\(\int \frac {(d x)^{3/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [722]

Optimal result

Integrand size = 28, antiderivative size = 389 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}-\frac {231 d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}-\frac {231 d^{3/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{19/4} b^{5/4}} \]

[Out]

-231/16384*d^(3/2)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(19/4)/b^(5/4)*2^(1/2)+231/16384*d^
(3/2)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(19/4)/b^(5/4)*2^(1/2)-231/32768*d^(3/2)*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^(19/4)/b^(5/4)*2^(1/2)+231/32768*d^(3/2)*
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^(19/4)/b^(5/4)*2^(1/2)-1/10*d*(d*x
)^(1/2)/b/(b*x^2+a)^5+1/160*d*(d*x)^(1/2)/a/b/(b*x^2+a)^4+1/128*d*(d*x)^(1/2)/a^2/b/(b*x^2+a)^3+11/1024*d*(d*x
)^(1/2)/a^3/b/(b*x^2+a)^2+77/4096*d*(d*x)^(1/2)/a^4/b/(b*x^2+a)

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, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {231 d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}-\frac {231 d^{3/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{19/4} b^{5/4}}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}-\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5} \]

[In]

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

[Out]

-1/10*(d*Sqrt[d*x])/(b*(a + b*x^2)^5) + (d*Sqrt[d*x])/(160*a*b*(a + b*x^2)^4) + (d*Sqrt[d*x])/(128*a^2*b*(a +
b*x^2)^3) + (11*d*Sqrt[d*x])/(1024*a^3*b*(a + b*x^2)^2) + (77*d*Sqrt[d*x])/(4096*a^4*b*(a + b*x^2)) - (231*d^(
3/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(19/4)*b^(5/4)) + (231*d^(3/2)
*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(19/4)*b^(5/4)) - (231*d^(3/2)*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^(19/4)*b^(5/4)) +
(231*d^(3/2)*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
9/4)*b^(5/4))

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)^{3/2}}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (b^4 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^5} \, dx \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {\left (3 b^3 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx}{64 a} \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {\left (11 b^2 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^2} \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {\left (77 b d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^3} \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {\left (231 d^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^4} \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {(231 d) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^4} \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}+\frac {231 \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^{9/2}}+\frac {231 \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^{9/2}} \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}-\frac {\left (231 d^{3/2}\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^{19/4} b^{5/4}}-\frac {\left (231 d^{3/2}\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^{19/4} b^{5/4}}+\frac {\left (231 d^2\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^{9/2} b^{3/2}}+\frac {\left (231 d^2\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^{9/2} b^{3/2}} \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}-\frac {231 d^{3/2} \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^{19/4} b^{5/4}}+\frac {231 d^{3/2} \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^{19/4} b^{5/4}}+\frac {\left (231 d^{3/2}\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^{19/4} b^{5/4}}-\frac {\left (231 d^{3/2}\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^{19/4} b^{5/4}} \\ & = -\frac {d \sqrt {d x}}{10 b \left (a+b x^2\right )^5}+\frac {d \sqrt {d x}}{160 a b \left (a+b x^2\right )^4}+\frac {d \sqrt {d x}}{128 a^2 b \left (a+b x^2\right )^3}+\frac {11 d \sqrt {d x}}{1024 a^3 b \left (a+b x^2\right )^2}+\frac {77 d \sqrt {d x}}{4096 a^4 b \left (a+b x^2\right )}-\frac {231 d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}+\frac {231 d^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{19/4} b^{5/4}}-\frac {231 d^{3/2} \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^{19/4} b^{5/4}}+\frac {231 d^{3/2} \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^{19/4} b^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.47 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{3/2} \left (\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-1155 a^4+2648 a^3 b x^2+3130 a^2 b^2 x^4+1760 a b^3 x^6+385 b^4 x^8\right )}{\left (a+b x^2\right )^5}-1155 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+1155 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{19/4} b^{5/4} x^{3/2}} \]

[In]

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

[Out]

((d*x)^(3/2)*((4*a^(3/4)*b^(1/4)*Sqrt[x]*(-1155*a^4 + 2648*a^3*b*x^2 + 3130*a^2*b^2*x^4 + 1760*a*b^3*x^6 + 385
*b^4*x^8))/(a + b*x^2)^5 - 1155*Sqrt[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 1155
*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81920*a^(19/4)*b^(5/4)*x^(3/2))

