3.5.84 \(\int \frac {1}{\sqrt {x} (a+b x^2) (c+d x^2)^3} \, dx\) [484]

Optimal. Leaf size=633 \[ -\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {b^{11/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^3} \]

[Out]

-1/2*b^(11/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/2*b^(11/4)*arctan(1+b^(
1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/64*d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*b^2*c^2)*arc
tan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/64*d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*b^
2*c^2)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/4*b^(11/4)*ln(a^(1/2)+x*b^(1/
2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/4*b^(11/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(
1/4)*2^(1/2)*x^(1/2))/a^(3/4)/(-a*d+b*c)^3*2^(1/2)+1/128*d^(3/4)*(21*a^2*d^2-66*a*b*c*d+77*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^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/128*d^(3/4)*(21*a^2*d^2-66*a*b*c*d
+77*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^(11/4)/(-a*d+b*c)^3*2^(1/2)-1/4*d*x^(1/2)
/c/(-a*d+b*c)/(d*x^2+c)^2-1/16*d*(-7*a*d+15*b*c)*x^(1/2)/c^2/(-a*d+b*c)^2/(d*x^2+c)

________________________________________________________________________________________

Rubi [A]
time = 0.57, antiderivative size = 633, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 425, 541, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {b^{11/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}-\frac {b^{11/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 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^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 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^{11/4} (b c-a d)^3}+\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 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^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (21 a^2 d^2-66 a b c d+77 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^{11/4} (b c-a d)^3}-\frac {d \sqrt {x} (15 b c-7 a d)}{16 c^2 \left (c+d x^2\right ) (b c-a d)^2}-\frac {d \sqrt {x}}{4 c \left (c+d x^2\right )^2 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

-1/4*(d*Sqrt[x])/(c*(b*c - a*d)*(c + d*x^2)^2) - (d*(15*b*c - 7*a*d)*Sqrt[x])/(16*c^2*(b*c - a*d)^2*(c + d*x^2
)) - (b^(11/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (b^(11/4)*ArcT
an[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*(b*c - a*d)^3) + (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d
+ 21*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (d^(3/4)*(
77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c
- a*d)^3) - (b^(11/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*(b*c - a
*d)^3) + (b^(11/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*(b*c - a*d)^
3) + (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x
])/(64*Sqrt[2]*c^(11/4)*(b*c - a*d)^3) - (d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]
*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(11/4)*(b*c - a*d)^3)

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 217

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

Rule 425

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

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 536

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

Rule 541

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

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 {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {1}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}+\frac {\text {Subst}\left (\int \frac {8 b c-7 a d-7 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )}{4 c (b c-a d)}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {32 b^2 c^2-45 a b c d+21 a^2 d^2-3 b d (15 b c-7 a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^2}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{(b c-a d)^3}-\frac {\left (d \left (77 b^2 c^2-66 a b c d+21 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^2 (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^3 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} (b c-a d)^3}+\frac {b^3 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {a} (b c-a d)^3}-\frac {\left (d \left (77 b^2 c^2-66 a b c d+21 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^{5/2} (b c-a d)^3}-\frac {\left (d \left (77 b^2 c^2-66 a b c d+21 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^{5/2} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {b^{5/2} \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 \sqrt {a} (b c-a d)^3}+\frac {b^{5/2} \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 \sqrt {a} (b c-a d)^3}-\frac {b^{11/4} \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} a^{3/4} (b c-a d)^3}-\frac {b^{11/4} \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} a^{3/4} (b c-a d)^3}-\frac {\left (\sqrt {d} \left (77 b^2 c^2-66 a b c d+21 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^{5/2} (b c-a d)^3}-\frac {\left (\sqrt {d} \left (77 b^2 c^2-66 a b c d+21 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^{5/2} (b c-a d)^3}+\frac {\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}+\frac {\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{11/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}+\frac {b^{11/4} \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} a^{3/4} (b c-a d)^3}-\frac {b^{11/4} \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} a^{3/4} (b c-a d)^3}-\frac {\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}+\frac {\left (d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}\\ &=-\frac {d \sqrt {x}}{4 c (b c-a d) \left (c+d x^2\right )^2}-\frac {d (15 b c-7 a d) \sqrt {x}}{16 c^2 (b c-a d)^2 \left (c+d x^2\right )}-\frac {b^{11/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac {b^{11/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {b^{11/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{3/4} (b c-a d)^3}+\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac {d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 1.41, size = 362, normalized size = 0.57 \begin {gather*} \frac {1}{64} \left (\frac {4 d \sqrt {x} \left (a d \left (11 c+7 d x^2\right )-b c \left (19 c+15 d x^2\right )\right )}{c^2 (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {32 \sqrt {2} b^{11/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{3/4} (-b c+a d)^3}+\frac {\sqrt {2} d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}-\frac {32 \sqrt {2} b^{11/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{3/4} (-b c+a d)^3}-\frac {\sqrt {2} d^{3/4} \left (77 b^2 c^2-66 a b c d+21 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^{11/4} (b c-a d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + b*x^2)*(c + d*x^2)^3),x]

