3.30.63 \(\int \frac {(b x+a x^2) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx\)

Optimal. Leaf size=366 \[ \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4 a+2 a^2-b\& ,\frac {-\text {$\#$1}^4 a^4 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 a^4 \log (x)+\text {$\#$1}^4 a^2 b \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-\text {$\#$1}^4 a^2 b \log (x)-\text {$\#$1}^4 b^2 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 b^2 \log (x)+2 a^5 \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-a^3 b \log \left (\sqrt [4]{a x^4+b x^3}-\text {$\#$1} x\right )-2 a^5 \log (x)+a^3 b \log (x)}{3 \text {$\#$1}^3 a-2 \text {$\#$1}^7}\& \right ]+\frac {1}{8} \sqrt [4]{a x^4+b x^3} \left (-8 a^2+4 a x+9 b\right )+\frac {\left (-32 a^4+8 a^2 b-5 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{16 a^{3/4}}+\frac {\left (32 a^4-8 a^2 b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b x^3}}\right )}{16 a^{3/4}} \]

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Rubi [B]  time = 4.19, antiderivative size = 1264, normalized size of antiderivative = 3.45, number of steps used = 32, number of rules used = 15, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {1593, 2056, 903, 80, 50, 63, 331, 298, 203, 206, 6728, 105, 93, 205, 208} \begin {gather*} -\frac {3 b \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 (8 a^2-9 b\right )}{16 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 b \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 (8 a^2-9 b\right )}{16 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {1}{8} \sqrt [4]{a x^4+b x^3} \left (8 a^2-9 b\right )-\frac {\left (a^4-b a^2-\frac {\left (a^4+b a^2-3 b^2\right ) a}{\sqrt {a^2+4 b}}+b^2\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 )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^4-b a^2+\frac {\left (a^4+b a^2-3 b^2\right ) a}{\sqrt {a^2+4 b}}+b^2\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 )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-b a^2-\frac {\left (a^4+b a^2-3 b^2\right ) a}{\sqrt {a^2+4 b}}+b^2\right ) \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^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-b a^2+\frac {\left (a^4+b a^2-3 b^2\right ) a}{\sqrt {a^2+4 b}}+b^2\right ) \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^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^4-b a^2-\frac {\left (a^4+b a^2-3 b^2\right ) a}{\sqrt {a^2+4 b}}+b^2\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 )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^4-b a^2+\frac {\left (a^4+b a^2-3 b^2\right ) a}{\sqrt {a^2+4 b}}+b^2\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 )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-b a^2-\frac {\left (a^4+b a^2-3 b^2\right ) a}{\sqrt {a^2+4 b}}+b^2\right ) \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^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-b a^2+\frac {\left (a^4+b a^2-3 b^2\right ) a}{\sqrt {a^2+4 b}}+b^2\right ) \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^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {1}{2} a x \sqrt [4]{a x^4+b x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b*x + a*x^2)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]

[Out]

-1/8*((8*a^2 - 9*b)*(b*x^3 + a*x^4)^(1/4)) + (a*x*(b*x^3 + a*x^4)^(1/4))/2 - (3*(8*a^2 - 9*b)*b*(b*x^3 + a*x^4
)^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(16*a^(3/4)*x^(3/4)*(b + a*x)^(1/4)) - ((a^4 - a^2*b + b^2
- (a*(a^4 + a^2*b - 3*b^2))/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)])/
(a^(3/4)*x^(3/4)*(b + a*x)^(1/4)) - ((a^4 - a^2*b + b^2 + (a*(a^4 + a^2*b - 3*b^2))/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)])/(a^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + ((a - Sqrt[a^2 + 4
*b])^(3/4)*(a^4 - a^2*b + b^2 - (a*(a^4 + a^2*b - 3*b^2))/Sqrt[a^2 + 4*b])*(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))])/((a^2 - 2*b - a*Sqrt
[a^2 + 4*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + ((a + Sqrt[a^2 + 4*b])^(3/4)*(a^4 - a^2*b + b^2 + (a*(a^4 + a^2*
b - 3*b^2))/Sqrt[a^2 + 4*b])*(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))])/((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4)) +
(3*(8*a^2 - 9*b)*b*(b*x^3 + a*x^4)^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(16*a^(3/4)*x^(3/4)*(b +
a*x)^(1/4)) + ((a^4 - a^2*b + b^2 - (a*(a^4 + a^2*b - 3*b^2))/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)])/(a^(3/4)*x^(3/4)*(b + a*x)^(1/4)) + ((a^4 - a^2*b + b^2 + (a*(a^4 + a^2*b -
3*b^2))/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)])/(a^(3/4)*x^(3/4)*(
b + a*x)^(1/4)) - ((a - Sqrt[a^2 + 4*b])^(3/4)*(a^4 - a^2*b + b^2 - (a*(a^4 + a^2*b - 3*b^2))/Sqrt[a^2 + 4*b])
*(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))])/((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4)) - ((a + Sqrt[a^2 + 4*b])^(3/4
)*(a^4 - a^2*b + b^2 + (a*(a^4 + a^2*b - 3*b^2))/Sqrt[a^2 + 4*b])*(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))])/((a^2 - 2*b + a*Sqrt[a^2 + 4
*b])^(3/4)*x^(3/4)*(b + a*x)^(1/4))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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 903

