\(\int \frac {\sqrt {x}}{(a+b x^2) (c+d x^2)} \, dx\) [465]

Optimal result

Integrand size = 24, antiderivative size = 463 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {\sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {\sqrt [4]{d} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt [4]{d} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{d} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{d} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)} \]

[Out]

-1/2*b^(1/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(1/4)/(-a*d+b*c)*2^(1/2)+1/2*b^(1/4)*arctan(1+b^(1/4)
*2^(1/2)*x^(1/2)/a^(1/4))/a^(1/4)/(-a*d+b*c)*2^(1/2)+1/2*d^(1/4)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(
1/4)/(-a*d+b*c)*2^(1/2)-1/2*d^(1/4)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(1/4)/(-a*d+b*c)*2^(1/2)+1/4*b
^(1/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/4)/(-a*d+b*c)*2^(1/2)-1/4*b^(1/4)*ln(a^(1/2)
+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(1/4)/(-a*d+b*c)*2^(1/2)-1/4*d^(1/4)*ln(c^(1/2)+x*d^(1/2)-c^(1/4
)*d^(1/4)*2^(1/2)*x^(1/2))/c^(1/4)/(-a*d+b*c)*2^(1/2)+1/4*d^(1/4)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)
*x^(1/2))/c^(1/4)/(-a*d+b*c)*2^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 493, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {\sqrt [4]{d} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt [4]{d} \arctan \left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{d} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{d} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)} \]

[In]

Int[Sqrt[x]/((a + b*x^2)*(c + d*x^2)),x]

[Out]

-((b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d))) + (b^(1/4)*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*(b*c - a*d)) + (d^(1/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt
[x])/c^(1/4)])/(Sqrt[2]*c^(1/4)*(b*c - a*d)) - (d^(1/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2
]*c^(1/4)*(b*c - a*d)) + (b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/
4)*(b*c - a*d)) - (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*(b*c
- a*d)) - (d^(1/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d)
) + (d^(1/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(1/4)*(b*c - a*d))

Rule 210

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

Rule 303

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

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 493

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right ) \\ & = \frac {(2 b) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b c-a d}-\frac {(2 d) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{b c-a d} \\ & = -\frac {\sqrt {b} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b c-a d}+\frac {\sqrt {b} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b c-a d}+\frac {\sqrt {d} \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{b c-a d}-\frac {\sqrt {d} \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{b c-a d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}+\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}-\frac {\text {Subst}\left (\int \frac {1}{\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt {x}\right )}{2 (b c-a d)}+\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{d} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac {\sqrt {c}}{\sqrt {d}}-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt [4]{d} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac {\sqrt {c}}{\sqrt {d}}+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)} \\ & = \frac {\sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{d} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{d} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{b} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{d} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{d} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} (b c-a d)} \\ & = -\frac {\sqrt [4]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {\sqrt [4]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} (b c-a d)}+\frac {\sqrt [4]{d} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} (b c-a d)}-\frac {\sqrt [4]{d} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} (b c-a d)}-\frac {\sqrt [4]{d} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)}+\frac {\sqrt [4]{d} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} \sqrt [4]{c} (b c-a d)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {-\sqrt [4]{b} \sqrt [4]{c} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+\sqrt [4]{a} \sqrt [4]{d} \arctan \left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )-\sqrt [4]{b} \sqrt [4]{c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )+\sqrt [4]{a} \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} (b c-a d)} \]

[In]

Integrate[Sqrt[x]/((a + b*x^2)*(c + d*x^2)),x]

[Out]

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

Maple [A] (verified)

Time = 2.75 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.49

method	result	size
derivativedivides	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(226\)
default	\(\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(226\)

[In]

int(x^(1/2)/(b*x^2+a)/(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

1/4/(a*d-b*c)/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/
2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1))-1/4/(a*d-b*c
)/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/
2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 1347, normalized size of antiderivative = 2.91 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \]

