3.5.99 \(\int \frac {\sqrt {x}}{(a+b x^2)^2 (c+d x^2)^3} \, dx\) [499]

Optimal. Leaf size=739 \[ \frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {b^{9/4} (b c-13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {b^{9/4} (b c-13 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}-\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4} \]

[Out]

1/4*d*(a*d+2*b*c)*x^(3/2)/a/c/(-a*d+b*c)^2/(d*x^2+c)^2+1/2*b*x^(3/2)/a/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^2+1/16*d
*(-5*a^2*d^2+21*a*b*c*d+8*b^2*c^2)*x^(3/2)/a/c^2/(-a*d+b*c)^3/(d*x^2+c)-1/8*b^(9/4)*(-13*a*d+b*c)*arctan(1-b^(
1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^4*2^(1/2)+1/8*b^(9/4)*(-13*a*d+b*c)*arctan(1+b^(1/4)*2^(1/2)*
x^(1/2)/a^(1/4))/a^(5/4)/(-a*d+b*c)^4*2^(1/2)-1/64*d^(5/4)*(5*a^2*d^2-26*a*b*c*d+117*b^2*c^2)*arctan(1-d^(1/4)
*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^4*2^(1/2)+1/64*d^(5/4)*(5*a^2*d^2-26*a*b*c*d+117*b^2*c^2)*arctan(
1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)^4*2^(1/2)+1/16*b^(9/4)*(-13*a*d+b*c)*ln(a^(1/2)+x*b^(1/2
)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(5/4)/(-a*d+b*c)^4*2^(1/2)-1/16*b^(9/4)*(-13*a*d+b*c)*ln(a^(1/2)+x*b^(1/2
)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(5/4)/(-a*d+b*c)^4*2^(1/2)+1/128*d^(5/4)*(5*a^2*d^2-26*a*b*c*d+117*b^2*c^
2)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(9/4)/(-a*d+b*c)^4*2^(1/2)-1/128*d^(5/4)*(5*a^2*d^2
-26*a*b*c*d+117*b^2*c^2)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(9/4)/(-a*d+b*c)^4*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.76, antiderivative size = 739, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 483, 593, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {b^{9/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-13 a d)}{4 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {b^{9/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-13 a d)}{4 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {b^{9/4} (b c-13 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {d x^{3/2} \left (-5 a^2 d^2+21 a b c d+8 b^2 c^2\right )}{16 a c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac {d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}-\frac {d^{5/4} \left (5 a^2 d^2-26 a b c d+117 b^2 c^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {b x^{3/2}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac {d x^{3/2} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(d*(2*b*c + a*d)*x^(3/2))/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*x^(3/2))/(2*a*(b*c - a*d)*(a + b*x^2)*(c +
d*x^2)^2) + (d*(8*b^2*c^2 + 21*a*b*c*d - 5*a^2*d^2)*x^(3/2))/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (b^(9/4)*(
b*c - 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (b^(9/4)*(b*c
- 13*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) - (d^(5/4)*(117*b^
2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(9/4)*(b*c - a*d)
^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[
2]*c^(9/4)*(b*c - a*d)^4) + (b^(9/4)*(b*c - 13*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]
)/(8*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) - (b^(9/4)*(b*c - 13*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*(b*c - a*d)^4) + (d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] -
Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(9/4)*(b*c - a*d)^4) - (d^(5/4)*(117*b^2*c^2 - 26*
a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(9/4)*(b*c - a*
d)^4)

