\(\int \frac {-b+a x^2}{(b+a x^4) \sqrt [4]{b x^2+a x^4}} \, dx\) [1506]

Optimal result

Integrand size = 34, antiderivative size = 105 \[ \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {1}{4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a \log (x)-2 a \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-\text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(445\) vs. \(2(105)=210\).

Time = 0.53 (sec) , antiderivative size = 445, normalized size of antiderivative = 4.24, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {2081, 6847, 6857, 385, 218, 212, 209} \[ \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\sqrt {x} \left (\sqrt {-a}-\sqrt {b}\right ) \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \left (\sqrt {-a}+\sqrt {b}\right ) \sqrt [4]{a x^2+b} \arctan \left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (\sqrt {-a}-\sqrt {b}\right ) \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}-\frac {\sqrt {x} \left (\sqrt {-a}+\sqrt {b}\right ) \sqrt [4]{a x^2+b} \text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \sqrt [4]{a x^4+b x^2}} \]

[In]

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

[Out]

((Sqrt[-a] - Sqrt[b])*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[((a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4
)])/(2*(a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[b]*(b*x^2 + a*x^4)^(1/4)) - ((Sqrt[-a] + Sqrt[b])*Sqrt[x]*(b + a*x^2)
^(1/4)*ArcTan[((a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(2*(a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[
b]*(b*x^2 + a*x^4)^(1/4)) + ((Sqrt[-a] - Sqrt[b])*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[((a - Sqrt[-a]*Sqrt[b])^(1
/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(2*(a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[b]*(b*x^2 + a*x^4)^(1/4)) - ((Sqrt[-a] +
Sqrt[b])*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[((a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(2*(a +
Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[b]*(b*x^2 + a*x^4)^(1/4))

Rule 209

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[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]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {-b+a x^2}{\sqrt {x} \sqrt [4]{b+a x^2} \left (b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {-b+a x^4}{\sqrt [4]{b+a x^4} \left (b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \left (-\frac {\sqrt {-a} \left (a \sqrt {b}-\sqrt {-a} b\right )}{2 a \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{b+a x^4}}+\frac {\sqrt {-a} \left (a \sqrt {b}+\sqrt {-a} b\right )}{2 a \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^4\right ) \sqrt [4]{b+a x^4}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}+\sqrt {-a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}-\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-\left (a \sqrt {b}+\sqrt {-a} b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{\sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ & = \frac {\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2} \arctan \left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt {x} \sqrt [4]{b+a x^2} \text {arctanh}\left (\frac {\sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt [4]{a+\sqrt {-a} \sqrt {b}} \sqrt {b} \sqrt [4]{b x^2+a x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.30 \[ \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a \log (x)-4 a \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-\text {$\#$1}^5}\&\right ]}{8 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]

[In]

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

[Out]

-1/8*(Sqrt[x]*(b + a*x^2)^(1/4)*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (2*a*Log[x] - 4*a*Log[(b + a*x^2)^(1/4
) - Sqrt[x]*#1] - Log[x]*#1^4 + 2*Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1]*#1^4)/(a*#1 - #1^5) & ])/(x^2*(b + a*x^2
))^(1/4)

Maple [N/A] (verified)

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 a \,\textit {\_Z}^{4}+a^{2}+a b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 a \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (\textit {\_R}^{4}-a \right )}\right )}{4}\)	\(66\)

[In]

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

[Out]

1/4*sum(1/_R*(_R^4-2*a)*ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/(_R^4-a),_R=RootOf(_Z^8-2*_Z^4*a+a^2+a*b))

Fricas [F(-1)]

Timed out. \[ \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 5.36 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.26 \[ \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{2} - b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} + b\right )}\, dx \]

[In]

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

[Out]

Integral((a*x**2 - b)/((x**2*(a*x**2 + b))**(1/4)*(a*x**4 + b)), x)

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} - b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}} \,d x } \]

[In]

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

[Out]

integrate((a*x^2 - b)/((a*x^4 + b*x^2)^(1/4)*(a*x^4 + b)), x)

Giac [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} - b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}} \,d x } \]

[In]

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

[Out]

integrate((a*x^2 - b)/((a*x^4 + b*x^2)^(1/4)*(a*x^4 + b)), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int \frac {-b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int -\frac {b-a\,x^2}{\left (a\,x^4+b\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,d x \]

[In]

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

[Out]

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