3.8.24 \(\int \frac {1}{\sqrt {d x} (a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [724]

Optimal. Leaf size=387 \[ \frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}} \]

[Out]

-4389/16384*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(23/4)/b^(1/4)*2^(1/2)/d^(1/2)+4389/16384*
arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(23/4)/b^(1/4)*2^(1/2)/d^(1/2)-4389/32768*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^(23/4)/b^(1/4)*2^(1/2)/d^(1/2)+4389/32768*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^(23/4)/b^(1/4)*2^(1/2)/d^(1/2)+1/10*(d
*x)^(1/2)/a/d/(b*x^2+a)^5+19/160*(d*x)^(1/2)/a^2/d/(b*x^2+a)^4+19/128*(d*x)^(1/2)/a^3/d/(b*x^2+a)^3+209/1024*(
d*x)^(1/2)/a^4/d/(b*x^2+a)^2+1463/4096*(d*x)^(1/2)/a^5/d/(b*x^2+a)

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Rubi [A]
time = 0.29, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 296, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {4389 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[d*x]/(10*a*d*(a + b*x^2)^5) + (19*Sqrt[d*x])/(160*a^2*d*(a + b*x^2)^4) + (19*Sqrt[d*x])/(128*a^3*d*(a + b
*x^2)^3) + (209*Sqrt[d*x])/(1024*a^4*d*(a + b*x^2)^2) + (1463*Sqrt[d*x])/(4096*a^5*d*(a + b*x^2)) - (4389*ArcT
an[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(23/4)*b^(1/4)*Sqrt[d]) + (4389*ArcTan[
1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(23/4)*b^(1/4)*Sqrt[d]) - (4389*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^(23/4)*b^(1/4)*Sqrt[d]) +
(4389*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^(23/4)*b
^(1/4)*Sqrt[d])

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 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*} \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {\left (19 b^5\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {\left (57 b^4\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^4} \, dx}{64 a^2}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {\left (209 b^3\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^3}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {\left (1463 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^5 d}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {(4389 b) \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^{11/2} d^2}+\frac {(4389 b) \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^{11/2} d^2}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac {4389 \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^{11/2} \sqrt {b}}+\frac {4389 \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^{11/2} \sqrt {b}}-\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}\\ &=\frac {\sqrt {d x}}{10 a d \left (a+b x^2\right )^5}+\frac {19 \sqrt {d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac {19 \sqrt {d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac {209 \sqrt {d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac {1463 \sqrt {d x}}{4096 a^5 d \left (a+b x^2\right )}-\frac {4389 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{23/4} \sqrt [4]{b} \sqrt {d}}-\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}+\frac {4389 \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^{23/4} \sqrt [4]{b} \sqrt {d}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 183, normalized size = 0.47 \begin {gather*} \frac {\sqrt {x} \left (\frac {4 a^{3/4} \sqrt {x} \left (19015 a^4+50312 a^3 b x^2+59470 a^2 b^2 x^4+33440 a b^3 x^6+7315 b^4 x^8\right )}{\left (a+b x^2\right )^5}-\frac {21945 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {21945 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}\right )}{81920 a^{23/4} \sqrt {d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*((4*a^(3/4)*Sqrt[x]*(19015*a^4 + 50312*a^3*b*x^2 + 59470*a^2*b^2*x^4 + 33440*a*b^3*x^6 + 7315*b^4*x^8
))/(a + b*x^2)^5 - (21945*Sqrt[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(1/4) + (
21945*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(1/4)))/(81920*a^(23/4)*Sqrt
[d*x])