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 483

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

Rule 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && 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*} \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {x^2}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (-b c+4 a d-9 b d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)}\\ &=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (-4 \left (2 b^2 c^2-16 a b c d+5 a^2 d^2\right )-20 b d (2 b c+a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{16 a c (b c-a d)^2}\\ &=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (-4 \left (8 b^3 c^3-96 a b^2 c^2 d+21 a^2 b c d^2-5 a^3 d^3\right )-4 b d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{64 a c^2 (b c-a d)^3}\\ &=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \left (-\frac {32 b^3 c^2 (b c-13 a d) x^2}{(b c-a d) \left (a+b x^4\right )}+\frac {4 a d^2 \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{64 a c^2 (b c-a d)^3}\\ &=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^3 (b c-13 a d)\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)^4}+\frac {\left (d^2 \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^4}\\ &=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\left (b^{5/2} (b c-13 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a (b c-a d)^4}+\frac {\left (b^{5/2} (b c-13 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{4 a (b c-a d)^4}-\frac {\left (d^{3/2} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^2 (b c-a d)^4}+\frac {\left (d^{3/2} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{32 c^2 (b c-a d)^4}\\ &=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (b^2 (b c-13 a d)\right ) \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 )}{8 a (b c-a d)^4}+\frac {\left (b^2 (b c-13 a d)\right ) \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 )}{8 a (b c-a d)^4}+\frac {\left (b^{9/4} (b c-13 a d)\right ) \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 )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {\left (b^{9/4} (b c-13 a d)\right ) \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 )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {\left (d \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \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 )}{64 c^2 (b c-a d)^4}+\frac {\left (d \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \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 )}{64 c^2 (b c-a d)^4}+\frac {\left (d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \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 )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {\left (d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \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 )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}\\ &=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}-\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {\left (b^{9/4} (b c-13 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {\left (b^{9/4} (b c-13 a d)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {\left (d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}-\frac {\left (d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}\\ &=\frac {d (2 b c+a d) x^{3/2}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b x^{3/2}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+21 a b c d-5 a^2 d^2\right ) x^{3/2}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {b^{9/4} (b c-13 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {b^{9/4} (b c-13 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{9/4} (b c-a d)^4}+\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}-\frac {b^{9/4} (b c-13 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{5/4} (b c-a d)^4}+\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}-\frac {d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{9/4} (b c-a d)^4}\\ \end {align*}

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Mathematica [A]
time = 2.05, size = 451, normalized size = 0.61 \begin {gather*} \frac {1}{64} \left (-\frac {4 x^{3/2} \left (8 b^3 c^2 \left (c+d x^2\right )^2-a^3 d^3 \left (9 c+5 d x^2\right )+a b^2 c d^2 x^2 \left (25 c+21 d x^2\right )+a^2 b d^2 \left (25 c^2+12 c d x^2-5 d^2 x^4\right )\right )}{a c^2 (-b c+a d)^3 \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {8 \sqrt {2} b^{9/4} (-b c+13 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{5/4} (b c-a d)^4}-\frac {\sqrt {2} d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{9/4} (b c-a d)^4}+\frac {8 \sqrt {2} b^{9/4} (-b c+13 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{5/4} (b c-a d)^4}-\frac {\sqrt {2} d^{5/4} \left (117 b^2 c^2-26 a b c d+5 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{9/4} (b c-a d)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/((a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

((-4*x^(3/2)*(8*b^3*c^2*(c + d*x^2)^2 - a^3*d^3*(9*c + 5*d*x^2) + a*b^2*c*d^2*x^2*(25*c + 21*d*x^2) + a^2*b*d^
2*(25*c^2 + 12*c*d*x^2 - 5*d^2*x^4)))/(a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + (8*Sqrt[2]*b^(9/4)*
(-(b*c) + 13*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(5/4)*(b*c - a*d)^4) - (
Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*S
qrt[x])])/(c^(9/4)*(b*c - a*d)^4) + (8*Sqrt[2]*b^(9/4)*(-(b*c) + 13*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt
[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(5/4)*(b*c - a*d)^4) - (Sqrt[2]*d^(5/4)*(117*b^2*c^2 - 26*a*b*c*d + 5*a^2*d^2)
*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(9/4)*(b*c - a*d)^4))/64

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Maple [A]
time = 0.18, size = 381, normalized size = 0.52

method	result	size
derivativedivides	\(-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (13 a d -b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{2} \left (\frac {\frac {d \left (5 a^{2} d^{2}-26 a b c d +21 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c^{2}}+\frac {\left (9 a^{2} d^{2}-34 a b c d +25 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-26 a b c d +117 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}\)	\(381\)
default	\(-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) x^{\frac {3}{2}}}{4 a \left (b \,x^{2}+a \right )}+\frac {\left (13 a d -b c \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{2} \left (\frac {\frac {d \left (5 a^{2} d^{2}-26 a b c d +21 b^{2} c^{2}\right ) x^{\frac {7}{2}}}{32 c^{2}}+\frac {\left (9 a^{2} d^{2}-34 a b c d +25 b^{2} c^{2}\right ) x^{\frac {3}{2}}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (5 a^{2} d^{2}-26 a b c d +117 b^{2} c^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{2} d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{\left (a d -b c \right )^{4}}\)	\(381\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(b*x^2+a)^2/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