[In]

integrate(x^(1/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

1/2*(-b/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^5*d^4))^(1/4)*log((a*b^3*c^3 - 3*
a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*(-b/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3
+ a^5*d^4))^(3/4) + b*sqrt(x)) - 1/2*(-b/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^
5*d^4))^(1/4)*log(-(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*(-b/(a*b^4*c^4 - 4*a^2*b^3*c^3*d +
6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^5*d^4))^(3/4) + b*sqrt(x)) + 1/2*I*(-b/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*
a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^5*d^4))^(1/4)*log(-(I*a*b^3*c^3 - 3*I*a^2*b^2*c^2*d + 3*I*a^3*b*c*d^2 - I*
a^4*d^3)*(-b/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^5*d^4))^(3/4) + b*sqrt(x)) -
1/2*I*(-b/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3 + a^5*d^4))^(1/4)*log(-(-I*a*b^3*c
^3 + 3*I*a^2*b^2*c^2*d - 3*I*a^3*b*c*d^2 + I*a^4*d^3)*(-b/(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4
*a^4*b*c*d^3 + a^5*d^4))^(3/4) + b*sqrt(x)) - 1/2*(-d/(b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c
^2*d^3 + a^4*c*d^4))^(1/4)*log((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(-d/(b^4*c^5 - 4*a*b^3*
c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4))^(3/4) + d*sqrt(x)) + 1/2*(-d/(b^4*c^5 - 4*a*b^3*c^4*
d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4))^(1/4)*log(-(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 -
a^3*c*d^3)*(-d/(b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4))^(3/4) + d*sqrt(x))
- 1/2*I*(-d/(b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4))^(1/4)*log(-(I*b^3*c^
4 - 3*I*a*b^2*c^3*d + 3*I*a^2*b*c^2*d^2 - I*a^3*c*d^3)*(-d/(b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^
3*b*c^2*d^3 + a^4*c*d^4))^(3/4) + d*sqrt(x)) + 1/2*I*(-d/(b^4*c^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*
b*c^2*d^3 + a^4*c*d^4))^(1/4)*log(-(-I*b^3*c^4 + 3*I*a*b^2*c^3*d - 3*I*a^2*b*c^2*d^2 + I*a^3*c*d^3)*(-d/(b^4*c
^5 - 4*a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 4*a^3*b*c^2*d^3 + a^4*c*d^4))^(3/4) + d*sqrt(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (b c - a d\right )}} - \frac {d {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, {\left (b c - a d\right )}} \]

[In]

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

[Out]

1/4*b*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt
(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(
sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x +
sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3
/4)))/(b*c - a*d) - 1/4*d*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqr
t(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sq
rt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sq
rt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sq
rt(c))/(c^(1/4)*d^(3/4)))/(b*c - a*d)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} c - \sqrt {2} a^{2} b^{2} d} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a b^{3} c - \sqrt {2} a^{2} b^{2} d} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c^{2} d^{2} - \sqrt {2} a c d^{3}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {c}{d}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} b c^{2} d^{2} - \sqrt {2} a c d^{3}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a b^{3} c - \sqrt {2} a^{2} b^{2} d\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a b^{3} c - \sqrt {2} a^{2} b^{2} d\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{2} d^{2} - \sqrt {2} a c d^{3}\right )}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{2} d^{2} - \sqrt {2} a c d^{3}\right )}} \]

[In]

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

[Out]

(a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^3*c - sqrt(2)*a^2
*b^2*d) + (a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^3*c -
sqrt(2)*a^2*b^2*d) - (c*d^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*
b*c^2*d^2 - sqrt(2)*a*c*d^3) - (c*d^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4)
)/(sqrt(2)*b*c^2*d^2 - sqrt(2)*a*c*d^3) - 1/2*(a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(
sqrt(2)*a*b^3*c - sqrt(2)*a^2*b^2*d) + 1/2*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sq
rt(2)*a*b^3*c - sqrt(2)*a^2*b^2*d) + 1/2*(c*d^3)^(3/4)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(
2)*b*c^2*d^2 - sqrt(2)*a*c*d^3) - 1/2*(c*d^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)
*b*c^2*d^2 - sqrt(2)*a*c*d^3)

Mupad [B] (verification not implemented)

