3.15.81 \(\int \frac {1}{(b-a x^2+x^4) \sqrt [4]{-b x^2+a x^4}} \, dx\)

Optimal. Leaf size=104 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4 a+b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))-a \log \left (\sqrt [4]{a x^4-b x^2}-\text {$\#$1} x\right )+a \log (x)}{\text {$\#$1} a-2 \text {$\#$1}^5}\& \right ]}{2 b} \]

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Rubi [B]  time = 0.63, antiderivative size = 609, normalized size of antiderivative = 5.86, number of steps used = 11, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2056, 1269, 1428, 377, 212, 208, 205} \begin {gather*} \frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4-b x^2}}-\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4-b x^2}}+\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2-b}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4-b x^2}}-\frac {2 \sqrt {x} \sqrt [4]{a x^2-b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2-b}}\right )}{\left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4*b])^(1
/4)*(-b + a*x^2)^(1/4))])/((a - Sqrt[a^2 - 4*b])^(3/4)*Sqrt[a^2 - 4*b]*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*(
-(b*x^2) + a*x^4)^(1/4)) - (2*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTan[((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x]
)/((a + Sqrt[a^2 - 4*b])^(1/4)*(-b + a*x^2)^(1/4))])/((a + Sqrt[a^2 - 4*b])^(3/4)*Sqrt[a^2 - 4*b]*(a^2 + a*Sqr
t[a^2 - 4*b] - 2*b)^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) + (2*Sqrt[x]*(-b + a*x^2)^(1/4)*ArcTanh[((a^2 - a*Sqrt[a^2
- 4*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4*b])^(1/4)*(-b + a*x^2)^(1/4))])/((a - Sqrt[a^2 - 4*b])^(3/4)*
Sqrt[a^2 - 4*b]*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*(-(b*x^2) + a*x^4)^(1/4)) - (2*Sqrt[x]*(-b + a*x^2)^(1/4
)*ArcTanh[((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a + Sqrt[a^2 - 4*b])^(1/4)*(-b + a*x^2)^(1/4))])/(
(a + Sqrt[a^2 - 4*b])^(3/4)*Sqrt[a^2 - 4*b]*(a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*(-(b*x^2) + a*x^4)^(1/4))

Rule 205

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

Rule 208

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

Rule 212

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] &&
!GtQ[a/b, 0]

Rule 377

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 1269

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With
[{k = Denominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + (e*x^(2*k))/f^2)^q*(a + (b*x^(2*k))/f^k + (c
*x^(4*k))/f^4)^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && Fra
ctionQ[m] && IntegerQ[p]

Rule 1428

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[b^2 -
4*a*c, 2]}, Dist[(2*c)/r, Int[(d + e*x^n)^q/(b - r + 2*c*x^n), x], x] - Dist[(2*c)/r, Int[(d + e*x^n)^q/(b +
r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] &&  !IntegerQ[q]

Rule 2056

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]

Rubi steps

\begin {align*} \int \frac {1}{\left (b-a x^2+x^4\right ) \sqrt [4]{-b x^2+a x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{-b+a x^2} \left (b-a x^2+x^4\right )} \, dx}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^4} \left (b-a x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a-\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-a+\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-a-\sqrt {a^2-4 b}-\left (a \left (-a-\sqrt {a^2-4 b}\right )+2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (4 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+\sqrt {a^2-4 b}-\left (a \left (-a+\sqrt {a^2-4 b}\right )+2 b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}+\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}+\sqrt {a^2-a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a-\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}-\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}+\sqrt {a^2+a \sqrt {a^2-4 b}-2 b} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-b+a x^2}}\right )}{\sqrt {a+\sqrt {a^2-4 b}} \sqrt {a^2-4 b} \sqrt [4]{-b x^2+a x^4}}\\ &=\frac {2 \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{-b x^2+a x^4}}-\frac {2 \sqrt {x} \sqrt [4]{-b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a+\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{-b x^2+a x^4}}+\frac {2 \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{-b x^2+a x^4}}-\frac {2 \sqrt {x} \sqrt [4]{-b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt {x}}{\sqrt [4]{a+\sqrt {a^2-4 b}} \sqrt [4]{-b+a x^2}}\right )}{\left (a+\sqrt {a^2-4 b}\right )^{3/4} \sqrt {a^2-4 b} \sqrt [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{-b x^2+a x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 165, normalized size = 1.59 \begin {gather*} \frac {4 x \left (\frac {\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (\frac {2}{a-\sqrt {a^2-4 b}}-\frac {a}{b}\right ) b x^2}{b-a x^2}\right )}{a \sqrt {a^2-4 b}-a^2+4 b}-\frac {\, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (\frac {2}{a+\sqrt {a^2-4 b}}-\frac {a}{b}\right ) b x^2}{b-a x^2}\right )}{a \sqrt {a^2-4 b}+a^2-4 b}\right )}{\sqrt [4]{a x^4-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x*(Hypergeometric2F1[1/4, 1, 5/4, ((2/(a - Sqrt[a^2 - 4*b]) - a/b)*b*x^2)/(b - a*x^2)]/(-a^2 + a*Sqrt[a^2 -
4*b] + 4*b) - Hypergeometric2F1[1/4, 1, 5/4, ((2/(a + Sqrt[a^2 - 4*b]) - a/b)*b*x^2)/(b - a*x^2)]/(a^2 + a*Sq
rt[a^2 - 4*b] - 4*b)))/(-(b*x^2) + a*x^4)^(1/4)

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IntegrateAlgebraic [A]  time = 0.00, size = 105, normalized size = 1.01 \begin {gather*} \frac {\text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[83,-68]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[-17,52]Evaluation time: 2.62Unable to convert to real 1/4
Error: Bad Argument Value

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{4}-a \,x^{2}+b \right ) \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a x^{4} - b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a\,x^4-b\,x^2\right )}^{1/4}\,\left (x^4-a\,x^2+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (- a x^{2} + b + x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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