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

Optimal. Leaf size=101 \[ \frac {\left (4 a x^2-3 b\right ) \left (a x^4+b x^2\right )^{3/4}}{21 b^2 x^5}-\frac {a \text {RootSum}\left [2 \text {$\#$1}^8-4 \text {$\#$1}^4 a+2 a^2+a b\& ,\frac {\log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{16 b} \]

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Rubi [B]  time = 1.79, antiderivative size = 538, normalized size of antiderivative = 5.33, number of steps used = 19, number of rules used = 11, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {2056, 6725, 271, 264, 1270, 1529, 1429, 377, 212, 206, 203} \begin {gather*} \frac {4 a \left (a x^2+b\right )}{21 b^2 x \sqrt [4]{a x^4+b x^2}}+\frac {a \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}+\frac {a \sqrt {x} \sqrt [4]{a x^2+b} \tan ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+\sqrt {2} a}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {-a} \sqrt {b}+\sqrt {2} a} \sqrt [4]{a x^4+b x^2}}+\frac {a \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} \sqrt [4]{a x^4+b x^2}}+\frac {a \sqrt {x} \sqrt [4]{a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+\sqrt {2} a}}{\sqrt [8]{2} \sqrt [4]{a x^2+b}}\right )}{4\ 2^{7/8} b \sqrt [4]{\sqrt {-a} \sqrt {b}+\sqrt {2} a} \sqrt [4]{a x^4+b x^2}}-\frac {a x^2+b}{7 b x^3 \sqrt [4]{a x^4+b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*(b + a*x^2)/(b*x^3*(b*x^2 + a*x^4)^(1/4)) + (4*a*(b + a*x^2))/(21*b^2*x*(b*x^2 + a*x^4)^(1/4)) + (a*Sqrt[
x]*(b + a*x^2)^(1/4)*ArcTan[((Sqrt[2]*a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))])/(4*2^
(7/8)*(Sqrt[2]*a - Sqrt[-a]*Sqrt[b])^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) + (a*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTan[((Sq
rt[2]*a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))])/(4*2^(7/8)*(Sqrt[2]*a + Sqrt[-a]*Sqrt
[b])^(1/4)*b*(b*x^2 + a*x^4)^(1/4)) + (a*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[((Sqrt[2]*a - Sqrt[-a]*Sqrt[b])^(1/
4)*Sqrt[x])/(2^(1/8)*(b + a*x^2)^(1/4))])/(4*2^(7/8)*(Sqrt[2]*a - Sqrt[-a]*Sqrt[b])^(1/4)*b*(b*x^2 + a*x^4)^(1
/4)) + (a*Sqrt[x]*(b + a*x^2)^(1/4)*ArcTanh[((Sqrt[2]*a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(2^(1/8)*(b + a*x^2
)^(1/4))])/(4*2^(7/8)*(Sqrt[2]*a + Sqrt[-a]*Sqrt[b])^(1/4)*b*(b*x^2 + a*x^4)^(1/4))

Rule 203

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

Rule 206

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

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 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

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 1270

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominat
or[m]}, Dist[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]

Rule 1429

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[-(a*c), 2]}, -Dist[c/(2
*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[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]

Rule 1529

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

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 6725

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*} \int \frac {b+a x^4}{x^4 \left (2 b+a 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^4}{x^{9/2} \sqrt [4]{b+a x^2} \left (2 b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \left (\frac {1}{x^{9/2} \sqrt [4]{b+a x^2}}-\frac {b}{x^{9/2} \sqrt [4]{b+a x^2} \left (2 b+a x^4\right )}\right ) \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{b+a x^2}} \, dx}{\sqrt [4]{b x^2+a x^4}}-\frac {\left (b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{9/2} \sqrt [4]{b+a x^2} \left (2 b+a x^4\right )} \, dx}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}-\frac {\left (4 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \int \frac {1}{x^{5/2} \sqrt [4]{b+a x^2}} \, dx}{7 b \sqrt [4]{b x^2+a x^4}}-\frac {\left (2 b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{b+a x^4} \left (2 b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}-\frac {\left (2 b \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 b x^8 \sqrt [4]{b+a x^4}}-\frac {a}{2 b \sqrt [4]{b+a x^4} \left (2 b+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {2 \left (b+a x^2\right )}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^8 \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{b+a x^4} \left (2 b+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^4}}\\ &=-\frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {8 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {\left (4 a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{7 b \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {-a} \sqrt {b}-a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {2} \sqrt {-a} \sqrt {b}+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}-a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}+\frac {\left (\sqrt {-a} a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {-a} \sqrt {b}-\left (\sqrt {2} \sqrt {-a} a \sqrt {b}+a b\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {2} \sqrt {b} \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a-\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}-\sqrt {\sqrt {2} a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}+\frac {\left (a \sqrt {x} \sqrt [4]{b+a x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{2}+\sqrt {\sqrt {2} a+\sqrt {-a} \sqrt {b}} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4\ 2^{3/4} b \sqrt [4]{b x^2+a x^4}}\\ &=-\frac {b+a x^2}{7 b x^3 \sqrt [4]{b x^2+a x^4}}+\frac {4 a \left (b+a x^2\right )}{21 b^2 x \sqrt [4]{b x^2+a x^4}}+\frac {a \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} b \sqrt [4]{b x^2+a x^4}}+\frac {a \sqrt {x} \sqrt [4]{b+a x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} a+\sqrt {-a} \sqrt {b}} b \sqrt [4]{b x^2+a x^4}}+\frac {a \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} a-\sqrt {-a} \sqrt {b}} b \sqrt [4]{b x^2+a x^4}}+\frac {a \sqrt {x} \sqrt [4]{b+a x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2} a+\sqrt {-a} \sqrt {b}} \sqrt {x}}{\sqrt [8]{2} \sqrt [4]{b+a x^2}}\right )}{4\ 2^{7/8} \sqrt [4]{\sqrt {2} a+\sqrt {-a} \sqrt {b}} b \sqrt [4]{b x^2+a x^4}}\\ \end {align*}