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Maple [A]
time = 0.08, size = 241, normalized size = 0.62

method	result	size
derivativedivides	\(2 d^{11} \left (\frac {\frac {3803 \sqrt {d x}}{8192 a \,d^{2}}+\frac {6289 b \left (d x \right )^{\frac {5}{2}}}{5120 a^{2} d^{4}}+\frac {5947 b^{2} \left (d x \right )^{\frac {9}{2}}}{4096 a^{3} d^{6}}+\frac {209 b^{3} \left (d x \right )^{\frac {13}{2}}}{256 a^{4} d^{8}}+\frac {1463 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192 a^{5} d^{10}}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{6} d^{12}}\right )\)	\(241\)
default	\(2 d^{11} \left (\frac {\frac {3803 \sqrt {d x}}{8192 a \,d^{2}}+\frac {6289 b \left (d x \right )^{\frac {5}{2}}}{5120 a^{2} d^{4}}+\frac {5947 b^{2} \left (d x \right )^{\frac {9}{2}}}{4096 a^{3} d^{6}}+\frac {209 b^{3} \left (d x \right )^{\frac {13}{2}}}{256 a^{4} d^{8}}+\frac {1463 b^{4} \left (d x \right )^{\frac {17}{2}}}{8192 a^{5} d^{10}}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {4389 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a^{6} d^{12}}\right )\)	\(241\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^11*((3803/8192/a/d^2*(d*x)^(1/2)+6289/5120/a^2/d^4*b*(d*x)^(5/2)+5947/4096/a^3/d^6*b^2*(d*x)^(9/2)+209/256
/a^4/d^8*b^3*(d*x)^(13/2)+1463/8192/a^5/d^10*b^4*(d*x)^(17/2))/(b*d^2*x^2+a*d^2)^5+4389/65536/a^6/d^12*(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)))

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Maxima [A]
time = 0.52, size = 382, normalized size = 0.99 \begin {gather*} \frac {\frac {8 \, {\left (7315 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{2} + 33440 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{4} + 59470 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{6} + 50312 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{8} + 19015 \, \sqrt {d x} a^{4} d^{10}\right )}}{a^{5} b^{5} d^{10} x^{10} + 5 \, a^{6} b^{4} d^{10} x^{8} + 10 \, a^{7} b^{3} d^{10} x^{6} + 10 \, a^{8} b^{2} d^{10} x^{4} + 5 \, a^{9} b d^{10} x^{2} + a^{10} d^{10}} + \frac {21945 \, {\left (\frac {\sqrt {2} d^{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 {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{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 {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d \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 \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^{5}}}{163840 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/163840*(8*(7315*(d*x)^(17/2)*b^4*d^2 + 33440*(d*x)^(13/2)*a*b^3*d^4 + 59470*(d*x)^(9/2)*a^2*b^2*d^6 + 50312*
(d*x)^(5/2)*a^3*b*d^8 + 19015*sqrt(d*x)*a^4*d^10)/(a^5*b^5*d^10*x^10 + 5*a^6*b^4*d^10*x^8 + 10*a^7*b^3*d^10*x^
6 + 10*a^8*b^2*d^10*x^4 + 5*a^9*b*d^10*x^2 + a^10*d^10) + 21945*(sqrt(2)*d^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)^(3/4)*b^(1/4)) - sqrt(2)*d^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)^(3/4)*b^(1/4)) + 2*sqrt(2)*d*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
*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)))/a^5)/d

