3.725 \(\int \frac{1}{(d x)^{3/2} (a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=404 \[ -\frac{13923 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5} \]

[Out]

-13923/(4096*a^6*d*Sqrt[d*x]) + 1/(10*a*d*Sqrt[d*x]*(a + b*x^2)^5) + 21/(160*a^2*d*Sqrt[d*x]*(a + b*x^2)^4) +
119/(640*a^3*d*Sqrt[d*x]*(a + b*x^2)^3) + 1547/(5120*a^4*d*Sqrt[d*x]*(a + b*x^2)^2) + 13923/(20480*a^5*d*Sqrt[
d*x]*(a + b*x^2)) + (13923*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^
(25/4)*d^(3/2)) - (13923*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(2
5/4)*d^(3/2)) - (13923*b^(1/4)*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^(25/4)*d^(3/2)) + (13923*b^(1/4)*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^(25/4)*d^(3/2))

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Rubi [A]  time = 0.529383, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {28, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{13923 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-13923/(4096*a^6*d*Sqrt[d*x]) + 1/(10*a*d*Sqrt[d*x]*(a + b*x^2)^5) + 21/(160*a^2*d*Sqrt[d*x]*(a + b*x^2)^4) +
119/(640*a^3*d*Sqrt[d*x]*(a + b*x^2)^3) + 1547/(5120*a^4*d*Sqrt[d*x]*(a + b*x^2)^2) + 13923/(20480*a^5*d*Sqrt[
d*x]*(a + b*x^2)) + (13923*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^
(25/4)*d^(3/2)) - (13923*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(2
5/4)*d^(3/2)) - (13923*b^(1/4)*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^(25/4)*d^(3/2)) + (13923*b^(1/4)*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^(25/4)*d^(3/2))

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 290

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{\left (21 b^5\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{\left (357 b^4\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^4} \, dx}{320 a^2}\\ &=\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{\left (1547 b^3\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^3} \, dx}{1280 a^3}\\ &=\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{\left (13923 b^2\right ) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )^2} \, dx}{10240 a^4}\\ &=\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}+\frac{(13923 b) \int \frac{1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5}\\ &=-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}-\frac{\left (13923 b^2\right ) \int \frac{\sqrt{d x}}{a b+b^2 x^2} \, dx}{8192 a^6 d^2}\\ &=-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}-\frac{\left (13923 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{4096 a^6 d^3}\\ &=-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}+\frac{\left (13923 b^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d-\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{8192 a^6 d^3}-\frac{\left (13923 b^{3/2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} d+\sqrt{b} x^2}{a b+\frac{b^2 x^4}{d^2}} \, dx,x,\sqrt{d x}\right )}{8192 a^6 d^3}\\ &=-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}-\frac{\left (13923 \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}-\frac{\left (13923 \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{d x}\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{16384 a^6 d}-\frac{13923 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a} d}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{d x}\right )}{16384 a^6 d}\\ &=-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}-\frac{13923 \sqrt [4]{b} \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^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \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^{25/4} d^{3/2}}-\frac{\left (13923 \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}+\frac{\left (13923 \sqrt [4]{b}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}\\ &=-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}+\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923 \sqrt [4]{b} \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^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \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^{25/4} d^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0133965, size = 30, normalized size = 0.07 \[ -\frac{2 x \, _2F_1\left (-\frac{1}{4},6;\frac{3}{4};-\frac{b x^2}{a}\right )}{a^6 (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

(-2*x*Hypergeometric2F1[-1/4, 6, 3/4, -((b*x^2)/a)])/(a^6*(d*x)^(3/2))