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Di
st[g/c^2, Int[Simp[2*c*e*f + c*d*g - b*e*g + c*e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x], x] + Dist[1/
c^2, Int[(Simp[c^2*d*f^2 - 2*a*c*e*f*g - a*c*d*g^2 + a*b*e*g^2 + (c^2*e*f^2 + 2*c^2*d*f*g - 2*b*c*e*f*g - b*c*
d*g^2 + b^2*e*g^2 - a*c*e*g^2)*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2))/(a + b*x + c*x^2), x], x] /; FreeQ[{
a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !Integer
Q[n] && GtQ[m, 0] && GtQ[n, 1]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx &=\int \frac {x (b+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^{7/4} (b+a x)^{5/4}}{-b+a x+x^2} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {\sqrt [4]{b x^3+a x^4} \int \frac {x^{3/4} \left (-a \left (a^2-2 b\right ) b+\left (a^4-a^2 b+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-b+a x+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4} \left (-a^2+2 b+a x\right )}{(b+a x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{b+a x}}\\ &=\frac {1}{2} a x \sqrt [4]{b x^3+a x^4}+\frac {\sqrt [4]{b x^3+a x^4} \int \left (\frac {\left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\sqrt {a^2+4 b}}\right ) x^{3/4}}{\left (a-\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}}+\frac {\left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\sqrt {a^2+4 b}}\right ) x^{3/4}}{\left (a+\sqrt {a^2+4 b}+2 x\right ) (b+a x)^{3/4}}\right ) \, dx}{x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a \left (-8 a^2+9 b\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4}}{(b+a x)^{3/4}} \, dx}{8 x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {1}{8} \left (8 a^2-9 b\right ) \sqrt [4]{b x^3+a x^4}+\frac {1}{2} a x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b \left (-8 a^2+9 b\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{32 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4}}{\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^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\sqrt {a^2+4 b}}\right ) \sqrt [4]{b x^3+a x^4}\right ) \int \frac {x^{3/4}}{\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 {1}{8} \left (8 a^2-9 b\right ) \sqrt [4]{b x^3+a x^4}+\frac {1}{2} a x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b \left (-8 a^2+9 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 )}{8 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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}{2 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a-\sqrt {a^2+4 b}\right ) \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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}{2 x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a+\sqrt {a^2+4 b}\right ) \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 {1}{8} \left (8 a^2-9 b\right ) \sqrt [4]{b x^3+a x^4}+\frac {1}{2} a x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b \left (-8 a^2+9 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 )}{8 x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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-\sqrt {a^2+4 b}\right ) \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 (2 \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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+\sqrt {a^2+4 b}\right ) \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 {1}{8} \left (8 a^2-9 b\right ) \sqrt [4]{b x^3+a x^4}+\frac {1}{2} a x \sqrt [4]{b x^3+a x^4}-\frac {\left (3 b \left (-8 a^2+9 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 )}{16 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (3 b \left (-8 a^2+9 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 )}{16 \sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (2 \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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-\sqrt {a^2+4 b}\right ) \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\sqrt {a^2+4 b}}\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-\sqrt {a^2+4 b}\right ) \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\sqrt {a^2+4 b}}\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 (2 \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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+\sqrt {a^2+4 b}\right ) \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\sqrt {a^2+4 b}}\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+\sqrt {a^2+4 b}\right ) \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\sqrt {a^2+4 b}}\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 {1}{8} \left (8 a^2-9 b\right ) \sqrt [4]{b x^3+a x^4}+\frac {1}{2} a x \sqrt [4]{b x^3+a x^4}-\frac {3 \left (8 a^2-9 b\right ) b \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\left (a^2-2 b-a \sqrt {a^2+4 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\left (a^2-2 b+a \sqrt {a^2+4 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 \left (8 a^2-9 b\right ) b \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\left (a^2-2 b-a \sqrt {a^2+4 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\left (a^2-2 b+a \sqrt {a^2+4 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\sqrt {a} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\sqrt {a} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (\left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\sqrt {a} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (\left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\sqrt {a} x^{3/4} \sqrt [4]{b+a x}}\\ &=-\frac {1}{8} \left (8 a^2-9 b\right ) \sqrt [4]{b x^3+a x^4}+\frac {1}{2} a x \sqrt [4]{b x^3+a x^4}-\frac {3 \left (8 a^2-9 b\right ) b \sqrt [4]{b x^3+a x^4} \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\left (a^2-2 b-a \sqrt {a^2+4 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a+\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\left (a^2-2 b+a \sqrt {a^2+4 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {3 \left (8 a^2-9 b\right ) b \sqrt [4]{b x^3+a x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{16 a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}+\frac {\left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{a^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a-\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-a^2 b+b^2-\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\left (a^2-2 b-a \sqrt {a^2+4 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}-\frac {\left (a+\sqrt {a^2+4 b}\right )^{3/4} \left (a^4-a^2 b+b^2+\frac {a \left (a^4+a^2 b-3 b^2\right )}{\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 )}{\left (a^2-2 b+a \sqrt {a^2+4 b}\right )^{3/4} x^{3/4} \sqrt [4]{b+a x}}\\ \end {align*}