-2*b^3/(a*d-b*c)^4*(1/4*(a*d-b*c)/a*x^(3/2)/(b*x^2+a)+1/32*(13*a*d-b*c)/a/b/(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)))+2*d^2/(a*d-b*c)^4*((1/32*d*(5*a^2*d^2-26*a*b*c*d+21*b^2*c^2
)/c^2*x^(7/2)+1/32*(9*a^2*d^2-34*a*b*c*d+25*b^2*c^2)/c*x^(3/2))/(d*x^2+c)^2+1/256*(5*a^2*d^2-26*a*b*c*d+117*b^
2*c^2)/c^2/d/(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)))

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Maxima [A]
time = 0.54, size = 845, normalized size = 1.14 \begin {gather*} \frac {{\left (b^{4} c - 13 \, a b^{3} d\right )} {\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 )}}{16 \, {\left (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}\right )}} + \frac {{\left (117 \, b^{2} c^{2} d^{2} - 26 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} {\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 )}}{128 \, {\left (b^{4} c^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + a^{4} c^{2} d^{4}\right )}} + \frac {{\left (8 \, b^{3} c^{2} d^{2} + 21 \, a b^{2} c d^{3} - 5 \, a^{2} b d^{4}\right )} x^{\frac {11}{2}} + {\left (16 \, b^{3} c^{3} d + 25 \, a b^{2} c^{2} d^{2} + 12 \, a^{2} b c d^{3} - 5 \, a^{3} d^{4}\right )} x^{\frac {7}{2}} + {\left (8 \, b^{3} c^{4} + 25 \, a^{2} b c^{2} d^{2} - 9 \, a^{3} c d^{3}\right )} x^{\frac {3}{2}}}{16 \, {\left (a^{2} b^{3} c^{7} - 3 \, a^{3} b^{2} c^{6} d + 3 \, a^{4} b c^{5} d^{2} - a^{5} c^{4} d^{3} + {\left (a b^{4} c^{5} d^{2} - 3 \, a^{2} b^{3} c^{4} d^{3} + 3 \, a^{3} b^{2} c^{3} d^{4} - a^{4} b c^{2} d^{5}\right )} x^{6} + {\left (2 \, a b^{4} c^{6} d - 5 \, a^{2} b^{3} c^{5} d^{2} + 3 \, a^{3} b^{2} c^{4} d^{3} + a^{4} b c^{3} d^{4} - a^{5} c^{2} d^{5}\right )} x^{4} + {\left (a b^{4} c^{7} - a^{2} b^{3} c^{6} d - 3 \, a^{3} b^{2} c^{5} d^{2} + 5 \, a^{4} b c^{4} d^{3} - 2 \, a^{5} c^{3} d^{4}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/16*(b^4*c - 13*a*b^3*d)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqr
t(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*sq
rt(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)*sq
rt(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 + sq
rt(a))/(a^(1/4)*b^(3/4)))/(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) + 1/128*
(117*b^2*c^2*d^2 - 26*a*b*c*d^3 + 5*a^2*d^4)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d
)*sqrt(x))/sqrt(sqrt(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*sqrt(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)*sqrt(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 + sqrt(c))/(c^(1/4)*d^(3/4)))/(b^4*c^6 - 4*a*b^3*c^5*d + 6*a^2*b^2*c^4*d^2 - 4*a^3*b*c^3*d^3 +
a^4*c^2*d^4) + 1/16*((8*b^3*c^2*d^2 + 21*a*b^2*c*d^3 - 5*a^2*b*d^4)*x^(11/2) + (16*b^3*c^3*d + 25*a*b^2*c^2*d^
2 + 12*a^2*b*c*d^3 - 5*a^3*d^4)*x^(7/2) + (8*b^3*c^4 + 25*a^2*b*c^2*d^2 - 9*a^3*c*d^3)*x^(3/2))/(a^2*b^3*c^7 -