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(2 d^{11} \left (\frac {-\frac {231 \sqrt {d x}}{8192 b}+\frac {331 \left (d x \right )^{\frac {5}{2}}}{5120 a \,d^{2}}+\frac {313 b \left (d x \right )^{\frac {9}{2}}}{4096 a^{2} d^{4}}+\frac {11 b^{2} \left (d x \right )^{\frac {13}{2}}}{256 a^{3} d^{6}}+\frac {77 b^{3} \left (d x \right )^{\frac {17}{2}}}{8192 a^{4} d^{8}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {231 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{5} d^{10} b}\right )\)	\(238\)
default	\(2 d^{11} \left (\frac {-\frac {231 \sqrt {d x}}{8192 b}+\frac {331 \left (d x \right )^{\frac {5}{2}}}{5120 a \,d^{2}}+\frac {313 b \left (d x \right )^{\frac {9}{2}}}{4096 a^{2} d^{4}}+\frac {11 b^{2} \left (d x \right )^{\frac {13}{2}}}{256 a^{3} d^{6}}+\frac {77 b^{3} \left (d x \right )^{\frac {17}{2}}}{8192 a^{4} d^{8}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{5}}+\frac {231 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{5} d^{10} b}\right )\)	\(238\)
pseudoelliptic	\(\frac {\left (\left (3080 a \,x^{8} b^{4}+14080 a^{2} x^{6} b^{3}+25040 a^{3} x^{4} b^{2}+21184 x^{2} a^{4} b -9240 a^{5}\right ) \sqrt {d x}+1155 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \,x^{2}+a \right )^{5} \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right ) d}{163840 b \,a^{5} \left (b \,x^{2}+a \right )^{5}}\)	\(238\)

[In]

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

[Out]

2*d^11*((-231/8192/b*(d*x)^(1/2)+331/5120/a/d^2*(d*x)^(5/2)+313/4096/a^2/d^4*b*(d*x)^(9/2)+11/256/a^3/d^6*b^2*
(d*x)^(13/2)+77/8192/a^4/d^8*b^3*(d*x)^(17/2))/(b*d^2*x^2+a*d^2)^5+231/65536/a^5/d^10/b*(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 = 540, normalized size of antiderivative = 1.39 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1155 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (231 \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) - 1155 \, {\left (-i \, a^{4} b^{6} x^{10} - 5 i \, a^{5} b^{5} x^{8} - 10 i \, a^{6} b^{4} x^{6} - 10 i \, a^{7} b^{3} x^{4} - 5 i \, a^{8} b^{2} x^{2} - i \, a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (231 i \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) - 1155 \, {\left (i \, a^{4} b^{6} x^{10} + 5 i \, a^{5} b^{5} x^{8} + 10 i \, a^{6} b^{4} x^{6} + 10 i \, a^{7} b^{3} x^{4} + 5 i \, a^{8} b^{2} x^{2} + i \, a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (-231 i \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) - 1155 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} \log \left (-231 \, a^{5} b \left (-\frac {d^{6}}{a^{19} b^{5}}\right )^{\frac {1}{4}} + 231 \, \sqrt {d x} d\right ) + 4 \, {\left (385 \, b^{4} d x^{8} + 1760 \, a b^{3} d x^{6} + 3130 \, a^{2} b^{2} d x^{4} + 2648 \, a^{3} b d x^{2} - 1155 \, a^{4} d\right )} \sqrt {d x}}{81920 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )}} \]

[In]

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

[Out]

1/81920*(1155*(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^6/(
a^19*b^5))^(1/4)*log(231*a^5*b*(-d^6/(a^19*b^5))^(1/4) + 231*sqrt(d*x)*d) - 1155*(-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^6/(a^19*b^5))^(1/4)*log(231*I*a^5
*b*(-d^6/(a^19*b^5))^(1/4) + 231*sqrt(d*x)*d) - 1155*(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^6/(a^19*b^5))^(1/4)*log(-231*I*a^5*b*(-d^6/(a^19*b^5))^(1/4) +
231*sqrt(d*x)*d) - 1155*(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^6/(a^19*b^5))^(1/4)*log(-231*a^5*b*(-d^6/(a^19*b^5))^(1/4) + 231*sqrt(d*x)*d) + 4*(385*b^4*d*x^8 + 176
0*a*b^3*d*x^6 + 3130*a^2*b^2*d*x^4 + 2648*a^3*b*d*x^2 - 1155*a^4*d)*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]