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Mathematica [B]  time = 4.21, size = 390, normalized size = 3.86 \begin {gather*} \frac {\sqrt {x} \left (\frac {8 \left (a x^2+b\right ) \left (4 a x^2-3 b\right )}{21 b^2 x^{7/2}}-\frac {a \sqrt {x} \sqrt [4]{a+\frac {b}{x^2}} \left (\sqrt [4]{\sqrt {2} \sqrt {-a} \sqrt {b}+2 a} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\frac {\sqrt {-a} \sqrt {b}}{\sqrt {2}}}}\right )+\sqrt [4]{2 a-\sqrt {2} \sqrt {-a} \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\frac {\sqrt {-a} \sqrt {b}}{\sqrt {2}}+a}}\right )-\sqrt [4]{\sqrt {2} \sqrt {-a} \sqrt {b}+2 a} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\frac {\sqrt {-a} \sqrt {b}}{\sqrt {2}}}}\right )-\sqrt [4]{2 a-\sqrt {2} \sqrt {-a} \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\frac {\sqrt {-a} \sqrt {b}}{\sqrt {2}}+a}}\right )\right )}{b \sqrt [4]{a-\frac {\sqrt {-a} \sqrt {b}}{\sqrt {2}}} \sqrt [4]{\sqrt {2} \sqrt {-a} \sqrt {b}+2 a}}\right )}{8 \sqrt [4]{x^2 \left (a x^2+b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*((8*(b + a*x^2)*(-3*b + 4*a*x^2))/(21*b^2*x^(7/2)) - (a*(a + b/x^2)^(1/4)*Sqrt[x]*((2*a + Sqrt[2]*Sqr
t[-a]*Sqrt[b])^(1/4)*ArcTan[(a + b/x^2)^(1/4)/(a - (Sqrt[-a]*Sqrt[b])/Sqrt[2])^(1/4)] + (2*a - Sqrt[2]*Sqrt[-a
]*Sqrt[b])^(1/4)*ArcTan[(a + b/x^2)^(1/4)/(a + (Sqrt[-a]*Sqrt[b])/Sqrt[2])^(1/4)] - (2*a + Sqrt[2]*Sqrt[-a]*Sq
rt[b])^(1/4)*ArcTanh[(a + b/x^2)^(1/4)/(a - (Sqrt[-a]*Sqrt[b])/Sqrt[2])^(1/4)] - (2*a - Sqrt[2]*Sqrt[-a]*Sqrt[
b])^(1/4)*ArcTanh[(a + b/x^2)^(1/4)/(a + (Sqrt[-a]*Sqrt[b])/Sqrt[2])^(1/4)]))/((a - (Sqrt[-a]*Sqrt[b])/Sqrt[2]
)^(1/4)*(2*a + Sqrt[2]*Sqrt[-a]*Sqrt[b])^(1/4)*b)))/(8*(x^2*(b + a*x^2))^(1/4))

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IntegrateAlgebraic [A]  time = 1.41, size = 101, normalized size = 1.00 \begin {gather*} \frac {\left (-3 b+4 a x^2\right ) \left (b x^2+a x^4\right )^{3/4}}{21 b^2 x^5}-\frac {a \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 ]}{16 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*b + 4*a*x^2)*(b*x^2 + a*x^4)^(3/4))/(21*b^2*x^5) - (a*RootSum[2*a^2 + a*b - 4*a*#1^4 + 2*#1^8 & , (-Log[x
] + Log[(b*x^2 + a*x^4)^(1/4) - x*#1])/#1 & ])/(16*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((a*x^4+b)/x^4/(a*x^4+2*b)/(a*x^4+b*x^2)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+b}{x^{4} \left (a \,x^{4}+2 b \right ) \left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^4+b)/x^4/(a*x^4+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^{4} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + 2 \, b\right )} x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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