3*a^3*b^2*c^6*d + 3*a^4*b*c^5*d^2 - a^5*c^4*d^3 + (a*b^4*c^5*d^2 - 3*a^2*b^3*c^4*d^3 + 3*a^3*b^2*c^3*d^4 - a^
4*b*c^2*d^5)*x^6 + (2*a*b^4*c^6*d - 5*a^2*b^3*c^5*d^2 + 3*a^3*b^2*c^4*d^3 + a^4*b*c^3*d^4 - a^5*c^2*d^5)*x^4 +
(a*b^4*c^7 - a^2*b^3*c^6*d - 3*a^3*b^2*c^5*d^2 + 5*a^4*b*c^4*d^3 - 2*a^5*c^3*d^4)*x^2)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1233 vs. \(2 (583) = 1166\).
time = 2.35, size = 1233, normalized size = 1.67 \begin {gather*} \frac {b^{3} x^{\frac {3}{2}}}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} {\left (b x^{2} + a\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} b c - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \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 )}{4 \, {\left (\sqrt {2} a^{2} b^{4} c^{4} - 4 \, \sqrt {2} a^{3} b^{3} c^{3} d + 6 \, \sqrt {2} a^{4} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{5} b c d^{3} + \sqrt {2} a^{6} d^{4}\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} b c - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \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 )}{4 \, {\left (\sqrt {2} a^{2} b^{4} c^{4} - 4 \, \sqrt {2} a^{3} b^{3} c^{3} d + 6 \, \sqrt {2} a^{4} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{5} b c d^{3} + \sqrt {2} a^{6} d^{4}\right )}} + \frac {{\left (117 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 26 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \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 )}{32 \, {\left (\sqrt {2} b^{4} c^{7} d - 4 \, \sqrt {2} a b^{3} c^{6} d^{2} + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{3} - 4 \, \sqrt {2} a^{3} b c^{4} d^{4} + \sqrt {2} a^{4} c^{3} d^{5}\right )}} + \frac {{\left (117 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 26 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \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 )}{32 \, {\left (\sqrt {2} b^{4} c^{7} d - 4 \, \sqrt {2} a b^{3} c^{6} d^{2} + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{3} - 4 \, \sqrt {2} a^{3} b c^{4} d^{4} + \sqrt {2} a^{4} c^{3} d^{5}\right )}} - \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} b c - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a^{2} b^{4} c^{4} - 4 \, \sqrt {2} a^{3} b^{3} c^{3} d + 6 \, \sqrt {2} a^{4} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{5} b c d^{3} + \sqrt {2} a^{6} d^{4}\right )}} + \frac {{\left (\left (a b^{3}\right )^{\frac {3}{4}} b c - 13 \, \left (a b^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{8 \, {\left (\sqrt {2} a^{2} b^{4} c^{4} - 4 \, \sqrt {2} a^{3} b^{3} c^{3} d + 6 \, \sqrt {2} a^{4} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{5} b c d^{3} + \sqrt {2} a^{6} d^{4}\right )}} - \frac {{\left (117 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 26 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{4} c^{7} d - 4 \, \sqrt {2} a b^{3} c^{6} d^{2} + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{3} - 4 \, \sqrt {2} a^{3} b c^{4} d^{4} + \sqrt {2} a^{4} c^{3} d^{5}\right )}} + \frac {{\left (117 \, \left (c d^{3}\right )^{\frac {3}{4}} b^{2} c^{2} - 26 \, \left (c d^{3}\right )^{\frac {3}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac {3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{64 \, {\left (\sqrt {2} b^{4} c^{7} d - 4 \, \sqrt {2} a b^{3} c^{6} d^{2} + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{3} - 4 \, \sqrt {2} a^{3} b c^{4} d^{4} + \sqrt {2} a^{4} c^{3} d^{5}\right )}} + \frac {21 \, b c d^{3} x^{\frac {7}{2}} - 5 \, a d^{4} x^{\frac {7}{2}} + 25 \, b c^{2} d^{2} x^{\frac {3}{2}} - 9 \, a c d^{3} x^{\frac {3}{2}}}{16 \, {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/2*b^3*x^(3/2)/((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*x^2 + a)) + 1/4*((a*b^3)^(3/4)*b*c
- 13*(a*b^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^4*c^
4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^2 - 4*sqrt(2)*a^5*b*c*d^3 + sqrt(2)*a^6*d^4) + 1/4*((a*b
^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt
(2)*a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^2 - 4*sqrt(2)*a^5*b*c*d^3 + sqrt(2)*a^6*d^
4) + 1/32*(117*(c*d^3)^(3/4)*b^2*c^2 - 26*(c*d^3)^(3/4)*a*b*c*d + 5*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*
(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^7*d - 4*sqrt(2)*a*b^3*c^6*d^2 + 6*sqrt(2)*a^2*b^
2*c^5*d^3 - 4*sqrt(2)*a^3*b*c^4*d^4 + sqrt(2)*a^4*c^3*d^5) + 1/32*(117*(c*d^3)^(3/4)*b^2*c^2 - 26*(c*d^3)^(3/4
)*a*b*c*d + 5*(c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(
2)*b^4*c^7*d - 4*sqrt(2)*a*b^3*c^6*d^2 + 6*sqrt(2)*a^2*b^2*c^5*d^3 - 4*sqrt(2)*a^3*b*c^4*d^4 + sqrt(2)*a^4*c^3
*d^5) - 1/8*((a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(
2)*a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^2 - 4*sqrt(2)*a^5*b*c*d^3 + sqrt(2)*a^6*d^4
) + 1/8*((a*b^3)^(3/4)*b*c - 13*(a*b^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*
a^2*b^4*c^4 - 4*sqrt(2)*a^3*b^3*c^3*d + 6*sqrt(2)*a^4*b^2*c^2*d^2 - 4*sqrt(2)*a^5*b*c*d^3 + sqrt(2)*a^6*d^4) -
1/64*(117*(c*d^3)^(3/4)*b^2*c^2 - 26*(c*d^3)^(3/4)*a*b*c*d + 5*(c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/
d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^7*d - 4*sqrt(2)*a*b^3*c^6*d^2 + 6*sqrt(2)*a^2*b^2*c^5*d^3 - 4*sqrt(2)
*a^3*b*c^4*d^4 + sqrt(2)*a^4*c^3*d^5) + 1/64*(117*(c*d^3)^(3/4)*b^2*c^2 - 26*(c*d^3)^(3/4)*a*b*c*d + 5*(c*d^3)
^(3/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^7*d - 4*sqrt(2)*a*b^3*c^6*d^2
+ 6*sqrt(2)*a^2*b^2*c^5*d^3 - 4*sqrt(2)*a^3*b*c^4*d^4 + sqrt(2)*a^4*c^3*d^5) + 1/16*(21*b*c*d^3*x^(7/2) - 5*a
*d^4*x^(7/2) + 25*b*c^2*d^2*x^(3/2) - 9*a*c*d^3*x^(3/2))/((b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2
*d^3)*(d*x^2 + c)^2)