[Out]

((4*d*Sqrt[x]*(a*d*(11*c + 7*d*x^2) - b*c*(19*c + 15*d*x^2)))/(c^2*(b*c - a*d)^2*(c + d*x^2)^2) + (32*Sqrt[2]*
b^(11/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(3/4)*(-(b*c) + a*d)^3) + (Sqrt[2
]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x]
)])/(c^(11/4)*(b*c - a*d)^3) - (32*Sqrt[2]*b^(11/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[
b]*x)])/(a^(3/4)*(-(b*c) + a*d)^3) - (Sqrt[2]*d^(3/4)*(77*b^2*c^2 - 66*a*b*c*d + 21*a^2*d^2)*ArcTanh[(Sqrt[2]*
c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(11/4)*(b*c - a*d)^3))/64

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Maple [A]
time = 0.09, size = 336, normalized size = 0.53

method	result	size
derivativedivides	\(-\frac {b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} a}+\frac {2 d \left (\frac {\frac {d \left (7 a^{2} d^{2}-22 a b c d +15 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-30 a b c d +19 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (21 a^{2} d^{2}-66 a b c d +77 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{3}}\)	\(336\)
default	\(-\frac {b^{3} \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (a d -b c \right )^{3} a}+\frac {2 d \left (\frac {\frac {d \left (7 a^{2} d^{2}-22 a b c d +15 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-30 a b c d +19 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (21 a^{2} d^{2}-66 a b c d +77 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{3}}\)	\(336\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b^3/(a*d-b*c)^3*(a/b)^(1/4)/a*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
/(a*d-b*c)^3*((1/32*d*(7*a^2*d^2-22*a*b*c*d+15*b^2*c^2)/c^2*x^(5/2)+1/32*(11*a^2*d^2-30*a*b*c*d+19*b^2*c^2)/c*
x^(1/2))/(d*x^2+c)^2+1/256*(21*a^2*d^2-66*a*b*c*d+77*b^2*c^2)/c^3*(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 = 0.52, size = 675, normalized size = 1.07 \begin {gather*} -\frac {{\left (15 \, b c d^{2} - 7 \, a d^{3}\right )} x^{\frac {5}{2}} + {\left (19 \, b c^{2} d - 11 \, a c d^{2}\right )} \sqrt {x}}{16 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2} + {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} x^{2}\right )}} + \frac {\frac {2 \, \sqrt {2} b^{3} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} b^{3} \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 {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {11}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}} - \frac {\sqrt {2} b^{\frac {11}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}}}}{4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} \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 {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {2 \, \sqrt {2} {\left (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} \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 {c} \sqrt {\sqrt {c} \sqrt {d}}} + \frac {\sqrt {2} {\left (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (77 \, b^{2} c^{2} d - 66 \, a b c d^{2} + 21 \, a^{2} d^{3}\right )} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {3}{4}} d^{\frac {1}{4}}}}{128 \, {\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*((15*b*c*d^2 - 7*a*d^3)*x^(5/2) + (19*b*c^2*d - 11*a*c*d^2)*sqrt(x))/(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^