Time = 5.96 (sec) , antiderivative size = 6701, normalized size of antiderivative = 14.47 \[ \int \frac {\sqrt {x}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

atan((((-b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(3/4)*(x^(1/2
)*(-b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*(4096*a*b^10
*c^7*d^4 + 4096*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672
*a^5*b^6*c^3*d^8 - 16384*a^6*b^5*c^2*d^9) + 2048*a*b^9*c^6*d^4 + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4
096*a^3*b^7*c^4*d^6 + 4096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8) + x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*
c*d^6))*(-b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*1i - (
(-b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(3/4)*(2048*a*b^9*c^
6*d^4 - x^(1/2)*(-b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4
)*(4096*a*b^10*c^7*d^4 + 4096*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c
^4*d^7 + 28672*a^5*b^6*c^3*d^8 - 16384*a^6*b^5*c^2*d^9) + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3
*b^7*c^4*d^6 + 4096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8) - x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))
*(-b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*1i)/(((-b/(16
*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(3/4)*(x^(1/2)*(-b/(16*a^5*
d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*(4096*a*b^10*c^7*d^4 + 409
6*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d
^8 - 16384*a^6*b^5*c^2*d^9) + 2048*a*b^9*c^6*d^4 + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^
4*d^6 + 4096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8) + x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-b/(1
6*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4) + ((-b/(16*a^5*d^4 +
16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(3/4)*(2048*a*b^9*c^6*d^4 - x^(1/2)*(
-b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*(4096*a*b^10*c^
7*d^4 + 4096*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^
5*b^6*c^3*d^8 - 16384*a^6*b^5*c^2*d^9) + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 + 40
96*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8) - x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-b/(16*a^5*d^4
+ 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4) + 256*a*b^5*c*d^5))*(-b/(16*a^
5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*2i - 2*atan((((-b/(16*a^
5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(3/4)*(2048*a*b^9*c^6*d^4 - x^
(1/2)*(-b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*(4096*a*
b^10*c^7*d^4 + 4096*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 2
8672*a^5*b^6*c^3*d^8 - 16384*a^6*b^5*c^2*d^9)*1i + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^
4*d^6 + 4096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8)*1i + x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-b
/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4) - ((-b/(16*a^5*d^
4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(3/4)*(x^(1/2)*(-b/(16*a^5*d^4 + 1
6*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*(4096*a*b^10*c^7*d^4 + 4096*a^7*b
^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d^8 - 16
384*a^6*b^5*c^2*d^9)*1i + 2048*a*b^9*c^6*d^4 + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^
6 + 4096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8)*1i - x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-b/(16
*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4))/(((-b/(16*a^5*d^4 +
16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(3/4)*(2048*a*b^9*c^6*d^4 - x^(1/2)*(-
b/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*(4096*a*b^10*c^7
*d^4 + 4096*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5
*b^6*c^3*d^8 - 16384*a^6*b^5*c^2*d^9)*1i + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 +
4096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8)*1i + x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-b/(16*a^5
*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*1i + ((-b/(16*a^5*d^4 + 1
6*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(3/4)*(x^(1/2)*(-b/(16*a^5*d^4 + 16*a*b
^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*(4096*a*b^10*c^7*d^4 + 4096*a^7*b^4*c*
d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d^8 - 16384*a
^6*b^5*c^2*d^9)*1i + 2048*a*b^9*c^6*d^4 + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 + 4
096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8)*1i - x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-b/(16*a^5*
d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4)*1i - 256*a*b^5*c*d^5))*(-b
/(16*a^5*d^4 + 16*a*b^4*c^4 - 64*a^2*b^3*c^3*d + 96*a^3*b^2*c^2*d^2 - 64*a^4*b*c*d^3))^(1/4) + atan((((-d/(16*
b^4*c^5 + 16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(3/4)*(x^(1/2)*(-d/(16*b^4*c
