3.30.11 \(\int \frac {(b+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx\)
Optimal. Leaf size=327 \[ -\text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4 a+2 a^2-b\& ,\frac {-2 \text {$\#$1}^4 a^3 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+2 \text {$\#$1}^4 a^3 \log (x)+3 \text {$\#$1}^4 a b \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-3 \text {$\#$1}^4 a b \log (x)+4 a^4 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-4 a^2 b \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+b^2 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-4 a^4 \log (x)+4 a^2 b \log (x)-b^2 \log (x)}{3 \text {$\#$1}^3 a-2 \text {$\#$1}^7}\& \right ]+\left (4 a^{9/4}-3 \sqrt [4]{a} b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )+\left (3 \sqrt [4]{a} b-4 a^{9/4}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )+2 a \sqrt [4]{a x^4+b x^3} \]
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Rubi [B] time = 1.62, antiderivative size = 1092, normalized size of antiderivative =
3.34, number of steps used = 25, number of rules used = 12, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\)
= 0.353, Rules used = {2056, 6728, 101, 157, 63, 331, 298, 203, 206, 93, 205, 208} \begin {gather*} \frac {\left (2 a^2-2 \sqrt {a^2+4 b} a-b\right ) \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right ) \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right ) \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 a^2-2 \sqrt {a^2+4 b} a-b\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right ) \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right ) \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\sqrt [4]{a x^4+b x^3} \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right )-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (b-2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4+b x^3} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (b-2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{a x^4+b x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{a x^4+b x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((b + 2*a*x)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]
[Out]
(a - (a^2 - b)/Sqrt[a^2 + 4*b])*(b*x^3 + a*x^4)^(1/4) + (a + (a^2 - b)/Sqrt[a^2 + 4*b])*(b*x^3 + a*x^4)^(1/4)
+ ((a - (a^2 - b)/Sqrt[a^2 + 4*b])*(2*a^2 - b - 2*a*Sqrt[a^2 + 4*b])*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(
1/4))/(b + a*x)^(1/4)])/(2*a^(3/4)*x^(3/4)*(b + a*x)^(1/4)) - ((a + (a^2 - b)/Sqrt[a^2 + 4*b])*(b - 2*a*(a + S
qrt[a^2 + 4*b]))*(b*x^3 + a*x^4)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(3/4)*x^(3/4)*(b + a*x)
^(1/4)) - ((a - (a^2 - b)/Sqrt[a^2 + 4*b])*(a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*(
b*x^3 + a*x^4)^(1/4)*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((a - Sqrt[a^2 + 4*b])^(1/4)*(b +
a*x)^(1/4))])/(x^(3/4)*(b + a*x)^(1/4)) - ((a + (a^2 - b)/Sqrt[a^2 + 4*b])*(a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 -
2*b + a*Sqrt[a^2 + 4*b])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((
a + Sqrt[a^2 + 4*b])^(1/4)*(b + a*x)^(1/4))])/(x^(3/4)*(b + a*x)^(1/4)) - ((a - (a^2 - b)/Sqrt[a^2 + 4*b])*(2*
a^2 - b - 2*a*Sqrt[a^2 + 4*b])*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(3/4)*x^
(3/4)*(b + a*x)^(1/4)) + ((a + (a^2 - b)/Sqrt[a^2 + 4*b])*(b - 2*a*(a + Sqrt[a^2 + 4*b]))*(b*x^3 + a*x^4)^(1/4
)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + ((a - (a^2 - b)/Sqrt[a^2 +
4*b])*(a - Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*(b*x^3 + a*x^4)^(1/4)*ArcTanh[((a^2 -
2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((a - Sqrt[a^2 + 4*b])^(1/4)*(b + a*x)^(1/4))])/(x^(3/4)*(b + a*x)^(1
/4)) + ((a + (a^2 - b)/Sqrt[a^2 + 4*b])*(a + Sqrt[a^2 + 4*b])^(3/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*(b*x