2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2) + 1/4*(
2*sqrt(2)*b^3*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)
*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*b^3*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt
(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*b^(11/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sq
rt(b)*x + sqrt(a))/a^(3/4) - sqrt(2)*b^(11/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/a^(3
/4))/(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3) - 1/128*(2*sqrt(2)*(77*b^2*c^2*d - 66*a*b*c*d^2 + 21*
a^2*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt
(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(77*b^2*c^2*d - 66*a*b*c*d^2 + 21*a^2*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)
*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + sqrt(2)*(77*b^2*c^2*d -
66*a*b*c*d^2 + 21*a^2*d^3)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqr
t(2)*(77*b^2*c^2*d - 66*a*b*c*d^2 + 21*a^2*d^3)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c
^(3/4)*d^(1/4)))/(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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5548 vs. \(2 (486) = 972\).
time = 257.05, size = 5548, normalized size = 8.76 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(4*(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*x^4 + 2*(b^2*c^5*d
- 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*(-(35153041*b^8*c^8*d^3 - 120524712*a*b^7*c^7*d^4 + 193309116*a^2*b^6*c^6*
d^5 - 187159896*a^3*b^5*c^5*d^6 + 119186694*a^4*b^4*c^4*d^7 - 51043608*a^5*b^3*c^3*d^8 + 14378364*a^6*b^2*c^2*
d^9 - 2444904*a^7*b*c*d^10 + 194481*a^8*d^11)/(b^12*c^23 - 12*a*b^11*c^22*d + 66*a^2*b^10*c^21*d^2 - 220*a^3*b
^9*c^20*d^3 + 495*a^4*b^8*c^19*d^4 - 792*a^5*b^7*c^18*d^5 + 924*a^6*b^6*c^17*d^6 - 792*a^7*b^5*c^16*d^7 + 495*
a^8*b^4*c^15*d^8 - 220*a^9*b^3*c^14*d^9 + 66*a^10*b^2*c^13*d^10 - 12*a^11*b*c^12*d^11 + a^12*c^11*d^12))^(1/4)
*arctan(-((b^9*c^17 - 9*a*b^8*c^16*d + 36*a^2*b^7*c^15*d^2 - 84*a^3*b^6*c^14*d^3 + 126*a^4*b^5*c^13*d^4 - 126*
a^5*b^4*c^12*d^5 + 84*a^6*b^3*c^11*d^6 - 36*a^7*b^2*c^10*d^7 + 9*a^8*b*c^9*d^8 - a^9*c^8*d^9)*sqrt((5929*b^4*c
^4*d^2 - 10164*a*b^3*c^3*d^3 + 7590*a^2*b^2*c^2*d^4 - 2772*a^3*b*c*d^5 + 441*a^4*d^6)*x + (b^6*c^12 - 6*a*b^5*
c^11*d + 15*a^2*b^4*c^10*d^2 - 20*a^3*b^3*c^9*d^3 + 15*a^4*b^2*c^8*d^4 - 6*a^5*b*c^7*d^5 + a^6*c^6*d^6)*sqrt(-
(35153041*b^8*c^8*d^3 - 120524712*a*b^7*c^7*d^4 + 193309116*a^2*b^6*c^6*d^5 - 187159896*a^3*b^5*c^5*d^6 + 1191
86694*a^4*b^4*c^4*d^7 - 51043608*a^5*b^3*c^3*d^8 + 14378364*a^6*b^2*c^2*d^9 - 2444904*a^7*b*c*d^10 + 194481*a^
8*d^11)/(b^12*c^23 - 12*a*b^11*c^22*d + 66*a^2*b^10*c^21*d^2 - 220*a^3*b^9*c^20*d^3 + 495*a^4*b^8*c^19*d^4 - 7
92*a^5*b^7*c^18*d^5 + 924*a^6*b^6*c^17*d^6 - 792*a^7*b^5*c^16*d^7 + 495*a^8*b^4*c^15*d^8 - 220*a^9*b^3*c^14*d^
9 + 66*a^10*b^2*c^13*d^10 - 12*a^11*b*c^12*d^11 + a^12*c^11*d^12)))*(-(35153041*b^8*c^8*d^3 - 120524712*a*b^7*
