3.27.25 \(\int \frac {\sqrt [4]{b x^2+a x^4} (b+a x^4+x^8)}{b+a x^4} \, dx\)

Optimal. Leaf size=230 \[ -\frac {b^2 \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2+a b\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^4-a}\& \right ]}{4 a^2}+\frac {\left (-32 a^3 b+32 a b^2-7 b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{128 a^{15/4}}+\frac {\left (32 a^3 b-32 a b^2+7 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^2}}\right )}{128 a^{15/4}}+\frac {\sqrt [4]{a x^4+b x^2} \left (96 a^3 x+32 a^2 x^5+4 a b x^3-96 a b x-7 b^2 x\right )}{192 a^3} \]

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Rubi [B]  time = 1.21, antiderivative size = 529, normalized size of antiderivative = 2.30, number of steps used = 24, number of rules used = 13, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2056, 6725, 279, 329, 331, 298, 203, 206, 321, 1270, 1529, 511, 510} \begin {gather*} -\frac {7 b^3 \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{128 a^{15/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {7 b^3 \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{128 a^{15/4} \sqrt {x} \sqrt [4]{a x^2+b}}-\frac {7 b^2 x \sqrt [4]{a x^4+b x^2}}{192 a^3}+\frac {b x \sqrt [4]{a x^4+b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{\frac {a x^2}{b}+1}}+\frac {b x \sqrt [4]{a x^4+b x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{\frac {a x^2}{b}+1}}+\frac {1}{2} x \left (1-\frac {b}{a^2}\right ) \sqrt [4]{a x^4+b x^2}+\frac {b x^3 \sqrt [4]{a x^4+b x^2}}{48 a^2}-\frac {b \left (a^2-b\right ) \sqrt [4]{a x^4+b x^2} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {b \left (a^2-b\right ) \sqrt [4]{a x^4+b x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{a x^2+b}}+\frac {x^5 \sqrt [4]{a x^4+b x^2}}{6 a} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((b*x^2 + a*x^4)^(1/4)*(b + a*x^4 + x^8))/(b + a*x^4),x]

[Out]

(-7*b^2*x*(b*x^2 + a*x^4)^(1/4))/(192*a^3) + ((1 - b/a^2)*x*(b*x^2 + a*x^4)^(1/4))/2 + (b*x^3*(b*x^2 + a*x^4)^
(1/4))/(48*a^2) + (x^5*(b*x^2 + a*x^4)^(1/4))/(6*a) + (b*x*(b*x^2 + a*x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, -
((Sqrt[-a]*x^2)/Sqrt[b]), -((a*x^2)/b)])/(3*a^2*(1 + (a*x^2)/b)^(1/4)) + (b*x*(b*x^2 + a*x^4)^(1/4)*AppellF1[3
/4, 1, -1/4, 7/4, (Sqrt[-a]*x^2)/Sqrt[b], -((a*x^2)/b)])/(3*a^2*(1 + (a*x^2)/b)^(1/4)) - ((a^2 - b)*b*(b*x^2 +
a*x^4)^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(4*a^(11/4)*Sqrt[x]*(b + a*x^2)^(1/4)) - (7*b^3*(b*
x^2 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(128*a^(15/4)*Sqrt[x]*(b + a*x^2)^(1/4)) + ((a
^2 - b)*b*(b*x^2 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(4*a^(11/4)*Sqrt[x]*(b + a*x^2)^
(1/4)) + (7*b^3*(b*x^2 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/(128*a^(15/4)*Sqrt[x]*(b +
a*x^2)^(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 279