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

[In]

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

[Out]

Integral((d*x)**(3/2)/(a + b*x**2)**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.01 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {8 \, {\left (385 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{4} + 1760 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{6} + 3130 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{8} + 2648 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{10} - 1155 \, \sqrt {d x} a^{4} d^{12}\right )}}{a^{4} b^{6} d^{10} x^{10} + 5 \, a^{5} b^{5} d^{10} x^{8} + 10 \, a^{6} b^{4} d^{10} x^{6} + 10 \, a^{7} b^{3} d^{10} x^{4} + 5 \, a^{8} b^{2} d^{10} x^{2} + a^{9} b d^{10}} + \frac {1155 \, {\left (\frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{4} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{4} b}}{163840 \, d} \]

[In]

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

[Out]

1/163840*(8*(385*(d*x)^(17/2)*b^4*d^4 + 1760*(d*x)^(13/2)*a*b^3*d^6 + 3130*(d*x)^(9/2)*a^2*b^2*d^8 + 2648*(d*x
)^(5/2)*a^3*b*d^10 - 1155*sqrt(d*x)*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) + 1155*(sqrt(2)*d^4*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^4*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^3*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
^3*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(sq
rt(a)*sqrt(b)*d)*sqrt(a)))/(a^4*b))/d

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 340, normalized size of antiderivative = 0.87 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {1}{163840} \, d {\left (\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{2}} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{5} b^{2}} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{2}} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{5} b^{2}} + \frac {8 \, {\left (385 \, \sqrt {d x} b^{4} d^{10} x^{8} + 1760 \, \sqrt {d x} a b^{3} d^{10} x^{6} + 3130 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} + 2648 \, \sqrt {d x} a^{3} b d^{10} x^{2} - 1155 \, \sqrt {d x} a^{4} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{4} b}\right )} \]

[In]

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

[Out]

1/163840*d*(2310*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^5*b^2) + 2310*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^5*b^2) + 1155*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq
rt(a*d^2/b))/(a^5*b^2) - 1155*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^5*b^2) + 8*(385*sqrt(d*x)*b^4*d^10*x^8 + 1760*sqrt(d*x)*a*b^3*d^10*x^6 + 3130*sqrt(d*x)*a^2*b^2*d^10
*x^4 + 2648*sqrt(d*x)*a^3*b*d^10*x^2 - 1155*sqrt(d*x)*a^4*d^10)/((b*d^2*x^2 + a*d^2)^5*a^4*b))

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.54 \[ \int \frac {(d x)^{3/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {331\,d^9\,{\left (d\,x\right )}^{5/2}}{2560\,a}-\frac {231\,d^{11}\,\sqrt {d\,x}}{4096\,b}+\frac {11\,b^2\,d^5\,{\left (d\,x\right )}^{13/2}}{128\,a^3}+\frac {77\,b^3\,d^3\,{\left (d\,x\right )}^{17/2}}{4096\,a^4}+\frac {313\,b\,d^7\,{\left (d\,x\right )}^{9/2}}{2048\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {231\,d^{3/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{19/4}\,b^{5/4}}-\frac {231\,d^{3/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{19/4}\,b^{5/4}} \]

[In]

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

[Out]

((331*d^9*(d*x)^(5/2))/(2560*a) - (231*d^11*(d*x)^(1/2))/(4096*b) + (11*b^2*d^5*(d*x)^(13/2))/(128*a^3) + (77*
b^3*d^3*(d*x)^(17/2))/(4096*a^4) + (313*b*d^7*(d*x)^(9/2))/(2048*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) - (231*d^(3/2)*atan((b^(1/4)*(d*x)^(1/2)
)/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(19/4)*b^(5/4)) - (231*d^(3/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(
1/2))))/(8192*(-a)^(19/4)*b^(5/4))