^5 + 16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*(4096*a*b^10*c^7*d^4 + 4096
*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d^
8 - 16384*a^6*b^5*c^2*d^9) + 2048*a*b^9*c^6*d^4 + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4
*d^6 + 4096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8) + x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-d/(16
*b^4*c^5 + 16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*1i - ((-d/(16*b^4*c^5
+ 16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(3/4)*(2048*a*b^9*c^6*d^4 - x^(1/2)
*(-d/(16*b^4*c^5 + 16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*(4096*a*b^10*
c^7*d^4 + 4096*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*
a^5*b^6*c^3*d^8 - 16384*a^6*b^5*c^2*d^9) + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 +
4096*a^4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8) - x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-d/(16*b^4*c^
5 + 16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*1i)/(((-d/(16*b^4*c^5 + 16*a
^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(3/4)*(x^(1/2)*(-d/(16*b^4*c^5 + 16*a^4*c*
d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*(4096*a*b^10*c^7*d^4 + 4096*a^7*b^4*c*d^1
0 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d^8 - 16384*a^6*
b^5*c^2*d^9) + 2048*a*b^9*c^6*d^4 + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 + 4096*a^
4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8) + x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-d/(16*b^4*c^5 + 16*
a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4) + ((-d/(16*b^4*c^5 + 16*a^4*c*d^4 -
64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(3/4)*(2048*a*b^9*c^6*d^4 - x^(1/2)*(-d/(16*b^4*c^5
+ 16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*(4096*a*b^10*c^7*d^4 + 4096*a^
7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d^8 -
16384*a^6*b^5*c^2*d^9) + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 + 4096*a^4*b^6*c^3*
d^7 - 6144*a^5*b^5*c^2*d^8) - x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-d/(16*b^4*c^5 + 16*a^4*c*d^4
- 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4) + 256*a*b^5*c*d^5))*(-d/(16*b^4*c^5 + 16*a^4*
c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*2i - 2*atan((((-d/(16*b^4*c^5 + 16*a^4*
c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(3/4)*(2048*a*b^9*c^6*d^4 - x^(1/2)*(-d/(16*b
^4*c^5 + 16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*(4096*a*b^10*c^7*d^4 +
4096*a^7*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^
3*d^8 - 16384*a^6*b^5*c^2*d^9)*1i + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 + 4096*a^
4*b^6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8)*1i + x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-d/(16*b^4*c^5 +
16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4) - ((-d/(16*b^4*c^5 + 16*a^4*c*d^
4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(3/4)*(x^(1/2)*(-d/(16*b^4*c^5 + 16*a^4*c*d^4 - 6
4*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*(4096*a*b^10*c^7*d^4 + 4096*a^7*b^4*c*d^10 - 163
84*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d^8 - 16384*a^6*b^5*c^2
*d^9)*1i + 2048*a*b^9*c^6*d^4 + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 + 4096*a^4*b^
6*c^3*d^7 - 6144*a^5*b^5*c^2*d^8)*1i - x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-d/(16*b^4*c^5 + 16*a
^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4))/(((-d/(16*b^4*c^5 + 16*a^4*c*d^4 -
64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(3/4)*(2048*a*b^9*c^6*d^4 - x^(1/2)*(-d/(16*b^4*c^5 +
16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*(4096*a*b^10*c^7*d^4 + 4096*a^7
*b^4*c*d^10 - 16384*a^2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d^8 -
16384*a^6*b^5*c^2*d^9)*1i + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 + 4096*a^4*b^6*c^
3*d^7 - 6144*a^5*b^5*c^2*d^8)*1i + x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-d/(16*b^4*c^5 + 16*a^4*c
*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*1i + ((-d/(16*b^4*c^5 + 16*a^4*c*d^4 - 6
4*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(3/4)*(x^(1/2)*(-d/(16*b^4*c^5 + 16*a^4*c*d^4 - 64*a^3
*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*(4096*a*b^10*c^7*d^4 + 4096*a^7*b^4*c*d^10 - 16384*a^
2*b^9*c^6*d^5 + 28672*a^3*b^8*c^5*d^6 - 32768*a^4*b^7*c^4*d^7 + 28672*a^5*b^6*c^3*d^8 - 16384*a^6*b^5*c^2*d^9)
*1i + 2048*a*b^9*c^6*d^4 + 2048*a^6*b^4*c*d^9 - 6144*a^2*b^8*c^5*d^5 + 4096*a^3*b^7*c^4*d^6 + 4096*a^4*b^6*c^3
*d^7 - 6144*a^5*b^5*c^2*d^8)*1i - x^(1/2)*(256*a*b^6*c^2*d^5 + 256*a^2*b^5*c*d^6))*(-d/(16*b^4*c^5 + 16*a^4*c*
d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)*1i - 256*a*b^5*c*d^5))*(-d/(16*b^4*c^5 +
16*a^4*c*d^4 - 64*a^3*b*c^2*d^3 + 96*a^2*b^2*c^3*d^2 - 64*a*b^3*c^4*d))^(1/4)