^3 + a*x^4)^(1/4)*ArcTanh[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((a + Sqrt[a^2 + 4*b])^(1/4)*(b + a*
x)^(1/4))])/(x^(3/4)*(b + a*x)^(1/4))
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 93
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
Rule 101
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Rule 157
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
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 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 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 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 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 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 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 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+2 a x) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \sqrt [4]{b+a x} (b+2 a x)}{-b+a x+x^2} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {\left (2 a-\frac {2 \left (a^2-b\right )}{\sqrt {a^2+4 b}}\right ) x^{3/4} \sqrt [4]{b+a x}}{a-\sqrt {a^2+4 b}+2 x}+\frac {\left (2 a+\frac {2 \left (a^2-b\right )}{\sqrt {a^2+4 b}}\right ) x^{3/4} \sqrt [4]{b+a x}}{a+\sqrt {a^2+4 b}+2 x}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\left (2 \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{b+a x}}{a-\sqrt {a^2+4 b}+2 x} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{b+a x}}{a+\sqrt {a^2+4 b}+2 x} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\frac {3}{4} b \left (a-\sqrt {a^2+4 b}\right )+\frac {1}{2} \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) x}{\sqrt [4]{x} \left (a-\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {\frac {3}{4} b \left (a+\sqrt {a^2+4 b}\right )+\frac {1}{2} \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) x}{\sqrt [4]{x} \left (a+\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{4 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right ) \left (a^2-2 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a+\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{4 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (\frac {3}{2} b \left (a-\sqrt {a^2+4 b}\right )-\frac {1}{2} \left (a-\sqrt {a^2+4 b}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (a-\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right ) \left (a^2-2 b+a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+\sqrt {a^2+4 b}-\left (-2 b+a \left (a+\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (2 \left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (\frac {3}{2} b \left (a-\sqrt {a^2+4 b}\right )-\frac {1}{2} \left (a-\sqrt {a^2+4 b}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a-\sqrt {a^2+4 b}-\left (-2 b+a \left (a-\sqrt {a^2+4 b}\right )\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right ) \sqrt {a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}-\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right ) \sqrt {a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2+4 b}}+\sqrt {a^2-2 b+a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (\frac {3}{2} b \left (a-\sqrt {a^2+4 b}\right )-\frac {1}{2} \left (a-\sqrt {a^2+4 b}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}-\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (\frac {3}{2} b \left (a-\sqrt {a^2+4 b}\right )-\frac {1}{2} \left (a-\sqrt {a^2+4 b}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2+4 b}}+\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{\sqrt {a^2-2 b-a \sqrt {a^2+4 b}} x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2-b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2-b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (-b+2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}\\ &=\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}+\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (b-2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2-b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (2 a^2-b-2 a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (b-2 a \left (a+\sqrt {a^2+4 b}\right )\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \left (a^2-b-a \sqrt {a^2+4 b}\right ) \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b-a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\frac {a^2-b}{\sqrt {a^2+4 b}}\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-2 b+a \sqrt {a^2+4 b}} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}
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Mathematica [A] time = 14.15, size = 320, normalized size = 0.98 \begin {gather*} \frac {3 \sqrt [4]{a} x^{9/4} \left (4 a^2-3 b\right ) (a x+b)^{3/4} \left (\tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )\right )+\frac {4 x^3 \left (2 a^4-a^2 b+\frac {-2 a^5-3 a^3 b+5 a b^2}{\sqrt {a^2+4 b}}+b^2\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {b \left (\frac {a}{b}+\frac {2}{\sqrt {a^2+4 b}-a}\right ) x}{b+a x}\right )}{a-\sqrt {a^2+4 b}}+\frac {4 x^3 \left (2 a^4-a^2 b+\frac {2 a^5+3 a^3 b-5 a b^2}{\sqrt {a^2+4 b}}+b^2\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {b \left (\frac {a}{b}-\frac {2}{a+\sqrt {a^2+4 b}}\right ) x}{b+a x}\right )}{\sqrt {a^2+4 b}+a}+6 a x^3 (a x+b)}{3 \left (x^3 (a x+b)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((b + 2*a*x)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]
[Out]
(6*a*x^3*(b + a*x) + 3*a^(1/4)*(4*a^2 - 3*b)*x^(9/4)*(b + a*x)^(3/4)*(ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)
] - ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]) + (4*(2*a^4 - a^2*b + b^2 + (-2*a^5 - 3*a^3*b + 5*a*b^2)/Sqrt[
a^2 + 4*b])*x^3*Hypergeometric2F1[3/4, 1, 7/4, (b*(a/b + 2/(-a + Sqrt[a^2 + 4*b]))*x)/(b + a*x)])/(a - Sqrt[a^
2 + 4*b]) + (4*(2*a^4 - a^2*b + b^2 + (2*a^5 + 3*a^3*b - 5*a*b^2)/Sqrt[a^2 + 4*b])*x^3*Hypergeometric2F1[3/4,
1, 7/4, (b*(a/b - 2/(a + Sqrt[a^2 + 4*b]))*x)/(b + a*x)])/(a + Sqrt[a^2 + 4*b]))/(3*(x^3*(b + a*x))^(3/4))
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IntegrateAlgebraic [A] time = 0.00, size = 327, normalized size = 1.00 \begin {gather*} 2 a \sqrt [4]{b x^3+a x^4}+\left (4 a^{9/4}-3 \sqrt [4]{a} b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )+\left (-4 a^{9/4}+3 \sqrt [4]{a} b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )-\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {4 a^4 \log (x)-4 a^2 b \log (x)+b^2 \log (x)-4 a^4 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+4 a^2 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-b^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-2 a^3 \log (x) \text {$\#$1}^4+3 a b \log (x) \text {$\#$1}^4+2 a^3 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-3 a b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((b + 2*a*x)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]
[Out]
2*a*(b*x^3 + a*x^4)^(1/4) + (4*a^(9/4) - 3*a^(1/4)*b)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] + (-4*a^(9/4)
+ 3*a^(1/4)*b)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)] - RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (4*a^4*Log
[x] - 4*a^2*b*Log[x] + b^2*Log[x] - 4*a^4*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] + 4*a^2*b*Log[(b*x^3 + a*x^4)^(1/4
) - x*#1] - b^2*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - 2*a^3*Log[x]*#1^4 + 3*a*b*Log[x]*#1^4 + 2*a^3*Log[(b*x^3 +
a*x^4)^(1/4) - x*#1]*#1^4 - 3*a*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4)/(-3*a*#1^3 + 2*#1^7) & ]
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fricas [B] time = 34.47, size = 8527, normalized size = 26.08