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Mathematica [F]  time = 13.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((b*x + a*x^2)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]

[Out]

Integrate[((b*x + a*x^2)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2), x]

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IntegrateAlgebraic [A]  time = 0.00, size = 366, normalized size = 1.00 \begin {gather*} \frac {1}{8} \left (-8 a^2+9 b+4 a x\right ) \sqrt [4]{b x^3+a x^4}+\frac {\left (-32 a^4+8 a^2 b-5 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{16 a^{3/4}}+\frac {\left (32 a^4-8 a^2 b+5 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{16 a^{3/4}}+\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a^5 \log (x)-a^3 b \log (x)-2 a^5 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a^3 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-a^4 \log (x) \text {$\#$1}^4+a^2 b \log (x) \text {$\#$1}^4-b^2 \log (x) \text {$\#$1}^4+a^4 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-a^2 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+b^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 a \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((b*x + a*x^2)*(b*x^3 + a*x^4)^(1/4))/(-b + a*x + x^2),x]

[Out]

((-8*a^2 + 9*b + 4*a*x)*(b*x^3 + a*x^4)^(1/4))/8 + ((-32*a^4 + 8*a^2*b - 5*b^2)*ArcTan[(a^(1/4)*x)/(b*x^3 + a*
x^4)^(1/4)])/(16*a^(3/4)) + ((32*a^4 - 8*a^2*b + 5*b^2)*ArcTanh[(a^(1/4)*x)/(b*x^3 + a*x^4)^(1/4)])/(16*a^(3/4
)) + RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (2*a^5*Log[x] - a^3*b*Log[x] - 2*a^5*Log[(b*x^3 + a*x^4)^(1/4) -
x*#1] + a^3*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1] - a^4*Log[x]*#1^4 + a^2*b*Log[x]*#1^4 - b^2*Log[x]*#1^4 + a^4*
Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 - a^2*b*Log[(b*x^3 + a*x^4)^(1/4) - x*#1]*#1^4 + b^2*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 = 88.67, size = 10609, normalized size = 28.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+b*x)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x, algorithm="fricas")