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Mupad [B]
time = 4.25, size = 2500, normalized size = 3.38 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/((a + b*x^2)^2*(c + d*x^2)^3),x)

[Out]

((x^(7/2)*(16*b^3*c^3*d - 5*a^3*d^4 + 25*a*b^2*c^2*d^2 + 12*a^2*b*c*d^3))/(16*a*c*(b^3*c^4 - a^3*c*d^3 + 3*a^2
*b*c^2*d^2 - 3*a*b^2*c^3*d)) - (x^(3/2)*(8*b^3*c^3 - 9*a^3*d^3 + 25*a^2*b*c*d^2))/(16*a*c*(a^3*d^3 - b^3*c^3 +
3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b*d^2*x^(11/2)*(8*b^2*c^2 - 5*a^2*d^2 + 21*a*b*c*d))/(16*a*c*(b^3*c^4 - a^
3*c*d^3 + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^3*d)))/(a*c^2 + x^2*(b*c^2 + 2*a*c*d) + x^4*(a*d^2 + 2*b*c*d) + b*d^2*x^
6) - atan(((-(625*a^8*d^13 + 187388721*b^8*c^8*d^5 - 166567752*a*b^7*c^7*d^6 + 87554844*a^2*b^6*c^6*d^7 - 2958
0408*a^3*b^5*c^5*d^8 + 7255846*a^4*b^4*c^4*d^9 - 1264120*a^5*b^3*c^3*d^10 + 159900*a^6*b^2*c^2*d^11 - 13000*a^
7*b*c*d^12)/(16777216*b^16*c^25 + 16777216*a^16*c^9*d^16 - 268435456*a^15*b*c^10*d^15 + 2013265920*a^2*b^14*c^
23*d^2 - 9395240960*a^3*b^13*c^22*d^3 + 30534533120*a^4*b^12*c^21*d^4 - 73282879488*a^5*b^11*c^20*d^5 + 134351
945728*a^6*b^10*c^19*d^6 - 191931351040*a^7*b^9*c^18*d^7 + 215922769920*a^8*b^8*c^17*d^8 - 191931351040*a^9*b^
7*c^16*d^9 + 134351945728*a^10*b^6*c^15*d^10 - 73282879488*a^11*b^5*c^14*d^11 + 30534533120*a^12*b^4*c^13*d^12
- 9395240960*a^13*b^3*c^12*d^13 + 2013265920*a^14*b^2*c^11*d^14 - 268435456*a*b^15*c^24*d))^(1/4)*((-(625*a^8
*d^13 + 187388721*b^8*c^8*d^5 - 166567752*a*b^7*c^7*d^6 + 87554844*a^2*b^6*c^6*d^7 - 29580408*a^3*b^5*c^5*d^8
+ 7255846*a^4*b^4*c^4*d^9 - 1264120*a^5*b^3*c^3*d^10 + 159900*a^6*b^2*c^2*d^11 - 13000*a^7*b*c*d^12)/(16777216
*b^16*c^25 + 16777216*a^16*c^9*d^16 - 268435456*a^15*b*c^10*d^15 + 2013265920*a^2*b^14*c^23*d^2 - 9395240960*a
^3*b^13*c^22*d^3 + 30534533120*a^4*b^12*c^21*d^4 - 73282879488*a^5*b^11*c^20*d^5 + 134351945728*a^6*b^10*c^19*
d^6 - 191931351040*a^7*b^9*c^18*d^7 + 215922769920*a^8*b^8*c^17*d^8 - 191931351040*a^9*b^7*c^16*d^9 + 13435194
5728*a^10*b^6*c^15*d^10 - 73282879488*a^11*b^5*c^14*d^11 + 30534533120*a^12*b^4*c^13*d^12 - 9395240960*a^13*b^
3*c^12*d^13 + 2013265920*a^14*b^2*c^11*d^14 - 268435456*a*b^15*c^24*d))^(3/4)*((32*b^30*c^27*d^4 - 1728*a*b^29
*c^26*d^5 - (125*a^26*b^4*c*d^30)/16 + 38304*a^2*b^28*c^25*d^6 - 459264*a^3*b^27*c^24*d^7 + 3369600*a^4*b^26*c
^23*d^8 - (263413683*a^5*b^25*c^22*d^9)/16 + (903579807*a^6*b^24*c^21*d^10)/16 - (1116788283*a^7*b^23*c^20*d^1
1)/8 + (1980689243*a^8*b^22*c^19*d^12)/8 - (4711274035*a^9*b^21*c^18*d^13)/16 + (2530187127*a^10*b^20*c^17*d^1
4)/16 + (409977699*a^11*b^19*c^16*d^15)/2 - (1337499867*a^12*b^18*c^15*d^16)/2 + (8002341693*a^13*b^17*c^14*d^