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*a*x+b)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x, algorithm="fricas")
[Out]
2*sqrt(2)*sqrt(sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b
^6 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4
224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)
/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*arctan(-1/16*sqrt(2)*(sqrt(2)*((8*a^16 +
124*a^14*b + 710*a^12*b^2 + 1717*a^10*b^3 + 1100*a^8*b^4 - 1358*a^6*b^5 - 424*a^4*b^6 + 1120*a^2*b^7 + 128*b^8
)*x*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b
^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*
b^2 + 64*b^3)) - (128*a^25 + 1600*a^23*b + 6752*a^21*b^2 + 8720*a^19*b^3 - 7928*a^17*b^4 - 14692*a^15*b^5 + 12
226*a^13*b^6 - 1093*a^11*b^7 - 9155*a^9*b^8 + 9513*a^7*b^9 - 4386*a^5*b^10 + 1344*a^3*b^11 + 96*a*b^12)*x)*sqr
t((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 + (a^4 + 8*a^2*b + 16*b
^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b
^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*
b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))*sqrt(-(sqrt(2)*((64*a^23 + 1024*a^21*b + 6192*a^19*b^2 + 16592*a^17*
b^3 + 14348*a^15*b^4 - 13512*a^13*b^5 - 18051*a^11*b^6 + 11013*a^9*b^7 + 3423*a^7*b^8 - 8428*a^5*b^9 + 2768*a^
3*b^10 + 192*a*b^11)*x^2*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a
^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6
+ 12*a^4*b + 48*a^2*b^2 + 64*b^3)) - (1024*a^32 + 13312*a^30*b + 60672*a^28*b^2 + 96256*a^26*b^3 - 43136*a^24
*b^4 - 176256*a^22*b^5 + 90912*a^20*b^6 + 115008*a^18*b^7 - 178476*a^16*b^8 + 42164*a^14*b^9 + 67697*a^12*b^10
- 69972*a^10*b^11 + 34035*a^8*b^12 - 7620*a^6*b^13 + 1017*a^4*b^14 + 230*a^2*b^15 + 8*b^16)*x^2)*sqrt((16*a^1
3 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 + (a^4 + 8*a^2*b + 16*b^2)*sqrt(
(256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488
*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*
b^3)))/(a^4 + 8*a^2*b + 16*b^2)) - 4*(9216*a^28*b^4 + 52224*a^26*b^5 + 37120*a^24*b^6 - 136704*a^22*b^7 - 1036
8*a^20*b^8 + 166016*a^18*b^9 - 125280*a^16*b^10 - 24960*a^14*b^11 + 91316*a^12*b^12 - 59988*a^10*b^13 + 18933*
a^8*b^14 - 1754*a^6*b^15 - 165*a^4*b^16 + 18*a^2*b^17 + b^18)*sqrt(a*x^4 + b*x^3))/x^2) + 2*sqrt(2)*(12288*a^3
9*b^2 + 188416*a^37*b^3 + 1058816*a^35*b^4 + 2344960*a^33*b^5 + 75008*a^31*b^6 - 6023168*a^29*b^7 - 1408000*a^
27*b^8 + 8562944*a^25*b^9 - 3888944*a^23*b^10 - 5739808*a^21*b^11 + 8664328*a^19*b^12 - 3247768*a^17*b^13 - 22
57157*a^15*b^14 + 3821356*a^13*b^15 - 2506591*a^11*b^16 + 908932*a^9*b^17 - 173193*a^7*b^18 - 4098*a^5*b^19 +
2208*a^3*b^20 + 96*a*b^21 - (768*a^30*b^2 + 14080*a^28*b^3 + 100352*a^26*b^4 + 332800*a^24*b^5 + 417312*a^22*b
^6 - 252576*a^20*b^7 - 761616*a^18*b^8 + 376832*a^16*b^9 + 674803*a^14*b^10 - 540469*a^12*b^11 - 131605*a^10*b
^12 + 277126*a^8*b^13 - 113494*a^6*b^14 - 6088*a^4*b^15 + 2272*a^2*b^16 + 128*b^17)*sqrt((256*a^24 + 1536*a^22
*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*
b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))*(a*x^4 + b*x^3
)^(1/4)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 + (a^4 + 8*a
^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 -
2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*
b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqrt(sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 11
2*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*
a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*
a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)
))/((663552*a^34*b^8 + 3207168*a^32*b^9 - 331776*a^30*b^10 - 11347968*a^28*b^11 + 7923200*a^26*b^12 + 10624256
*a^24*b^13 - 18989568*a^22*b^14 + 8054272*a^20*b^15 + 6152416*a^18*b^16 - 10022256*a^16*b^17 + 6265840*a^14*b^