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Maple [A]  time = 0.076, size = 349, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{{a}^{6}d\sqrt{dx}}}-{\frac{11743\,{d}^{7}b}{4096\,{a}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{1129\,{d}^{5}{b}^{2}}{128\,{a}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{22467\,{d}^{3}{b}^{3}}{2048\,{a}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{16169\,{b}^{4}d}{2560\,{a}^{5} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{15}{2}}}}-{\frac{5731\,{b}^{5}}{4096\,{a}^{6}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{19}{2}}}}-{\frac{13923\,\sqrt{2}}{32768\,{a}^{6}d}\ln \left ({ \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{13923\,\sqrt{2}}{16384\,{a}^{6}d}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{13923\,\sqrt{2}}{16384\,{a}^{6}d}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-2/a^6/d/(d*x)^(1/2)-11743/4096*d^7*b/a^2/(b*d^2*x^2+a*d^2)^5*(d*x)^(3/2)-1129/128*d^5*b^2/a^3/(b*d^2*x^2+a*d^
2)^5*(d*x)^(7/2)-22467/2048*d^3*b^3/a^4/(b*d^2*x^2+a*d^2)^5*(d*x)^(11/2)-16169/2560*d*b^4/a^5/(b*d^2*x^2+a*d^2
)^5*(d*x)^(15/2)-5731/4096/d*b^5/a^6/(b*d^2*x^2+a*d^2)^5*(d*x)^(19/2)-13923/32768/d/a^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)))-13923/16384/d/a^6/(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)-13923/1638
4/d/a^6/(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.75432, size = 1411, normalized size = 3.49 \begin{align*} \frac{278460 \,{\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{1}{4}} \arctan \left (-\frac{2698972561467 \, \sqrt{d x} a^{6} b d \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{1}{4}} - \sqrt{-7284452887551739093192089 \, a^{13} b d^{4} \sqrt{-\frac{b}{a^{25} d^{6}}} + 7284452887551739093192089 \, b^{2} d x} a^{6} d \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{1}{4}}}{2698972561467 \, b}\right ) - 69615 \,{\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{1}{4}} \log \left (2698972561467 \, a^{19} d^{5} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{3}{4}} + 2698972561467 \, \sqrt{d x} b\right ) + 69615 \,{\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{1}{4}} \log \left (-2698972561467 \, a^{19} d^{5} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{3}{4}} + 2698972561467 \, \sqrt{d x} b\right ) - 4 \,{\left (69615 \, b^{5} x^{10} + 334152 \, a b^{4} x^{8} + 634270 \, a^{2} b^{3} x^{6} + 590240 \, a^{3} b^{2} x^{4} + 263515 \, a^{4} b x^{2} + 40960 \, a^{5}\right )} \sqrt{d x}}{81920 \,{\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(278460*(a^6*b^5*d^2*x^11 + 5*a^7*b^4*d^2*x^9 + 10*a^8*b^3*d^2*x^7 + 10*a^9*b^2*d^2*x^5 + 5*a^10*b*d^2
*x^3 + a^11*d^2*x)*(-b/(a^25*d^6))^(1/4)*arctan(-1/2698972561467*(2698972561467*sqrt(d*x)*a^6*b*d*(-b/(a^25*d^
6))^(1/4) - sqrt(-7284452887551739093192089*a^13*b*d^4*sqrt(-b/(a^25*d^6)) + 7284452887551739093192089*b^2*d*x
)*a^6*d*(-b/(a^25*d^6))^(1/4))/b) - 69615*(a^6*b^5*d^2*x^11 + 5*a^7*b^4*d^2*x^9 + 10*a^8*b^3*d^2*x^7 + 10*a^9*
b^2*d^2*x^5 + 5*a^10*b*d^2*x^3 + a^11*d^2*x)*(-b/(a^25*d^6))^(1/4)*log(2698972561467*a^19*d^5*(-b/(a^25*d^6))^
(3/4) + 2698972561467*sqrt(d*x)*b) + 69615*(a^6*b^5*d^2*x^11 + 5*a^7*b^4*d^2*x^9 + 10*a^8*b^3*d^2*x^7 + 10*a^9
*b^2*d^2*x^5 + 5*a^10*b*d^2*x^3 + a^11*d^2*x)*(-b/(a^25*d^6))^(1/4)*log(-2698972561467*a^19*d^5*(-b/(a^25*d^6)
)^(3/4) + 2698972561467*sqrt(d*x)*b) - 4*(69615*b^5*x^10 + 334152*a*b^4*x^8 + 634270*a^2*b^3*x^6 + 590240*a^3*
b^2*x^4 + 263515*a^4*b*x^2 + 40960*a^5)*sqrt(d*x))/(a^6*b^5*d^2*x^11 + 5*a^7*b^4*d^2*x^9 + 10*a^8*b^3*d^2*x^7
+ 10*a^9*b^2*d^2*x^5 + 5*a^10*b*d^2*x^3 + a^11*d^2*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.2959, size = 493, normalized size = 1.22 \begin{align*} -\frac{\frac{327680}{\sqrt{d x} a^{6}} + \frac{139230 \, \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^{7} b^{2} d^{2}} + \frac{139230 \, \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^{7} b^{2} d^{2}} - \frac{69615 \, \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^{7} b^{2} d^{2}} + \frac{69615 \, \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^{7} b^{2} d^{2}} + \frac{8 \,{\left (28655 \, \sqrt{d x} b^{5} d^{9} x^{9} + 129352 \, \sqrt{d x} a b^{4} d^{9} x^{7} + 224670 \, \sqrt{d x} a^{2} b^{3} d^{9} x^{5} + 180640 \, \sqrt{d x} a^{3} b^{2} d^{9} x^{3} + 58715 \, \sqrt{d x} a^{4} b d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{6}}}{163840 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

-1/163840*(327680/(sqrt(d*x)*a^6) + 139230*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^7*b^2*d^2) + 139230*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^7*b^2*d^2) - 69615*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + s
qrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^7*b^2*d^2) + 69615*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sq
rt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^7*b^2*d^2) + 8*(28655*sqrt(d*x)*b^5*d^9*x^9 + 129352*sqrt(
d*x)*a*b^4*d^9*x^7 + 224670*sqrt(d*x)*a^2*b^3*d^9*x^5 + 180640*sqrt(d*x)*a^3*b^2*d^9*x^3 + 58715*sqrt(d*x)*a^4
*b*d^9*x)/((b*d^2*x^2 + a*d^2)^5*a^6))/d