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

[Out]

13923/16384*b^(1/4)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(25/4)/d^(3/2)*2^(1/2)-13923/16384
*b^(1/4)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(25/4)/d^(3/2)*2^(1/2)-13923/32768*b^(1/4)*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^(25/4)/d^(3/2)*2^(1/2)+13923/32768*b
^(1/4)*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^(25/4)/d^(3/2)*2^(1/2)-1392
3/4096/a^6/d/(d*x)^(1/2)+1/10/a/d/(b*x^2+a)^5/(d*x)^(1/2)+21/160/a^2/d/(b*x^2+a)^4/(d*x)^(1/2)+119/640/a^3/d/(
b*x^2+a)^3/(d*x)^(1/2)+1547/5120/a^4/d/(b*x^2+a)^2/(d*x)^(1/2)+13923/20480/a^5/d/(b*x^2+a)/(d*x)^(1/2)

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Rubi [A]  time = 0.53, antiderivative size = 404, normalized size of antiderivative = 1.00, 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 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 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 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 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 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 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]

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 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]

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*}

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Mathematica [C]  time = 0.01, 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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fricas [A]  time = 1.14, size = 544, normalized size = 1.35 \[ \frac {278460 \, {\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}} \arctan \left (-\frac {2698972561467 \, \sqrt {d x} a^{6} b d \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}} - \sqrt {-7284452887551739093192089 \, a^{13} b d^{4} \sqrt {-\frac {b}{a^{25} d^{6}}} + 7284452887551739093192089 \, b^{2} d x} a^{6} d \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}}}{2698972561467 \, b}\right ) - 69615 \, {\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}} \log \left (2698972561467 \, a^{19} d^{5} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b\right ) + 69615 \, {\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {1}{4}} \log \left (-2698972561467 \, a^{19} d^{5} \left (-\frac {b}{a^{25} d^{6}}\right )^{\frac {3}{4}} + 2698972561467 \, \sqrt {d x} b\right ) - 4 \, {\left (69615 \, b^{5} x^{10} + 334152 \, a b^{4} x^{8} + 634270 \, a^{2} b^{3} x^{6} + 590240 \, a^{3} b^{2} x^{4} + 263515 \, a^{4} b x^{2} + 40960 \, a^{5}\right )} \sqrt {d x}}{81920 \, {\left (a^{6} b^{5} d^{2} x^{11} + 5 \, a^{7} b^{4} d^{2} x^{9} + 10 \, a^{8} b^{3} d^{2} x^{7} + 10 \, a^{9} b^{2} d^{2} x^{5} + 5 \, a^{10} b d^{2} x^{3} + a^{11} d^{2} x\right )}} \]

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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giac [A]  time = 0.20, size = 365, normalized size = 0.90 \[ -\frac {\frac {327680}{\sqrt {d x} a^{6}} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} b^{2} d^{2}} + \frac {139230 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{7} b^{2} d^{2}} - \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{7} b^{2} d^{2}} + \frac {69615 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{7} b^{2} d^{2}} + \frac {8 \, {\left (28655 \, \sqrt {d x} b^{5} d^{9} x^{9} + 129352 \, \sqrt {d x} a b^{4} d^{9} x^{7} + 224670 \, \sqrt {d x} a^{2} b^{3} d^{9} x^{5} + 180640 \, \sqrt {d x} a^{3} b^{2} d^{9} x^{3} + 58715 \, \sqrt {d x} a^{4} b d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{6}}}{163840 \, d} \]

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

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maple [A]  time = 0.03, size = 349, normalized size = 0.86 \[ -\frac {11743 \left (d x \right )^{\frac {3}{2}} b \,d^{7}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}-\frac {1129 \left (d x \right )^{\frac {7}{2}} b^{2} d^{5}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}-\frac {22467 \left (d x \right )^{\frac {11}{2}} b^{3} d^{3}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{4}}-\frac {16169 \left (d x \right )^{\frac {15}{2}} b^{4} d}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{5}}-\frac {5731 \left (d x \right )^{\frac {19}{2}} b^{5}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{6} d}-\frac {13923 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{6} d}-\frac {13923 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{6} d}-\frac {13923 \sqrt {2}\, \ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{6} d}-\frac {2}{\sqrt {d x}\, a^{6} d} \]

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]