18 - 2193416*a^12*b^19 + 418410*a^10*b^20 - 32293*a^8*b^21 - 1564*a^6*b^22 + 357*a^4*b^23 - 4*a^2*b^24 - b^25)
*x)) - 2*sqrt(2)*sqrt(sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 +
15*a*b^6 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*
b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11
+ b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*arctan(-1/8*(((8*a^16 + 124*a^14*b
+ 710*a^12*b^2 + 1717*a^10*b^3 + 1100*a^8*b^4 - 1358*a^6*b^5 - 424*a^4*b^6 + 1120*a^2*b^7 + 128*b^8)*x*sqrt((
256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*
a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b
^3)) + (128*a^25 + 1600*a^23*b + 6752*a^21*b^2 + 8720*a^19*b^3 - 7928*a^17*b^4 - 14692*a^15*b^5 + 12226*a^13*b
^6 - 1093*a^11*b^7 - 9155*a^9*b^8 + 9513*a^7*b^9 - 4386*a^5*b^10 + 1344*a^3*b^11 + 96*a*b^12)*x)*sqrt(sqrt(2)*
sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 - (a^4 + 8*a^2*b + 1
6*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^1
2*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a
^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^
4 - 13*a^3*b^5 + 15*a*b^6 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*
b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^1
0 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))*sqrt((sqrt(2)*((64*
a^23 + 1024*a^21*b + 6192*a^19*b^2 + 16592*a^17*b^3 + 14348*a^15*b^4 - 13512*a^13*b^5 - 18051*a^11*b^6 + 11013
*a^9*b^7 + 3423*a^7*b^8 - 8428*a^5*b^9 + 2768*a^3*b^10 + 192*a*b^11)*x^2*sqrt((256*a^24 + 1536*a^22*b + 1536*a
^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a
^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) + (1024*a^32 + 13312*a^30*
b + 60672*a^28*b^2 + 96256*a^26*b^3 - 43136*a^24*b^4 - 176256*a^22*b^5 + 90912*a^20*b^6 + 115008*a^18*b^7 - 17
8476*a^16*b^8 + 42164*a^14*b^9 + 67697*a^12*b^10 - 69972*a^10*b^11 + 34035*a^8*b^12 - 7620*a^6*b^13 + 1017*a^4
*b^14 + 230*a^2*b^15 + 8*b^16)*x^2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3
*b^5 + 15*a*b^6 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440
*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2
*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)) + 4*(9216*a^28*b^4 + 52224*a^
26*b^5 + 37120*a^24*b^6 - 136704*a^22*b^7 - 10368*a^20*b^8 + 166016*a^18*b^9 - 125280*a^16*b^10 - 24960*a^14*b
^11 + 91316*a^12*b^12 - 59988*a^10*b^13 + 18933*a^8*b^14 - 1754*a^6*b^15 - 165*a^4*b^16 + 18*a^2*b^17 + b^18)*
sqrt(a*x^4 + b*x^3))/x^2) - 2*(12288*a^39*b^2 + 188416*a^37*b^3 + 1058816*a^35*b^4 + 2344960*a^33*b^5 + 75008*
a^31*b^6 - 6023168*a^29*b^7 - 1408000*a^27*b^8 + 8562944*a^25*b^9 - 3888944*a^23*b^10 - 5739808*a^21*b^11 + 86
64328*a^19*b^12 - 3247768*a^17*b^13 - 2257157*a^15*b^14 + 3821356*a^13*b^15 - 2506591*a^11*b^16 + 908932*a^9*b
^17 - 173193*a^7*b^18 - 4098*a^5*b^19 + 2208*a^3*b^20 + 96*a*b^21 + (768*a^30*b^2 + 14080*a^28*b^3 + 100352*a^
26*b^4 + 332800*a^24*b^5 + 417312*a^22*b^6 - 252576*a^20*b^7 - 761616*a^18*b^8 + 376832*a^16*b^9 + 674803*a^14
*b^10 - 540469*a^12*b^11 - 131605*a^10*b^12 + 277126*a^8*b^13 - 113494*a^6*b^14 - 6088*a^4*b^15 + 2272*a^2*b^1
6 + 128*b^17)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2
032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b
+ 48*a^2*b^2 + 64*b^3)))*(a*x^4 + b*x^3)^(1/4)*sqrt(sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*
b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b
^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^
9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqr
t((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 - (a^4 + 8*a^2*b + 16*b
^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b
^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*
b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))/((663552*a^34*b^8 + 3207168*a^32*b^9 - 331776*a^30*b^10 - 11347968*