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Fricas [A]
time = 0.38, size = 475, normalized size = 1.23 \begin {gather*} \frac {87780 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {a^{12} d^{2} \sqrt {-\frac {1}{a^{23} b d^{2}}} + d x} a^{17} b d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {3}{4}} - \sqrt {d x} a^{17} b d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {3}{4}}\right ) + 21945 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 21945 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{6} d \left (-\frac {1}{a^{23} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (7315 \, b^{4} x^{8} + 33440 \, a b^{3} x^{6} + 59470 \, a^{2} b^{2} x^{4} + 50312 \, a^{3} b x^{2} + 19015 \, a^{4}\right )} \sqrt {d x}}{81920 \, {\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/81920*(87780*(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^6 + 10*a^8*b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*
d)*(-1/(a^23*b*d^2))^(1/4)*arctan(sqrt(a^12*d^2*sqrt(-1/(a^23*b*d^2)) + d*x)*a^17*b*d*(-1/(a^23*b*d^2))^(3/4)
- sqrt(d*x)*a^17*b*d*(-1/(a^23*b*d^2))^(3/4)) + 21945*(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^6 + 1
0*a^8*b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)*(-1/(a^23*b*d^2))^(1/4)*log(a^6*d*(-1/(a^23*b*d^2))^(1/4) + sqrt(d*x
)) - 21945*(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^6 + 10*a^8*b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)*(
-1/(a^23*b*d^2))^(1/4)*log(-a^6*d*(-1/(a^23*b*d^2))^(1/4) + sqrt(d*x)) + 4*(7315*b^4*x^8 + 33440*a*b^3*x^6 + 5
9470*a^2*b^2*x^4 + 50312*a^3*b*x^2 + 19015*a^4)*sqrt(d*x))/(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^
6 + 10*a^8*b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d x} \left (a + b x^{2}\right )^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.90, size = 346, normalized size = 0.89 \begin {gather*} \frac {4389 \, \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 )}{16384 \, a^{6} b d} + \frac {4389 \, \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 )}{16384 \, a^{6} b d} + \frac {4389 \, \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 )}{32768 \, a^{6} b d} - \frac {4389 \, \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 )}{32768 \, a^{6} b d} + \frac {7315 \, \sqrt {d x} b^{4} d^{9} x^{8} + 33440 \, \sqrt {d x} a b^{3} d^{9} x^{6} + 59470 \, \sqrt {d x} a^{2} b^{2} d^{9} x^{4} + 50312 \, \sqrt {d x} a^{3} b d^{9} x^{2} + 19015 \, \sqrt {d x} a^{4} d^{9}}{20480 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4389/16384*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^6*b*d) + 4389/16384*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^6*b*d) + 4389/32768*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^6*b*d) - 4389/32768*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^6*b*d) + 1/20480*(7315*sqrt(d*x)*b^4*d^9*x^8 + 33440*sqrt(d*x)*a*b^3*d^9*x^6 + 59470*sqr
t(d*x)*a^2*b^2*d^9*x^4 + 50312*sqrt(d*x)*a^3*b*d^9*x^2 + 19015*sqrt(d*x)*a^4*d^9)/((b*d^2*x^2 + a*d^2)^5*a^5)

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Mupad [B]
time = 4.29, size = 210, normalized size = 0.54 \begin {gather*} \frac {\frac {3803\,d^9\,\sqrt {d\,x}}{4096\,a}+\frac {5947\,b^2\,d^5\,{\left (d\,x\right )}^{9/2}}{2048\,a^3}+\frac {209\,b^3\,d^3\,{\left (d\,x\right )}^{13/2}}{128\,a^4}+\frac {6289\,b\,d^7\,{\left (d\,x\right )}^{5/2}}{2560\,a^2}+\frac {1463\,b^4\,d\,{\left (d\,x\right )}^{17/2}}{4096\,a^5}}{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 {4389\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{23/4}\,b^{1/4}\,\sqrt {d}}+\frac {4389\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{23/4}\,b^{1/4}\,\sqrt {d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((3803*d^9*(d*x)^(1/2))/(4096*a) + (5947*b^2*d^5*(d*x)^(9/2))/(2048*a^3) + (209*b^3*d^3*(d*x)^(13/2))/(128*a^4
) + (6289*b*d^7*(d*x)^(5/2))/(2560*a^2) + (1463*b^4*d*(d*x)^(17/2))/(4096*a^5))/(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) + (4389*atan((b^(1/4)*(d*x)^(1/
2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(23/4)*b^(1/4)*d^(1/2)) + (4389*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*
d^(1/2))))/(8192*(-a)^(23/4)*b^(1/4)*d^(1/2))

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