[Out]

-2*sqrt(2)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^
6 - 4*a^3*b^7 + 3*a*b^8 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^
24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^
11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(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)*((a^19 + 17*a^17*b + 109*a^15*b^2 + 304*a^13*b^3 + 230*
a^11*b^4 - 437*a^9*b^5 - 447*a^7*b^6 + 492*a^5*b^7 + 48*a^3*b^8 - 192*a*b^9)*x*sqrt((a^32 + 10*a^30*b + 27*a^2
8*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 -
114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*
b + 48*a^2*b^2 + 64*b^3)) - (a^32 + 16*a^30*b + 93*a^28*b^2 + 202*a^26*b^3 - 86*a^24*b^4 - 822*a^22*b^5 - 91*a
^20*b^6 + 1706*a^18*b^7 - 342*a^16*b^8 - 1880*a^14*b^9 + 1534*a^12*b^10 + 180*a^10*b^11 - 1052*a^8*b^12 + 871*
a^6*b^13 - 393*a^4*b^14 + 104*a^2*b^15 - 16*b^16)*x)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9
*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^
28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9
- 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(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)*((a^29 + 19*a^27*b + 141*a^25*b^2 + 486*a
^23*b^3 + 591*a^21*b^4 - 732*a^19*b^5 - 2032*a^17*b^6 + 668*a^15*b^7 + 2667*a^13*b^8 - 1414*a^11*b^9 - 1126*a^
9*b^10 + 1272*a^7*b^11 - 544*a^5*b^12 + 128*a^3*b^13)*x^2*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 -
129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*
a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b
^3)) - (a^42 + 18*a^40*b + 123*a^38*b^2 + 354*a^36*b^3 + 105*a^34*b^4 - 1518*a^32*b^5 - 1668*a^30*b^6 + 3732*a
^28*b^7 + 4137*a^26*b^8 - 7674*a^24*b^9 - 3702*a^22*b^10 + 11208*a^20*b^11 - 2600*a^18*b^12 - 7584*a^16*b^13 +
7104*a^14*b^14 - 1064*a^12*b^15 - 2523*a^10*b^16 + 2496*a^8*b^17 - 1256*a^6*b^18 + 396*a^4*b^19 - 78*a^2*b^20
+ 8*b^21)*x^2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^
3*b^7 + 3*a*b^8 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 +
42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170
*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*
a^2*b + 16*b^2)) - 4*(4*a^36*b^6 + 36*a^34*b^7 + 69*a^32*b^8 - 170*a^30*b^9 - 417*a^28*b^10 + 666*a^26*b^11 +
847*a^24*b^12 - 1950*a^22*b^13 + 186*a^20*b^14 + 2248*a^18*b^15 - 2205*a^16*b^16 + 306*a^14*b^17 + 1094*a^12*b
^18 - 1212*a^10*b^19 + 714*a^8*b^20 - 276*a^6*b^21 + 72*a^4*b^22 - 12*a^2*b^23 + b^24)*sqrt(a*x^4 + b*x^3))/x^
2) + 2*sqrt(2)*(2*a^50*b^3 + 41*a^48*b^4 + 327*a^46*b^5 + 1164*a^44*b^6 + 927*a^42*b^7 - 5322*a^40*b^8 - 10738
*a^38*b^9 + 13334*a^36*b^10 + 37239*a^34*b^11 - 34816*a^32*b^12 - 71140*a^30*b^13 + 91296*a^28*b^14 + 55468*a^
26*b^15 - 150974*a^24*b^16 + 54297*a^22*b^17 + 91813*a^20*b^18 - 121826*a^18*b^19 + 49215*a^16*b^20 + 24381*a^
14*b^21 - 47547*a^12*b^22 + 35136*a^10*b^23 - 16712*a^8*b^24 + 5581*a^6*b^25 - 1305*a^4*b^26 + 200*a^2*b^27 -
16*b^28 - (2*a^37*b^3 + 43*a^35*b^4 + 368*a^33*b^5 + 1509*a^31*b^6 + 2400*a^29*b^7 - 2454*a^27*b^8 - 11492*a^2
5*b^9 + 41*a^23*b^10 + 25648*a^21*b^11 - 2762*a^19*b^12 - 35488*a^17*b^13 + 19822*a^15*b^14 + 18568*a^13*b^15
- 23842*a^11*b^16 + 6781*a^9*b^17 + 3225*a^7*b^18 - 3252*a^5*b^19 + 1200*a^3*b^20 - 192*a*b^21)*sqrt((a^32 + 1
0*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8
+ 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^1