c^7*d^4 + 193309116*a^2*b^6*c^6*d^5 - 187159896*a^3*b^5*c^5*d^6 + 119186694*a^4*b^4*c^4*d^7 - 51043608*a^5*b^3
*c^3*d^8 + 14378364*a^6*b^2*c^2*d^9 - 2444904*a^7*b*c*d^10 + 194481*a^8*d^11)/(b^12*c^23 - 12*a*b^11*c^22*d +
66*a^2*b^10*c^21*d^2 - 220*a^3*b^9*c^20*d^3 + 495*a^4*b^8*c^19*d^4 - 792*a^5*b^7*c^18*d^5 + 924*a^6*b^6*c^17*d
^6 - 792*a^7*b^5*c^16*d^7 + 495*a^8*b^4*c^15*d^8 - 220*a^9*b^3*c^14*d^9 + 66*a^10*b^2*c^13*d^10 - 12*a^11*b*c^
12*d^11 + a^12*c^11*d^12))^(3/4) - (77*b^11*c^19*d - 759*a*b^10*c^18*d^2 + 3387*a^2*b^9*c^17*d^3 - 9033*a^3*b^
8*c^16*d^4 + 16002*a^4*b^7*c^15*d^5 - 19782*a^5*b^6*c^14*d^6 + 17430*a^6*b^5*c^13*d^7 - 10962*a^7*b^4*c^12*d^8
+ 4833*a^8*b^3*c^11*d^9 - 1427*a^9*b^2*c^10*d^10 + 255*a^10*b*c^9*d^11 - 21*a^11*c^8*d^12)*sqrt(x)*(-(3515304
1*b^8*c^8*d^3 - 120524712*a*b^7*c^7*d^4 + 193309116*a^2*b^6*c^6*d^5 - 187159896*a^3*b^5*c^5*d^6 + 119186694*a^
4*b^4*c^4*d^7 - 51043608*a^5*b^3*c^3*d^8 + 14378364*a^6*b^2*c^2*d^9 - 2444904*a^7*b*c*d^10 + 194481*a^8*d^11)/
(b^12*c^23 - 12*a*b^11*c^22*d + 66*a^2*b^10*c^21*d^2 - 220*a^3*b^9*c^20*d^3 + 495*a^4*b^8*c^19*d^4 - 792*a^5*b
^7*c^18*d^5 + 924*a^6*b^6*c^17*d^6 - 792*a^7*b^5*c^16*d^7 + 495*a^8*b^4*c^15*d^8 - 220*a^9*b^3*c^14*d^9 + 66*a
^10*b^2*c^13*d^10 - 12*a^11*b*c^12*d^11 + a^12*c^11*d^12))^(3/4))/(35153041*b^8*c^8*d^3 - 120524712*a*b^7*c^7*
d^4 + 193309116*a^2*b^6*c^6*d^5 - 187159896*a^3*b^5*c^5*d^6 + 119186694*a^4*b^4*c^4*d^7 - 51043608*a^5*b^3*c^3
*d^8 + 14378364*a^6*b^2*c^2*d^9 - 2444904*a^7*b*c*d^10 + 194481*a^8*d^11)) - 128*(-b^11/(a^3*b^12*c^12 - 12*a^
4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a
^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 1
2*a^14*b*c*d^11 + a^15*d^12))^(1/4)*(b^2*c^6 - 2*a*b*c^5*d + a^2*c^4*d^2 + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*
c^2*d^4)*x^4 + 2*(b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*x^2)*arctan(-((a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a
^4*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 + 84*a^8*b^3*c^3*d^6 - 36*a^9*
b^2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*(-b^11/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 2
20*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 +
495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))^(3/4)*sqrt
(b^6*x + (a^2*b^6*c^6 - 6*a^3*b^5*c^5*d + 15*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 15*a^6*b^2*c^2*d^4 - 6*a^7
*b*c*d^5 + a^8*d^6)*sqrt(-b^11/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^
3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*
d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12))) - (a^2*b^12*c^9 - 9*a^3*b^
11*c^8*d + 36*a^4*b^10*c^7*d^2 - 84*a^5*b^9*c^6*d^3 + 126*a^6*b^8*c^5*d^4 - 126*a^7*b^7*c^4*d^5 + 84*a^8*b^6*c
^3*d^6 - 36*a^9*b^5*c^2*d^7 + 9*a^10*b^4*c*d^8 - a^11*b^3*d^9)*(-b^11/(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66
*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 7
92*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 +
a^15*d^12))^(3/4)*sqrt(x))/b^11) + 32*(-b^11/(a...