a^28*b^11 + 7923200*a^26*b^12 + 10624256*a^24*b^13 - 18989568*a^22*b^14 + 8054272*a^20*b^15 + 6152416*a^18*b^1
6 - 10022256*a^16*b^17 + 6265840*a^14*b^18 - 2193416*a^12*b^19 + 418410*a^10*b^20 - 32293*a^8*b^21 - 1564*a^6*
b^22 + 357*a^4*b^23 - 4*a^2*b^24 - b^25)*x)) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2
- 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b +
1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 -
926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16
*b^2)))*log((sqrt(2)*((2*a^7 + 19*a^5*b + 56*a^3*b^2 + 48*a*b^3)*x*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^
2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9
+ 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) - (32*a^16 + 208*a^14*b + 224*a^
12*b^2 - 424*a^10*b^3 - 62*a^8*b^4 + 329*a^6*b^5 - 239*a^4*b^6 + 53*a^2*b^7 + 4*b^8)*x)*sqrt(sqrt(2)*sqrt((16*
a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 + (a^4 + 8*a^2*b + 16*b^2)*sq
rt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1
488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 +
64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))) + 4*(96*a^14*b^2 + 272*a^12*b^3 - 192*a^10*b^4 - 168*a^8*b^5 + 230*a^6*b^
6 - 123*a^4*b^7 + 9*a^2*b^8 + b^9)*(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((16*a^13 + 80*a^1
1*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24
+ 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7
+ 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^
4 + 8*a^2*b + 16*b^2)))*log(-(sqrt(2)*((2*a^7 + 19*a^5*b + 56*a^3*b^2 + 48*a*b^3)*x*sqrt((256*a^24 + 1536*a^22
*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*
b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) - (32*a^16 + 20
8*a^14*b + 224*a^12*b^2 - 424*a^10*b^3 - 62*a^8*b^4 + 329*a^6*b^5 - 239*a^4*b^6 + 53*a^2*b^7 + 4*b^8)*x)*sqrt(
sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 + (a^4 + 8*a
^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 -
2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*
b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))) - 4*(96*a^14*b^2 + 272*a^12*b^3 - 192*a^10*b^4 - 168*a^8
*b^5 + 230*a^6*b^6 - 123*a^4*b^7 + 9*a^2*b^8 + b^9)*(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt(
(16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 - (a^4 + 8*a^2*b + 16*b^2
)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6
- 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^
2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*log((sqrt(2)*((2*a^7 + 19*a^5*b + 56*a^3*b^2 + 48*a*b^3)*x*sqrt((256*
a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10
*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3))
+ (32*a^16 + 208*a^14*b + 224*a^12*b^2 - 424*a^10*b^3 - 62*a^8*b^4 + 329*a^6*b^5 - 239*a^4*b^6 + 53*a^2*b^7 +
4*b^8)*x)*sqrt(sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*
b^6 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 +
4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12
)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))) + 4*(96*a^14*b^2 + 272*a^12*b^3 - 192*a^
10*b^4 - 168*a^8*b^5 + 230*a^6*b^6 - 123*a^4*b^7 + 9*a^2*b^8 + b^9)*(a*x^4 + b*x^3)^(1/4))/x) + 1/2*sqrt(2)*sq
rt(sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 13*a^3*b^5 + 15*a*b^6 - (a^4 +
8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5
- 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a
^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*log(-(sqrt(2)*((2*a^7 + 19*a^5*b + 56*a^3*b^2 + 48*a*
b^3)*x*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 - 1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^1
2*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 30*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a