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

Rule 298

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

Rule 321

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

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 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 {\sqrt [4]{b x^2+a x^4} \left (b+a x^4+x^8\right )}{b+a x^4} \, dx &=\frac {\sqrt [4]{b x^2+a x^4} \int \frac {\sqrt {x} \sqrt [4]{b+a x^2} \left (b+a x^4+x^8\right )}{b+a x^4} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\sqrt [4]{b x^2+a x^4} \int \left (\left (1-\frac {b}{a^2}\right ) \sqrt {x} \sqrt [4]{b+a x^2}+\frac {x^{9/2} \sqrt [4]{b+a x^2}}{a}+\frac {b^2 \sqrt {x} \sqrt [4]{b+a x^2}}{a^2 \left (b+a x^4\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {\sqrt [4]{b x^2+a x^4} \int x^{9/2} \sqrt [4]{b+a x^2} \, dx}{a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b^2 \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x} \sqrt [4]{b+a x^2}}{b+a x^4} \, dx}{a^2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \sqrt {x} \sqrt [4]{b+a x^2} \, dx}{\sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (b \sqrt [4]{b x^2+a x^4}\right ) \int \frac {x^{9/2}}{\left (b+a x^2\right )^{3/4}} \, dx}{12 a \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 b^2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{b+a x^8} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{4 \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}-\frac {\left (7 b^2 \sqrt [4]{b x^2+a x^4}\right ) \int \frac {x^{5/2}}{\left (b+a x^2\right )^{3/4}} \, dx}{96 a^2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (2 b^2 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {a x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}-a x^4\right )}-\frac {a x^2 \sqrt [4]{b+a x^4}}{2 \sqrt {-a} \sqrt {b} \left (\sqrt {-a} \sqrt {b}+a x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}+\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (\sqrt {-a} b^{3/2} \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {-a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\sqrt {-a} b^{3/2} \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{b+a x^4}}{\sqrt {-a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \int \frac {\sqrt {x}}{\left (b+a x^2\right )^{3/4}} \, dx}{128 a^3 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{2 \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}+\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{64 a^3 \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (b \left (1-\frac {b}{a^2}\right ) \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 \sqrt {a} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (\sqrt {-a} b^{3/2} \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {-a} \sqrt {b}-a x^4} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {\left (\sqrt {-a} b^{3/2} \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+\frac {a x^4}{b}}}{\sqrt {-a} \sqrt {b}+a x^4} \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {x} \sqrt [4]{1+\frac {a x^2}{b}}}\\ &=-\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}+\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {b x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {b x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{64 a^3 \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}+\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {b x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {b x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{128 a^{7/2} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {\left (7 b^3 \sqrt [4]{b x^2+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{128 a^{7/2} \sqrt {x} \sqrt [4]{b+a x^2}}\\ &=-\frac {7 b^2 x \sqrt [4]{b x^2+a x^4}}{192 a^3}+\frac {1}{2} \left (1-\frac {b}{a^2}\right ) x \sqrt [4]{b x^2+a x^4}+\frac {b x^3 \sqrt [4]{b x^2+a x^4}}{48 a^2}+\frac {x^5 \sqrt [4]{b x^2+a x^4}}{6 a}+\frac {b x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}+\frac {b x \sqrt [4]{b x^2+a x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {\sqrt {-a} x^2}{\sqrt {b}},-\frac {a x^2}{b}\right )}{3 a^2 \sqrt [4]{1+\frac {a x^2}{b}}}-\frac {\left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{b+a x^2}}-\frac {7 b^3 \sqrt [4]{b x^2+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{128 a^{15/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {\left (a^2-b\right ) b \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{4 a^{11/4} \sqrt {x} \sqrt [4]{b+a x^2}}+\frac {7 b^3 \sqrt [4]{b x^2+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )}{128 a^{15/4} \sqrt {x} \sqrt [4]{b+a x^2}}\\ \end {align*}

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Mathematica [A]  time = 1.25, size = 370, normalized size = 1.61 \begin {gather*} \frac {x \sqrt [4]{x^2 \left (a x^2+b\right )} \left (2 a^{3/4} \left (96 a^3+32 a^2 x^4+4 a b \left (x^2-24\right )-7 b^2\right )-\frac {3 b \left (a+\frac {b}{x^2}\right )^{3/4} \left (-64 a^{7/4} \sqrt {b} \left (\frac {\left (\sqrt {-a}+\sqrt {b}\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )\right )}{\left (a-\sqrt {-a} \sqrt {b}\right )^{3/4}}-\frac {\left (\sqrt {-a}-\sqrt {b}\right ) \left (\tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )\right )}{\left (\sqrt {-a} \sqrt {b}+a\right )^{3/4}}\right )-\left (32 a^3-32 a b+7 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a}}\right )-\left (32 a^3-32 a b+7 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a}}\right )\right )}{a x^2+b}\right )}{384 a^{15/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b*x^2 + a*x^4)^(1/4)*(b + a*x^4 + x^8))/(b + a*x^4),x]