6)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))*(a*x^4 + b*x^3)^(1/4)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11
*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a
^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 +
378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/
(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((a^17 + 7*a^15*b + 9*a^1
3*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 + (a^4 + 8*a^2*b + 16*b^2)*sq
rt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 -
237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^
2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))/((32*a^42*b^12 + 240*a^40*b
^13 + 144*a^38*b^14 - 1976*a^36*b^15 - 918*a^34*b^16 + 9243*a^32*b^17 - 3548*a^30*b^18 - 21351*a^28*b^19 + 293
04*a^26*b^20 + 3205*a^24*b^21 - 41550*a^22*b^22 + 42210*a^20*b^23 - 10398*a^18*b^24 - 18783*a^16*b^25 + 26514*
a^14*b^26 - 19142*a^12*b^27 + 9384*a^10*b^28 - 3330*a^8*b^29 + 860*a^6*b^30 - 156*a^4*b^31 + 18*a^2*b^32 - b^3
3)*x)) + 2*sqrt(2)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 1
0*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3
- 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132
*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(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*(((a^19 + 17*a^17*b + 109*a^15*b^2 + 304*a^13*b^3 + 230*a^11*b^4
- 437*a^9*b^5 - 447*a^7*b^6 + 492*a^5*b^7 + 48*a^3*b^8 - 192*a*b^9)*x*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 1
8*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12
*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^
2*b^2 + 64*b^3)) + (a^32 + 16*a^30*b + 93*a^28*b^2 + 202*a^26*b^3 - 86*a^24*b^4 - 822*a^22*b^5 - 91*a^20*b^6 +
1706*a^18*b^7 - 342*a^16*b^8 - 1880*a^14*b^9 + 1534*a^12*b^10 + 180*a^10*b^11 - 1052*a^8*b^12 + 871*a^6*b^13
- 393*a^4*b^14 + 104*a^2*b^15 - 16*b^16)*x)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15
*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 2
7*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*
b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12
*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 -
15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b +
27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^1
4*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(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)*((a^29 + 19*a^27*b + 141*a^25*b^2 +
486*a^23*b^3 + 591*a^21*b^4 - 732*a^19*b^5 - 2032*a^17*b^6 + 668*a^15*b^7 + 2667*a^13*b^8 - 1414*a^11*b^9 - 11
26*a^9*b^10 + 1272*a^7*b^11 - 544*a^5*b^12 + 128*a^3*b^13)*x^2*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*
b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 -
132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 +
64*b^3)) + (a^42 + 18*a^40*b + 123*a^38*b^2 + 354*a^36*b^3 + 105*a^34*b^4 - 1518*a^32*b^5 - 1668*a^30*b^6 + 3
732*a^28*b^7 + 4137*a^26*b^8 - 7674*a^24*b^9 - 3702*a^22*b^10 + 11208*a^20*b^11 - 2600*a^18*b^12 - 7584*a^16*b
^13 + 7104*a^14*b^14 - 1064*a^12*b^15 - 2523*a^10*b^16 + 2496*a^8*b^17 - 1256*a^6*b^18 + 396*a^4*b^19 - 78*a^2
*b^20 + 8*b^21)*x^2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 -
4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*
b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11