^2*b^2 + 64*b^3)) + (32*a^16 + 208*a^14*b + 224*a^12*b^2 - 424*a^10*b^3 - 62*a^8*b^4 + 329*a^6*b^5 - 239*a^4*b
^6 + 53*a^2*b^7 + 4*b^8)*x)*sqrt(sqrt(2)*sqrt((16*a^13 + 80*a^11*b + 40*a^9*b^2 - 112*a^7*b^3 + 49*a^5*b^4 - 1
3*a^3*b^5 + 15*a*b^6 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((256*a^24 + 1536*a^22*b + 1536*a^20*b^2 - 3328*a^18*b^3 -
1440*a^16*b^4 + 4224*a^14*b^5 - 2032*a^12*b^6 - 1488*a^10*b^7 + 2097*a^8*b^8 - 926*a^6*b^9 + 159*a^4*b^10 + 3
0*a^2*b^11 + b^12)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))) - 4*(96*a^14*b^2 + 272*
a^12*b^3 - 192*a^10*b^4 - 168*a^8*b^5 + 230*a^6*b^6 - 123*a^4*b^7 + 9*a^2*b^8 + b^9)*(a*x^4 + b*x^3)^(1/4))/x)
+ 2*(a*x^4 + b*x^3)^(1/4)*a - 2*(256*a^9 - 768*a^7*b + 864*a^5*b^2 - 432*a^3*b^3 + 81*a*b^4)^(1/4)*arctan(((2
56*a^9 - 768*a^7*b + 864*a^5*b^2 - 432*a^3*b^3 + 81*a*b^4)^(3/4)*x*sqrt((sqrt(256*a^9 - 768*a^7*b + 864*a^5*b^
2 - 432*a^3*b^3 + 81*a*b^4)*x^2 + sqrt(a*x^4 + b*x^3)*(16*a^4 - 24*a^2*b + 9*b^2))/x^2) + (256*a^9 - 768*a^7*b
+ 864*a^5*b^2 - 432*a^3*b^3 + 81*a*b^4)^(3/4)*(a*x^4 + b*x^3)^(1/4)*(4*a^2 - 3*b))/((256*a^9 - 768*a^7*b + 86
4*a^5*b^2 - 432*a^3*b^3 + 81*a*b^4)*x)) - 1/2*(256*a^9 - 768*a^7*b + 864*a^5*b^2 - 432*a^3*b^3 + 81*a*b^4)^(1/
4)*log(-((a*x^4 + b*x^3)^(1/4)*(4*a^2 - 3*b) + (256*a^9 - 768*a^7*b + 864*a^5*b^2 - 432*a^3*b^3 + 81*a*b^4)^(1
/4)*x)/x) + 1/2*(256*a^9 - 768*a^7*b + 864*a^5*b^2 - 432*a^3*b^3 + 81*a*b^4)^(1/4)*log(-((a*x^4 + b*x^3)^(1/4)
*(4*a^2 - 3*b) - (256*a^9 - 768*a^7*b + 864*a^5*b^2 - 432*a^3*b^3 + 81*a*b^4)^(1/4)*x)/x)
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giac [B] time = 5.24, size = 243, normalized size = 0.74 \begin {gather*} -\frac {1}{2} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{2} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) + \frac {1}{4} \, \sqrt {2} {\left (4 \, \left (-a\right )^{\frac {1}{4}} a^{2} - 3 \, \left (-a\right )^{\frac {1}{4}} b\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right ) + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*a*x+b)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x, algorithm="giac")
[Out]
-1/2*sqrt(2)*(4*(-a)^(1/4)*a^2 - 3*(-a)^(1/4)*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(
-a)^(1/4)) - 1/2*sqrt(2)*(4*(-a)^(1/4)*a^2 - 3*(-a)^(1/4)*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a +
b/x)^(1/4))/(-a)^(1/4)) - 1/4*sqrt(2)*(4*(-a)^(1/4)*a^2 - 3*(-a)^(1/4)*b)*log(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/
4) + sqrt(-a) + sqrt(a + b/x)) + 1/4*sqrt(2)*(4*(-a)^(1/4)*a^2 - 3*(-a)^(1/4)*b)*log(-sqrt(2)*(-a)^(1/4)*(a +
b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x)) + 2*(a + b/x)^(1/4)*a*x
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (2 a x +b \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a x +x^{2}-b}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*a*x+b)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x)
[Out]
int((2*a*x+b)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (2 \, a x + b\right )}}{a x + x^{2} - b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*a*x+b)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x, algorithm="maxima")
[Out]
integrate((a*x^4 + b*x^3)^(1/4)*(2*a*x + b)/(a*x + x^2 - 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^3\right )}^{1/4}\,\left (b+2\,a\,x\right )}{x^2+a\,x-b} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a*x^4 + b*x^3)^(1/4)*(b + 2*a*x))/(a*x - b + x^2),x)
[Out]
int(((a*x^4 + b*x^3)^(1/4)*(b + 2*a*x))/(a*x - b + x^2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (2 a x + b\right )}{a x - b + x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*a*x+b)*(a*x**4+b*x**3)**(1/4)/(a*x+x**2-b),x)
[Out]
Integral((x**3*(a*x + b))**(1/4)*(2*a*x + b)/(a*x - b + x**2), x)
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