17)/8 - (8341892385*a^14*b^16*c^13*d^18)/8 + (3315895143*a^15*b^15*c^12*d^19)/4 - (2079521847*a^16*b^14*c^11*d
^20)/4 + (2088923057*a^17*b^13*c^10*d^21)/8 - (845943917*a^18*b^12*c^9*d^22)/8 + (69181515*a^19*b^11*c^8*d^23)
/2 - (18239091*a^20*b^10*c^7*d^24)/2 + (30778137*a^21*b^9*c^6*d^25)/16 - (5119101*a^22*b^8*c^5*d^26)/16 + (327
093*a^23*b^7*c^4*d^27)/8 - (30645*a^24*b^6*c^3*d^28)/8 + (3825*a^25*b^5*c^2*d^29)/16)/(a^2*b^21*c^27 - a^23*c^
6*d^21 - 21*a^3*b^20*c^26*d + 21*a^22*b*c^7*d^20 + 210*a^4*b^19*c^25*d^2 - 1330*a^5*b^18*c^24*d^3 + 5985*a^6*b
^17*c^23*d^4 - 20349*a^7*b^16*c^22*d^5 + 54264*a^8*b^15*c^21*d^6 - 116280*a^9*b^14*c^20*d^7 + 203490*a^10*b^13
*c^19*d^8 - 293930*a^11*b^12*c^18*d^9 + 352716*a^12*b^11*c^17*d^10 - 352716*a^13*b^10*c^16*d^11 + 293930*a^14*
b^9*c^15*d^12 - 203490*a^15*b^8*c^14*d^13 + 116280*a^16*b^7*c^13*d^14 - 54264*a^17*b^6*c^12*d^15 + 20349*a^18*
b^5*c^11*d^16 - 5985*a^19*b^4*c^10*d^17 + 1330*a^20*b^3*c^9*d^18 - 210*a^21*b^2*c^8*d^19) - (x^(1/2)*(-(625*a^
8*d^13 + 187388721*b^8*c^8*d^5 - 166567752*a*b^7*c^7*d^6 + 87554844*a^2*b^6*c^6*d^7 - 29580408*a^3*b^5*c^5*d^8
+ 7255846*a^4*b^4*c^4*d^9 - 1264120*a^5*b^3*c^3*d^10 + 159900*a^6*b^2*c^2*d^11 - 13000*a^7*b*c*d^12)/(1677721
6*b^16*c^25 + 16777216*a^16*c^9*d^16 - 268435456*a^15*b*c^10*d^15 + 2013265920*a^2*b^14*c^23*d^2 - 9395240960*
a^3*b^13*c^22*d^3 + 30534533120*a^4*b^12*c^21*d^4 - 73282879488*a^5*b^11*c^20*d^5 + 134351945728*a^6*b^10*c^19
*d^6 - 191931351040*a^7*b^9*c^18*d^7 + 215922769920*a^8*b^8*c^17*d^8 - 191931351040*a^9*b^7*c^16*d^9 + 1343519
45728*a^10*b^6*c^15*d^10 - 73282879488*a^11*b^5*c^14*d^11 + 30534533120*a^12*b^4*c^13*d^12 - 9395240960*a^13*b
^3*c^12*d^13 + 2013265920*a^14*b^2*c^11*d^14 - 268435456*a*b^15*c^24*d))^(1/4)*(16777216*a*b^28*c^27*d^4 - 704
643072*a^2*b^27*c^26*d^5 + 11827937280*a^3*b^26*c^25*d^6 - 107105746944*a^4*b^25*c^24*d^7 + 618641227776*a^5*b
^24*c^23*d^8 - 2513987174400*a^6*b^23*c^22*d^9 + 7656663678976*a^7*b^22*c^21*d^10 - 18278639468544*a^8*b^21*c^
20*d^11 + 35394969403392*a^9*b^20*c^19*d^12 - 57098809376768*a^10*b^19*c^18*d^13 + 78238275600384*a^11*b^18*c^
17*d^14 - 92068449878016*a^12*b^17*c^16*d^15 + 93255551680512*a^13*b^16*c^15*d^16 - 80877025492992*a^14*b^15*c
^14*d^17 + 59448946065408*a^15*b^14*c^13*d^18 - 36574941151232*a^16*b^13*c^12*d^19 + 18584022024192*a^17*b^12*
c^11*d^20 - 7692575834112*a^18*b^11*c^10*d^21 + 2557512515584*a^19*b^10*c^9*d^22 - 672468566016*a^20*b^9*c^8*d
^23 + 137272492032*a^21*b^8*c^7*d^24 - 21186478080*a^22*b^7*c^6*d^25 + 2360868864*a^23*b^6*c^5*d^26 - 17301504
0*a^24*b^5*c^4*d^27 + 6553600*a^25*b^4*c^3*d^28...

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