+ 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4
+ 8*a^2*b + 16*b^2)) + 4*(4*a^36*b^6 + 36*a^34*b^7 + 69*a^32*b^8 - 170*a^30*b^9 - 417*a^28*b^10 + 666*a^26*b^
11 + 847*a^24*b^12 - 1950*a^22*b^13 + 186*a^20*b^14 + 2248*a^18*b^15 - 2205*a^16*b^16 + 306*a^14*b^17 + 1094*a
^12*b^18 - 1212*a^10*b^19 + 714*a^8*b^20 - 276*a^6*b^21 + 72*a^4*b^22 - 12*a^2*b^23 + b^24)*sqrt(a*x^4 + b*x^3
))/x^2) - 2*(2*a^50*b^3 + 41*a^48*b^4 + 327*a^46*b^5 + 1164*a^44*b^6 + 927*a^42*b^7 - 5322*a^40*b^8 - 10738*a^
38*b^9 + 13334*a^36*b^10 + 37239*a^34*b^11 - 34816*a^32*b^12 - 71140*a^30*b^13 + 91296*a^28*b^14 + 55468*a^26*
b^15 - 150974*a^24*b^16 + 54297*a^22*b^17 + 91813*a^20*b^18 - 121826*a^18*b^19 + 49215*a^16*b^20 + 24381*a^14*
b^21 - 47547*a^12*b^22 + 35136*a^10*b^23 - 16712*a^8*b^24 + 5581*a^6*b^25 - 1305*a^4*b^26 + 200*a^2*b^27 - 16*
b^28 + (2*a^37*b^3 + 43*a^35*b^4 + 368*a^33*b^5 + 1509*a^31*b^6 + 2400*a^29*b^7 - 2454*a^27*b^8 - 11492*a^25*b
^9 + 41*a^23*b^10 + 25648*a^21*b^11 - 2762*a^19*b^12 - 35488*a^17*b^13 + 19822*a^15*b^14 + 18568*a^13*b^15 - 2
3842*a^11*b^16 + 6781*a^9*b^17 + 3225*a^7*b^18 - 3252*a^5*b^19 + 1200*a^3*b^20 - 192*a*b^21)*sqrt((a^32 + 10*a
^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 +
378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/
(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((a^17 + 7*a^15*b + 9*a^13*b^2
- 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a
^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a
^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^1
5 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*sqrt((a^17 + 7*a^15*b + 9*a^13*b
^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt(
(a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237
*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b
^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))/((32*a^42*b^12 + 240*a^40*b^13
+ 144*a^38*b^14 - 1976*a^36*b^15 - 918*a^34*b^16 + 9243*a^32*b^17 - 3548*a^30*b^18 - 21351*a^28*b^19 + 29304*
a^26*b^20 + 3205*a^24*b^21 - 41550*a^22*b^22 + 42210*a^20*b^23 - 10398*a^18*b^24 - 18783*a^16*b^25 + 26514*a^1
4*b^26 - 19142*a^12*b^27 + 9384*a^10*b^28 - 3330*a^8*b^29 + 860*a^6*b^30 - 156*a^4*b^31 + 18*a^2*b^32 - b^33)*
x)) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10
*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 -
129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*
a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(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)*((a^8 + 10*a^6*b + 30*a^4*b^2 + 16*a^2*b^3 - 32*b^4)*x*sqrt((a^3
2 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^1
6*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15
+ b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) - (a^21 + 9*a^19*b + 21*a^17*b^2 - 10*a^15*b^3 - 51*a^13*b^4 +
30*a^11*b^5 + 30*a^9*b^6 - 36*a^7*b^7 + 15*a^5*b^8 - 4*a^3*b^9)*x)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^1
3*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 + (a^4 + 8*a^2*b + 16*b^2)*sq
rt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 -
237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^
2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))) + 4*(2*a^18*b^3 + 9*a^16*b^
4 - 3*a^14*b^5 - 29*a^12*b^6 + 24*a^10*b^7 + 15*a^8*b^8 - 30*a^6*b^9 + 18*a^4*b^10 - 6*a^2*b^11 + b^12)*(a*x^4