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

[Out]

Timed out

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Giac [A]
time = 2.85, size = 960, normalized size = 1.52 \begin {gather*} \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{2} \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^{3} - 3 \, \sqrt {2} a^{2} b^{2} c^{2} d + 3 \, \sqrt {2} a^{3} b c d^{2} - \sqrt {2} a^{4} d^{3}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{2} \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^{3} - 3 \, \sqrt {2} a^{2} b^{2} c^{2} d + 3 \, \sqrt {2} a^{3} b c d^{2} - \sqrt {2} a^{4} d^{3}} + \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{2} \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^{3} - 3 \, \sqrt {2} a^{2} b^{2} c^{2} d + 3 \, \sqrt {2} a^{3} b c d^{2} - \sqrt {2} a^{4} d^{3}\right )}} - \frac {\left (a b^{3}\right )^{\frac {1}{4}} b^{2} \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^{3} - 3 \, \sqrt {2} a^{2} b^{2} c^{2} d + 3 \, \sqrt {2} a^{3} b c d^{2} - \sqrt {2} a^{4} d^{3}\right )}} - \frac {{\left (77 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 66 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{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^{3} c^{6} - 3 \, \sqrt {2} a b^{2} c^{5} d + 3 \, \sqrt {2} a^{2} b c^{4} d^{2} - \sqrt {2} a^{3} c^{3} d^{3}\right )}} - \frac {{\left (77 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 66 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{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^{3} c^{6} - 3 \, \sqrt {2} a b^{2} c^{5} d + 3 \, \sqrt {2} a^{2} b c^{4} d^{2} - \sqrt {2} a^{3} c^{3} d^{3}\right )}} - \frac {{\left (77 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 66 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{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^{3} c^{6} - 3 \, \sqrt {2} a b^{2} c^{5} d + 3 \, \sqrt {2} a^{2} b c^{4} d^{2} - \sqrt {2} a^{3} c^{3} d^{3}\right )}} + \frac {{\left (77 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} - 66 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d + 21 \, \left (c d^{3}\right )^{\frac {1}{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^{3} c^{6} - 3 \, \sqrt {2} a b^{2} c^{5} d + 3 \, \sqrt {2} a^{2} b c^{4} d^{2} - \sqrt {2} a^{3} c^{3} d^{3}\right )}} - \frac {15 \, b c d^{2} x^{\frac {5}{2}} - 7 \, a d^{3} x^{\frac {5}{2}} + 19 \, b c^{2} d \sqrt {x} - 11 \, a c d^{2} \sqrt {x}}{16 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*b^3)^(1/4)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^3*c^3 - 3*sqr
t(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) + (a*b^3)^(1/4)*b^2*arctan(-1/2*sqrt(2)*(sqrt(2)
*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 -
sqrt(2)*a^4*d^3) + 1/2*(a*b^3)^(1/4)*b^2*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^3*c^3 -
3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)*a^4*d^3) - 1/2*(a*b^3)^(1/4)*b^2*log(-sqrt(2)*sqrt(
x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b^3*c^3 - 3*sqrt(2)*a^2*b^2*c^2*d + 3*sqrt(2)*a^3*b*c*d^2 - sqrt(2)
*a^4*d^3) - 1/32*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*arctan(1/2*s
qrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^3*c^6 - 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2
*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/32*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/
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^3*c^6 - 3*sqrt(2)*a*
b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/64*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3)^(1/4)
*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^6 - 3*sqr
t(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) + 1/64*(77*(c*d^3)^(1/4)*b^2*c^2 - 66*(c*d^3
)^(1/4)*a*b*c*d + 21*(c*d^3)^(1/4)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^3*c^6
- 3*sqrt(2)*a*b^2*c^5*d + 3*sqrt(2)*a^2*b*c^4*d^2 - sqrt(2)*a^3*c^3*d^3) - 1/16*(15*b*c*d^2*x^(5/2) - 7*a*d^3
*x^(5/2) + 19*b*c^2*d*sqrt(x) - 11*a*c*d^2*sqrt(x))/((b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*(d*x^2 + c)^2)

