-d^10/(a^17*b^7))^(3/4) + 1601613*sqrt(d*x)*d^7) - 585*(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x
^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)*(-d^10/(a^17*b^7))^(1/4)*log(-1601613*a^13*b^5*(-d^10/(a^17*b^7))
^(3/4) + 1601613*sqrt(d*x)*d^7) + 4*(585*b^4*d^2*x^9 + 2808*a*b^3*d^2*x^7 + 5330*a^2*b^2*d^2*x^5 + 4960*a^3*b*
d^2*x^3 - 195*a^4*d^2*x)*sqrt(d*x))/(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^
2*x^2 + a^9*b)

Sympy [F]

\[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\left (d x\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{6}}\, dx \]

[In]

integrate((d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Integral((d*x)**(5/2)/(a + b*x**2)**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.98 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (585 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{4} + 2808 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{6} + 5330 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{8} + 4960 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{10} - 195 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{12}\right )}}{a^{4} b^{6} d^{10} x^{10} + 5 \, a^{5} b^{5} d^{10} x^{8} + 10 \, a^{6} b^{4} d^{10} x^{6} + 10 \, a^{7} b^{3} d^{10} x^{4} + 5 \, a^{8} b^{2} d^{10} x^{2} + a^{9} b d^{10}} + \frac {585 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{4} b}}{163840 \, d} \]

[In]

integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/163840*(8*(585*(d*x)^(19/2)*b^4*d^4 + 2808*(d*x)^(15/2)*a*b^3*d^6 + 5330*(d*x)^(11/2)*a^2*b^2*d^8 + 4960*(d*
x)^(7/2)*a^3*b*d^10 - 195*(d*x)^(3/2)*a^4*d^12)/(a^4*b^6*d^10*x^10 + 5*a^5*b^5*d^10*x^8 + 10*a^6*b^4*d^10*x^6
+ 10*a^7*b^3*d^10*x^4 + 5*a^8*b^2*d^10*x^2 + a^9*b*d^10) + 585*d^4*(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(sqr
t(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)))/(a^4*b))/d

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.91 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{163840} \, d^{2} {\left (\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{4} d} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{4} d} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{4} d} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{4} d} + \frac {8 \, {\left (585 \, \sqrt {d x} b^{4} d^{10} x^{9} + 2808 \, \sqrt {d x} a b^{3} d^{10} x^{7} + 5330 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{5} + 4960 \, \sqrt {d x} a^{3} b d^{10} x^{3} - 195 \, \sqrt {d x} a^{4} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{4} b}\right )} \]

[In]

integrate((d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^2*(1170*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^5*b^4*d) + 1170*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^5*b^4*d) - 585*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^5*b^4*d) + 585*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq
rt(a*d^2/b))/(a^5*b^4*d) + 8*(585*sqrt(d*x)*b^4*d^10*x^9 + 2808*sqrt(d*x)*a*b^3*d^10*x^7 + 5330*sqrt(d*x)*a^2*
b^2*d^10*x^5 + 4960*sqrt(d*x)*a^3*b*d^10*x^3 - 195*sqrt(d*x)*a^4*d^10*x)/((b*d^2*x^2 + a*d^2)^5*a^4*b))

Mupad [B] (verification not implemented)

Time = 14.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.54 \[ \int \frac {(d x)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {31\,d^9\,{\left (d\,x\right )}^{7/2}}{128\,a}-\frac {39\,d^{11}\,{\left (d\,x\right )}^{3/2}}{4096\,b}+\frac {351\,b^2\,d^5\,{\left (d\,x\right )}^{15/2}}{2560\,a^3}+\frac {117\,b^3\,d^3\,{\left (d\,x\right )}^{19/2}}{4096\,a^4}+\frac {533\,b\,d^7\,{\left (d\,x\right )}^{11/2}}{2048\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {117\,d^{5/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{17/4}\,b^{7/4}}-\frac {117\,d^{5/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{17/4}\,b^{7/4}} \]

[In]

int((d*x)^(5/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

((31*d^9*(d*x)^(7/2))/(128*a) - (39*d^11*(d*x)^(3/2))/(4096*b) + (351*b^2*d^5*(d*x)^(15/2))/(2560*a^3) + (117*
b^3*d^3*(d*x)^(19/2))/(4096*a^4) + (533*b*d^7*(d*x)^(11/2))/(2048*a^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) + (117*d^(5/2)*atan((b^(1/4)*(d*x)^(1/2
))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(17/4)*b^(7/4)) - (117*d^(5/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^
(1/2))))/(8192*(-a)^(17/4)*b^(7/4))