+ b*x^3)^(1/4))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 +
30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 + (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2
- 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*
a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 4
8*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2)))*log(-(sqrt(2)*((a^8 + 10*a^6*b + 30*a^4*b^2 + 16*a^2*b^3 - 32
*b^4)*x*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a
^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b
^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) - (a^21 + 9*a^19*b + 21*a^17*b^2 - 10*a^15*b^
3 - 51*a^13*b^4 + 30*a^11*b^5 + 30*a^9*b^6 - 36*a^7*b^7 + 15*a^5*b^8 - 4*a^3*b^9)*x)*sqrt(sqrt(2)*sqrt((a^17 +
7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 + (a^4 + 8*a
^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6
- 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 3
6*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b^2))) - 4*(2*a^1
8*b^3 + 9*a^16*b^4 - 3*a^14*b^5 - 29*a^12*b^6 + 24*a^10*b^7 + 15*a^8*b^8 - 30*a^6*b^9 + 18*a^4*b^10 - 6*a^2*b^
11 + b^12)*(a*x^4 + b*x^3)^(1/4))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b
^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^3
0*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 37
8*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(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)*((a^8 + 10*a^6*b + 30*a^4*b^2 +
16*a^2*b^3 - 32*b^4)*x*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*
a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6
*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) + (a^21 + 9*a^19*b + 21*a^17*
b^2 - 10*a^15*b^3 - 51*a^13*b^4 + 30*a^11*b^5 + 30*a^9*b^6 - 36*a^7*b^7 + 15*a^5*b^8 - 4*a^3*b^9)*x)*sqrt(sqrt
(2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*
b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5
+ 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 -
100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4 + 8*a^2*b + 16*b
^2))) + 4*(2*a^18*b^3 + 9*a^16*b^4 - 3*a^14*b^5 - 29*a^12*b^6 + 24*a^10*b^7 + 15*a^8*b^8 - 30*a^6*b^9 + 18*a^4
*b^10 - 6*a^2*b^11 + b^12)*(a*x^4 + b*x^3)^(1/4))/x) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^13
*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 - 4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqr
t((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 2
37*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2
*b^15 + b^16)/(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)*((a^8 + 10*a^6
*b + 30*a^4*b^2 + 16*a^2*b^3 - 32*b^4)*x*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*b^4 + 4
2*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11 + 170*a
^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)) + (a^21 + 9*
a^19*b + 21*a^17*b^2 - 10*a^15*b^3 - 51*a^13*b^4 + 30*a^11*b^5 + 30*a^9*b^6 - 36*a^7*b^7 + 15*a^5*b^8 - 4*a^3*
b^9)*x)*sqrt(sqrt(2)*sqrt((a^17 + 7*a^15*b + 9*a^13*b^2 - 18*a^11*b^3 - 15*a^9*b^4 + 30*a^7*b^5 - 10*a^5*b^6 -
4*a^3*b^7 + 3*a*b^8 - (a^4 + 8*a^2*b + 16*b^2)*sqrt((a^32 + 10*a^30*b + 27*a^28*b^2 - 18*a^26*b^3 - 129*a^24*
b^4 + 42*a^22*b^5 + 286*a^20*b^6 - 212*a^18*b^7 - 237*a^16*b^8 + 378*a^14*b^9 - 114*a^12*b^10 - 132*a^10*b^11
+ 170*a^8*b^12 - 100*a^6*b^13 + 36*a^4*b^14 - 8*a^2*b^15 + b^16)/(a^6 + 12*a^4*b + 48*a^2*b^2 + 64*b^3)))/(a^4
