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

Optimal. Leaf size=100 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4 a+b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))-2 a \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+2 a \log (x)}{\text {$\#$1} a-2 \text {$\#$1}^5}\& \right ] \]

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Rubi [B]  time = 2.23, antiderivative size = 627, normalized size of antiderivative = 6.27, number of steps used = 12, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {2056, 6715, 6728, 377, 212, 208, 205} \begin {gather*} \frac {\sqrt {x} \left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (\frac {a^2+2 b}{\sqrt {a^2-4 b}}+a\right ) \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 [4]{a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{-a \sqrt {a^2-4 b}+a^2-2 b} \sqrt [4]{a x^4+b x^2}}+\frac {\sqrt {x} \left (\frac {a^2+2 b}{\sqrt {a^2-4 b}}+a\right ) \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 [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[(-b + a*x^2)/((b + a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]

[Out]

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

Rule 6715

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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-b+a x^2}{\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 {-b+a x^2}{\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 {-b+a x^4}{\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 (2 \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {a+\frac {-a^2-2 b}{\sqrt {a^2-4 b}}}{\left (a-\sqrt {a^2-4 b}+2 x^4\right ) \sqrt [4]{b+a x^4}}+\frac {a-\frac {-a^2-2 b}{\sqrt {a^2-4 b}}}{\left (a+\sqrt {a^2-4 b}+2 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 (2 \left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{b x^2+a x^4}}+\frac {\left (2 \left (a+\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{b x^2+a x^4}}\\ &=\frac {\left (2 \left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{b x^2+a x^4}}+\frac {\left (2 \left (a+\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{b x^2+a x^4}}\\ &=\frac {\left (\left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{b x^2+a x^4}}+\frac {\left (\left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{b x^2+a x^4}}+\frac {\left (\left (a+\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{b x^2+a x^4}}+\frac {\left (\left (a+\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{b x^2+a x^4}}\\ &=\frac {\left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a+\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{a^2+a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{a^2-a \sqrt {a^2-4 b}-2 b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (a+\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [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 [B]  time = 0.92, size = 535, normalized size = 5.35 \begin {gather*} \frac {\sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}+\frac {\left (\frac {a^2+2 b}{\sqrt {a^2-4 b}}+a\right ) \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 [4]{a \sqrt {a^2-4 b}+a^2-2 b}}+\frac {\left (a-\frac {a^2+2 b}{\sqrt {a^2-4 b}}\right ) \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 [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}+\frac {\left (\frac {a^2+2 b}{\sqrt {a^2-4 b}}+a\right ) \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 [4]{a \sqrt {a^2-4 b}+a^2-2 b}}\right )}{\sqrt [4]{x^2 \left (a x^2+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(b + a*x^2)^(1/4)*(((a - (a^2 + 2*b)/Sqrt[a^2 - 4*b])*ArcTan[((a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*S
qrt[x])/((a - Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))])/((a - Sqrt[a^2 - 4*b])^(3/4)*(a^2 - a*Sqrt[a^2 - 4*b
] - 2*b)^(1/4)) + ((a + (a^2 + 2*b)/Sqrt[a^2 - 4*b])*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)*(a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(
1/4)) + ((a - (a^2 + 2*b)/Sqrt[a^2 - 4*b])*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)*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)) + (
(a + (a^2 + 2*b)/Sqrt[a^2 - 4*b])*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)*(a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4))))/(x^2*(b +
a*x^2))^(1/4)

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IntegrateAlgebraic [A]  time = 0.62, size = 101, normalized size = 1.01 \begin {gather*} -\frac {1}{2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a \log (x)+2 a \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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((a*x^2-b)/(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((a*x^2-b)/(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]=[-59,28]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]=[7,11]Evaluation time: 2.58Unable to convert to real 1/4 E
rror: Bad Argument Value

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b}{\left (x^{4}+a \,x^{2}+b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

int(-(b - a*x^2)/((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 {a x^{2} - b}{\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((a*x**2-b)/(x**4+a*x**2+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**2 + b + x**4)), x)

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