-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)-224
67/2048*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)-573
1/4096/d*b^5/a^6/(b*d^2*x^2+a*d^2)^5*(d*x)^(19/2)-13923/32768/d/a^6/(a/b*d^2)^(1/4)*2^(1/2)*ln((d*x-(a/b*d^2)^
(1/4)*(d*x)^(1/2)*2^(1/2)+(a/b*d^2)^(1/2))/(d*x+(a/b*d^2)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a/b*d^2)^(1/2)))-13923/16
384/d/a^6/(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)-13923/16384/d/a^6/(a/b*d^2)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)-1)-2/a^6/d/(d*x)^(1/2)

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maxima [A]  time = 3.23, size = 388, normalized size = 0.96 \[ -\frac {\frac {8 \, {\left (69615 \, b^{5} d^{10} x^{10} + 334152 \, a b^{4} d^{10} x^{8} + 634270 \, a^{2} b^{3} d^{10} x^{6} + 590240 \, a^{3} b^{2} d^{10} x^{4} + 263515 \, a^{4} b d^{10} x^{2} + 40960 \, a^{5} d^{10}\right )}}{\left (d x\right )^{\frac {21}{2}} a^{6} b^{5} + 5 \, \left (d x\right )^{\frac {17}{2}} a^{7} b^{4} d^{2} + 10 \, \left (d x\right )^{\frac {13}{2}} a^{8} b^{3} d^{4} + 10 \, \left (d x\right )^{\frac {9}{2}} a^{9} b^{2} d^{6} + 5 \, \left (d x\right )^{\frac {5}{2}} a^{10} b d^{8} + \sqrt {d x} a^{11} d^{10}} + \frac {69615 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{6}}}{163840 \, d} \]

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]

-1/163840*(8*(69615*b^5*d^10*x^10 + 334152*a*b^4*d^10*x^8 + 634270*a^2*b^3*d^10*x^6 + 590240*a^3*b^2*d^10*x^4
+ 263515*a^4*b*d^10*x^2 + 40960*a^5*d^10)/((d*x)^(21/2)*a^6*b^5 + 5*(d*x)^(17/2)*a^7*b^4*d^2 + 10*(d*x)^(13/2)
*a^8*b^3*d^4 + 10*(d*x)^(9/2)*a^9*b^2*d^6 + 5*(d*x)^(5/2)*a^10*b*d^8 + sqrt(d*x)*a^11*d^10) + 69615*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)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))
/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) - sqrt(2)*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*
sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(
d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)))/a^6)/d

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mupad [B]  time = 0.21, size = 226, normalized size = 0.56 \[ \frac {13923\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{25/4}\,d^{3/2}}-\frac {13923\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{25/4}\,d^{3/2}}-\frac {\frac {2\,d^9}{a}+\frac {52703\,b\,d^9\,x^2}{4096\,a^2}+\frac {3689\,b^2\,d^9\,x^4}{128\,a^3}+\frac {63427\,b^3\,d^9\,x^6}{2048\,a^4}+\frac {41769\,b^4\,d^9\,x^8}{2560\,a^5}+\frac {13923\,b^5\,d^9\,x^{10}}{4096\,a^6}}{b^5\,{\left (d\,x\right )}^{21/2}+a^5\,d^{10}\,\sqrt {d\,x}+10\,a^3\,b^2\,d^6\,{\left (d\,x\right )}^{9/2}+10\,a^2\,b^3\,d^4\,{\left (d\,x\right )}^{13/2}+5\,a^4\,b\,d^8\,{\left (d\,x\right )}^{5/2}+5\,a\,b^4\,d^2\,{\left (d\,x\right )}^{17/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(13923*(-b)^(1/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192*a^(25/4)*d^(3/2)) - (13923*(-b)^(1/
4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192*a^(25/4)*d^(3/2)) - ((2*d^9)/a + (52703*b*d^9*x^2)/
(4096*a^2) + (3689*b^2*d^9*x^4)/(128*a^3) + (63427*b^3*d^9*x^6)/(2048*a^4) + (41769*b^4*d^9*x^8)/(2560*a^5) +
(13923*b^5*d^9*x^10)/(4096*a^6))/(b^5*(d*x)^(21/2) + a^5*d^10*(d*x)^(1/2) + 10*a^3*b^2*d^6*(d*x)^(9/2) + 10*a^
2*b^3*d^4*(d*x)^(13/2) + 5*a^4*b*d^8*(d*x)^(5/2) + 5*a*b^4*d^2*(d*x)^(17/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{6}}\, dx \]

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]

Integral(1/((d*x)**(3/2)*(a + b*x**2)**6), x)

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