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Mupad [B]
time = 2.33, size = 2500, normalized size = 3.95 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^
9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a
^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*((((194481*a^8*b^
8*d^14)/2048 + 1232*b^16*c^8*d^6 - (34792593*a*b^15*c^7*d^7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (86420247*
a^2*b^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/2048 + (80271027*a^4*b^12*c^4*d^10)/2048 - (38915667*a^5
*b^11*c^3*d^11)/2048 + (12127941*a^6*b^10*c^2*d^12)/2048)/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b
^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*
c^15*d) + ((x^(1/2)*(16777216*b^23*c^23*d^4 - 201326592*a*b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 35938
46784*a^3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8 - 4753588224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17*c
^17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 181503787008*a^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13
- 352258621440*a^10*b^13*c^13*d^14 + 334222688256*a^11*b^12*c^12*d^15 - 249961119744*a^12*b^11*c^11*d^16 + 147
248775168*a^13*b^10*c^10*d^17 - 67718086656*a^14*b^9*c^9*d^18 + 23871029248*a^15*b^8*c^8*d^19 - 6245842944*a^1
6*b^7*c^7*d^20 + 1146224640*a^17*b^6*c^6*d^21 - 132120576*a^18*b^5*c^5*d^22 + 7225344*a^19*b^4*c^4*d^23))/(409
6*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*
c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9
*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d)) - ((-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^
4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 +
14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^
2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*(8192*a*b^19*c^22*d^4 - 90112*a^2*b^18*c^21*d^5 + 430848*a^3*b^17*c^20*
d^6 - 1117952*a^4*b^16*c^19*d^7 + 1427968*a^5*b^15*c^18*d^8 + 456192*a^6*b^14*c^17*d^9 - 5803776*a^7*b^13*c^16
*d^10 + 12866304*a^8*b^12*c^15*d^11 - 17335296*a^9*b^11*c^14*d^12 + 16344064*a^10*b^10*c^13*d^13 - 11221760*a^
11*b^9*c^12*d^14 + 5637888*a^12*b^8*c^11*d^15 - 2033152*a^13*b^7*c^10*d^16 + 501248*a^14*b^6*c^9*d^17 - 76032*
a^15*b^5*c^8*d^18 + 5376*a^16*b^4*c^7*d^19))/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 -