[Out]

(x*(x^2*(b + a*x^2))^(1/4)*(2*a^(3/4)*(96*a^3 - 7*b^2 + 32*a^2*x^4 + 4*a*b*(-24 + x^2)) - (3*b*(a + b/x^2)^(3/
4)*(-((32*a^3 - 32*a*b + 7*b^2)*ArcTan[(a + b/x^2)^(1/4)/a^(1/4)]) - (32*a^3 - 32*a*b + 7*b^2)*ArcTanh[(a + b/
x^2)^(1/4)/a^(1/4)] - 64*a^(7/4)*Sqrt[b]*(((Sqrt[-a] + Sqrt[b])*(ArcTan[(a + b/x^2)^(1/4)/(a - Sqrt[-a]*Sqrt[b
])^(1/4)] + ArcTanh[(a + b/x^2)^(1/4)/(a - Sqrt[-a]*Sqrt[b])^(1/4)]))/(a - Sqrt[-a]*Sqrt[b])^(3/4) - ((Sqrt[-a
] - Sqrt[b])*(ArcTan[(a + b/x^2)^(1/4)/(a + Sqrt[-a]*Sqrt[b])^(1/4)] + ArcTanh[(a + b/x^2)^(1/4)/(a + Sqrt[-a]
*Sqrt[b])^(1/4)]))/(a + Sqrt[-a]*Sqrt[b])^(3/4))))/(b + a*x^2)))/(384*a^(15/4))

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IntegrateAlgebraic [A]  time = 0.00, size = 230, normalized size = 1.00 \begin {gather*} \frac {\sqrt [4]{b x^2+a x^4} \left (96 a^3 x-96 a b x-7 b^2 x+4 a b x^3+32 a^2 x^5\right )}{192 a^3}+\frac {\left (-32 a^3 b+32 a b^2-7 b^3\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{128 a^{15/4}}+\frac {\left (32 a^3 b-32 a b^2+7 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{128 a^{15/4}}-\frac {b^2 \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-a+\text {$\#$1}^4}\&\right ]}{4 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((b*x^2 + a*x^4)^(1/4)*(b + a*x^4 + x^8))/(b + a*x^4),x]

[Out]

((b*x^2 + a*x^4)^(1/4)*(96*a^3*x - 96*a*b*x - 7*b^2*x + 4*a*b*x^3 + 32*a^2*x^5))/(192*a^3) + ((-32*a^3*b + 32*
a*b^2 - 7*b^3)*ArcTan[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) + ((32*a^3*b - 32*a*b^2 + 7*b^3)*ArcT
anh[(a^(1/4)*x)/(b*x^2 + a*x^4)^(1/4)])/(128*a^(15/4)) - (b^2*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (-(Log[x
]*#1) + Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1)/(-a + #1^4) & ])/(4*a^2)

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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^2)^(1/4)*(x^8+a*x^4+b)/(a*x^4+b),x, algorithm="fricas")

[Out]

Timed out

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giac [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^2)^(1/4)*(x^8+a*x^4+b)/(a*x^4+b),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \,x^{2}\right )^{\frac {1}{4}} \left (x^{8}+a \,x^{4}+b \right )}{a \,x^{4}+b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4+b*x^2)^(1/4)*(x^8+a*x^4+b)/(a*x^4+b),x)

[Out]

int((a*x^4+b*x^2)^(1/4)*(x^8+a*x^4+b)/(a*x^4+b),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} + a x^{4} + b\right )} {\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}}}{a x^{4} + b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4+b*x^2)^(1/4)*(x^8+a*x^4+b)/(a*x^4+b),x, algorithm="maxima")

[Out]

integrate((x^8 + a*x^4 + b)*(a*x^4 + b*x^2)^(1/4)/(a*x^4 + b), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (x^8+a\,x^4+b\right )}{a\,x^4+b} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*x^4 + b*x^2)^(1/4)*(b + a*x^4 + x^8))/(b + a*x^4),x)

[Out]

int(((a*x^4 + b*x^2)^(1/4)*(b + a*x^4 + x^8))/(b + a*x^4), x)

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sympy [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**2)**(1/4)*(x**8+a*x**4+b)/(a*x**4+b),x)

[Out]

Timed out

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