+ 8*a^2*b + 16*b^2))) - 4*(2*a^18*b^3 + 9*a^16*b^4 - 3*a^14*b^5 - 29*a^12*b^6 + 24*a^10*b^7 + 15*a^8*b^8 - 30
*a^6*b^9 + 18*a^4*b^10 - 6*a^2*b^11 + b^12)*(a*x^4 + b*x^3)^(1/4))/x) - 1/8*(a*x^4 + b*x^3)^(1/4)*(8*a^2 - 4*a
*x - 9*b) - 1/8*((1048576*a^16 - 1048576*a^14*b + 1048576*a^12*b^2 - 557056*a^10*b^3 + 280576*a^8*b^4 - 87040*
a^6*b^5 + 25600*a^4*b^6 - 4000*a^2*b^7 + 625*b^8)/a^3)^(1/4)*arctan((a^2*x*sqrt((a^2*x^2*sqrt((1048576*a^16 -
1048576*a^14*b + 1048576*a^12*b^2 - 557056*a^10*b^3 + 280576*a^8*b^4 - 87040*a^6*b^5 + 25600*a^4*b^6 - 4000*a^
2*b^7 + 625*b^8)/a^3) + (1024*a^8 - 512*a^6*b + 384*a^4*b^2 - 80*a^2*b^3 + 25*b^4)*sqrt(a*x^4 + b*x^3))/x^2)*(
(1048576*a^16 - 1048576*a^14*b + 1048576*a^12*b^2 - 557056*a^10*b^3 + 280576*a^8*b^4 - 87040*a^6*b^5 + 25600*a
^4*b^6 - 4000*a^2*b^7 + 625*b^8)/a^3)^(3/4) - (32*a^6 - 8*a^4*b + 5*a^2*b^2)*(a*x^4 + b*x^3)^(1/4)*((1048576*a
^16 - 1048576*a^14*b + 1048576*a^12*b^2 - 557056*a^10*b^3 + 280576*a^8*b^4 - 87040*a^6*b^5 + 25600*a^4*b^6 - 4
000*a^2*b^7 + 625*b^8)/a^3)^(3/4))/((1048576*a^16 - 1048576*a^14*b + 1048576*a^12*b^2 - 557056*a^10*b^3 + 2805
76*a^8*b^4 - 87040*a^6*b^5 + 25600*a^4*b^6 - 4000*a^2*b^7 + 625*b^8)*x)) + 1/32*((1048576*a^16 - 1048576*a^14*
b + 1048576*a^12*b^2 - 557056*a^10*b^3 + 280576*a^8*b^4 - 87040*a^6*b^5 + 25600*a^4*b^6 - 4000*a^2*b^7 + 625*b
^8)/a^3)^(1/4)*log((a*x*((1048576*a^16 - 1048576*a^14*b + 1048576*a^12*b^2 - 557056*a^10*b^3 + 280576*a^8*b^4
- 87040*a^6*b^5 + 25600*a^4*b^6 - 4000*a^2*b^7 + 625*b^8)/a^3)^(1/4) + (a*x^4 + b*x^3)^(1/4)*(32*a^4 - 8*a^2*b
+ 5*b^2))/x) - 1/32*((1048576*a^16 - 1048576*a^14*b + 1048576*a^12*b^2 - 557056*a^10*b^3 + 280576*a^8*b^4 - 8
7040*a^6*b^5 + 25600*a^4*b^6 - 4000*a^2*b^7 + 625*b^8)/a^3)^(1/4)*log(-(a*x*((1048576*a^16 - 1048576*a^14*b +
1048576*a^12*b^2 - 557056*a^10*b^3 + 280576*a^8*b^4 - 87040*a^6*b^5 + 25600*a^4*b^6 - 4000*a^2*b^7 + 625*b^8)/
a^3)^(1/4) - (a*x^4 + b*x^3)^(1/4)*(32*a^4 - 8*a^2*b + 5*b^2))/x)

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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)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 5.82Unable to convert to real 1/4 Error: Bad Argument Value

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{2}+b x \right ) \left (a \,x^{4}+b \,x^{3}\right )^{\frac {1}{4}}}{a x +x^{2}-b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2+b*x)*(a*x^4+b*x^3)^(1/4)/(a*x+x^2-b),x)

[Out]

int((a*x^2+b*x)*(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 (a x^{2} + b x\right )}}{a x + x^{2} - b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^2+b*x)*(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)*(a*x^2 + b*x)/(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^2+b\,x\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^2+a\,x-b} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + a*x^2)*(a*x^4 + b*x^3)^(1/4))/(a*x - b + x^2),x)

[Out]

int(((b*x + a*x^2)*(a*x^4 + b*x^3)^(1/4))/(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 {x \sqrt [4]{x^{3} \left (a x + b\right )} \left (a x + b\right )}{a x - b + x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**2+b*x)*(a*x**4+b*x**3)**(1/4)/(a*x+x**2-b),x)

[Out]

Integral(x*(x**3*(a*x + b))**(1/4)*(a*x + b)/(a*x - b + x**2), x)

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