56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d))*(-b^
11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 79
20*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^
4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(3/4))*(-b^11/(16*a^15*d^12 + 16*
a^3*b^12*c^12 - 192*a^4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 1
2672*a^8*b^7*c^7*d^5 + 14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*
c^3*d^9 + 1056*a^13*b^2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*1i + (x^(1/2)*(194481*a^8*b^11*d^15 + 41224337*b^
19*c^8*d^7 - 130932648*a*b^18*c^7*d^8 - 2444904*a^7*b^12*c*d^14 + 201081276*a^2*b^17*c^6*d^9 - 189998424*a^3*b
^16*c^5*d^10 + 119638278*a^4*b^15*c^4*d^11 - 51043608*a^5*b^14*c^3*d^12 + 14378364*a^6*b^13*c^2*d^13)*1i)/(409
6*(b^12*c^20 + a^12*c^8*d^12 - 12*a^11*b*c^9*d^11 + 66*a^2*b^10*c^18*d^2 - 220*a^3*b^9*c^17*d^3 + 495*a^4*b^8*
c^16*d^4 - 792*a^5*b^7*c^15*d^5 + 924*a^6*b^6*c^14*d^6 - 792*a^7*b^5*c^13*d^7 + 495*a^8*b^4*c^12*d^8 - 220*a^9
*b^3*c^11*d^9 + 66*a^10*b^2*c^10*d^10 - 12*a*b^11*c^19*d))) - (-b^11/(16*a^15*d^12 + 16*a^3*b^12*c^12 - 192*a^
4*b^11*c^11*d + 1056*a^5*b^10*c^10*d^2 - 3520*a^6*b^9*c^9*d^3 + 7920*a^7*b^8*c^8*d^4 - 12672*a^8*b^7*c^7*d^5 +
14784*a^9*b^6*c^6*d^6 - 12672*a^10*b^5*c^5*d^7 + 7920*a^11*b^4*c^4*d^8 - 3520*a^12*b^3*c^3*d^9 + 1056*a^13*b^
2*c^2*d^10 - 192*a^14*b*c*d^11))^(1/4)*((((194481*a^8*b^8*d^14)/2048 + 1232*b^16*c^8*d^6 - (34792593*a*b^15*c^
7*d^7)/2048 - (2250423*a^7*b^9*c*d^13)/2048 + (86420247*a^2*b^14*c^6*d^8)/2048 - (106888869*a^3*b^13*c^5*d^9)/
2048 + (80271027*a^4*b^12*c^4*d^10)/2048 - (38915667*a^5*b^11*c^3*d^11)/2048 + (12127941*a^6*b^10*c^2*d^12)/20
48)/(b^8*c^16 + a^8*c^8*d^8 - 8*a^7*b*c^9*d^7 + 28*a^2*b^6*c^14*d^2 - 56*a^3*b^5*c^13*d^3 + 70*a^4*b^4*c^12*d^
4 - 56*a^5*b^3*c^11*d^5 + 28*a^6*b^2*c^10*d^6 - 8*a*b^7*c^15*d) - ((x^(1/2)*(16777216*b^23*c^23*d^4 - 20132659
2*a*b^22*c^22*d^5 + 1107296256*a^2*b^21*c^21*d^6 - 3593846784*a^3*b^20*c^20*d^7 + 6972506112*a^4*b^19*c^19*d^8
- 4753588224*a^5*b^18*c^18*d^9 - 18397265920*a^6*b^17*c^17*d^10 + 80192667648*a^7*b^16*c^16*d^11 - 1815037870
08*a^8*b^15*c^15*d^12 + 289980416000*a^9*b^14*c^14*d^13 - 352258621440*a^10*b^13*c^13*d^14 + 334222688256*a^11
*b^12*c^12*d^15 - 249961119744*a^12*b^11*c^11*d...

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