Integrand size = 27, antiderivative size = 65 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b} \]
Time = 0.34 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [2 a^2-a b-4 a \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 b \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
(Sqrt[x]*(b + a*x^2)^(1/4)*RootSum[2*a^2 - a*b - 4*a*#1^4 + 2*#1^8 & , (-L og[Sqrt[x]] + Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1])/#1 & ])/(8*b*(x^2*(b + a*x^2))^(1/4))
Leaf count is larger than twice the leaf count of optimal. \(428\) vs. \(2(65)=130\).
Time = 0.76 (sec) , antiderivative size = 428, normalized size of antiderivative = 6.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2467, 25, 1593, 1759, 27, 902, 756, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a x^4-2 b\right ) \sqrt [4]{a x^4+b x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2+b} \int -\frac {1}{\sqrt {x} \sqrt [4]{a x^2+b} \left (2 b-a x^4\right )}dx}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x^2+b} \int \frac {1}{\sqrt {x} \sqrt [4]{a x^2+b} \left (2 b-a x^4\right )}dx}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 1593 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \int \frac {1}{\sqrt [4]{a x^2+b} \left (2 b-a x^4\right )}d\sqrt {x}}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 1759 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {a} \left (\sqrt {2} \sqrt {b}-\sqrt {a} x^2\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {b}}+\frac {\sqrt {a} \int \frac {1}{\sqrt {a} \left (\sqrt {a} x^2+\sqrt {2} \sqrt {b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\int \frac {1}{\left (\sqrt {2} \sqrt {b}-\sqrt {a} x^2\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {b}}+\frac {\int \frac {1}{\left (\sqrt {a} x^2+\sqrt {2} \sqrt {b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}}{2 \sqrt {2} \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 902 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\int \frac {1}{\sqrt {2} \sqrt {b}-\left (\sqrt {2} a \sqrt {b}-\sqrt {a} b\right ) x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {2} \sqrt {b}}+\frac {\int \frac {1}{\sqrt {2} \sqrt {b}-\left (\sqrt {2} \sqrt {b} a+b \sqrt {a}\right ) x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {2} \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\frac {\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt [4]{2} \sqrt {b}}+\frac {\int \frac {1}{\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x+\sqrt [4]{2}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt [4]{2} \sqrt {b}}}{2 \sqrt {2} \sqrt {b}}+\frac {\frac {\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt [4]{2} \sqrt {b}}+\frac {\int \frac {1}{\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x+\sqrt [4]{2}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt [4]{2} \sqrt {b}}}{2 \sqrt {2} \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\frac {\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}-\sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt [4]{2} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}}{2 \sqrt {2} \sqrt {b}}+\frac {\frac {\int \frac {1}{\sqrt [4]{2}-\sqrt [4]{a} \sqrt {\sqrt {2} \sqrt {a}+\sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt [4]{2} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}}{2 \sqrt {2} \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\frac {\arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}-\sqrt {b}}}}{2 \sqrt {2} \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt {x} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{2\ 2^{3/8} \sqrt [8]{a} \sqrt {b} \sqrt [4]{\sqrt {2} \sqrt {a}+\sqrt {b}}}}{2 \sqrt {2} \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
(-2*Sqrt[x]*(b + a*x^2)^(1/4)*((ArcTan[(a^(1/8)*(Sqrt[2]*Sqrt[a] - Sqrt[b] )^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))]/(2*2^(3/8)*a^(1/8)*(Sqrt[2]* Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[b]) + ArcTanh[(a^(1/8)*(Sqrt[2]*Sqrt[a] - Sq rt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))]/(2*2^(3/8)*a^(1/8)*(Sqr t[2]*Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[b]))/(2*Sqrt[2]*Sqrt[b]) + (ArcTan[(a^( 1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4) )]/(2*2^(3/8)*a^(1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[b]) + ArcTanh [(a^(1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^ (1/4))]/(2*2^(3/8)*a^(1/8)*(Sqrt[2]*Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[b]))/(2* Sqrt[2]*Sqrt[b])))/(b*x^2 + a*x^4)^(1/4)
3.9.56.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_ ), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f Subst[Int[x^(k*(m + 1 ) - 1)*(d + e*(x^(2*k)/f))^q*(a + c*(x^(4*k)/f))^p, x], x, (f*x)^(1/k)], x] ] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> W ith[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r) Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Simp[c/(2*r) Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 1.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-4 \textit {\_Z}^{4} a +2 a^{2}-a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}}{8 b}\) | \(58\) |
Timed out. \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]
Not integrable
Time = 2.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.37 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} - 2 b\right )}\, dx \]
Not integrable
Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}} \,d x } \]
Not integrable
Time = 2.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {1}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - 2 \, b\right )}} \,d x } \]
Not integrable
Time = 5.47 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\int \frac {1}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (2\,b-a\,x^4\right )} \,d x \]