3.5.100 \(\int \frac {1}{\sqrt {x} (a+b x^2)^2 (c+d x^2)^3} \, dx\) [500]

Optimal. Leaf size=739 \[ \frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{11/4} (b c-a d)^4}-\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{11/4} (b c-a d)^4} \]

[Out]

-3/8*b^(11/4)*(-5*a*d+b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^4*2^(1/2)+3/8*b^(11/4)
*(-5*a*d+b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(7/4)/(-a*d+b*c)^4*2^(1/2)-3/64*d^(7/4)*(7*a^2*d^2-3
0*a*b*c*d+55*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)^4*2^(1/2)+3/64*d^(7/4)*(7*
a^2*d^2-30*a*b*c*d+55*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)^4*2^(1/2)-3/16*b^
(11/4)*(-5*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^(7/4)/(-a*d+b*c)^4*2^(1/2)+3/16*b^
(11/4)*(-5*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^(7/4)/(-a*d+b*c)^4*2^(1/2)-3/128*d
^(7/4)*(7*a^2*d^2-30*a*b*c*d+55*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)^4*2^(1/2)+3/128*d^(7/4)*(7*a^2*d^2-30*a*b*c*d+55*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)^4*2^(1/2)+1/4*d*(a*d+2*b*c)*x^(1/2)/a/c/(-a*d+b*c)^2/(d*x^2+c)^2+1/2*b*x^(1/2)/a/(-
a*d+b*c)/(b*x^2+a)/(d*x^2+c)^2+1/16*d*(-7*a^2*d^2+23*a*b*c*d+8*b^2*c^2)*x^(1/2)/a/c^2/(-a*d+b*c)^3/(d*x^2+c)

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 739, normalized size of antiderivative = 1.00, number of steps used = 23, 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 {3 b^{11/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) (b c-5 a d)}{4 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) (b c-5 a d)}{4 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 b^{11/4} (b c-5 a d) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 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)^4}+\frac {3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 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)^4}+\frac {d \sqrt {x} \left (-7 a^2 d^2+23 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 {3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 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)^4}+\frac {3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 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)^4}+\frac {b \sqrt {x}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac {d \sqrt {x} (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[1/(Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

(d*(2*b*c + a*d)*Sqrt[x])/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*Sqrt[x])/(2*a*(b*c - a*d)*(a + b*x^2)*(c +
d*x^2)^2) + (d*(8*b^2*c^2 + 23*a*b*c*d - 7*a^2*d^2)*Sqrt[x])/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(11/4
)*(b*c - 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (3*b^(11/4)
*(b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(
55*b^2*c^2 - 30*a*b*c*d + 7*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)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(3
2*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) - (3*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (3*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*L
og[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)^4) + (3*d^(7/4)*(5
5*b^2*c^2 - 30*a*b*c*d + 7*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)^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 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 )^2 \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {1}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {b \sqrt {x}}{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 {-3 b c+4 a d-11 b d x^4}{\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) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{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 {-4 \left (6 b^2 c^2-16 a b c d+7 a^2 d^2\right )-28 b d (2 b c+a d) x^4}{\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) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{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+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-12 \left (8 b^3 c^3-32 a b^2 c^2 d+23 a^2 b c d^2-7 a^3 d^3\right )-12 b d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) x^4}{\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) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{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+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (3 b^3 (b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a (b c-a d)^4}+\frac {\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 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)^4}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{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+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (3 b^3 (b c-5 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^{3/2} (b c-a d)^4}+\frac {\left (3 b^3 (b c-5 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^{3/2} (b c-a d)^4}+\frac {\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 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)^4}+\frac {\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 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)^4}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{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+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac {\left (3 b^{5/2} (b c-5 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^{3/2} (b c-a d)^4}+\frac {\left (3 b^{5/2} (b c-5 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^{3/2} (b c-a d)^4}-\frac {\left (3 b^{11/4} (b c-5 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^{7/4} (b c-a d)^4}-\frac {\left (3 b^{11/4} (b c-5 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^{7/4} (b c-a d)^4}+\frac {\left (3 d^{3/2} \left (55 b^2 c^2-30 a b c d+7 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)^4}+\frac {\left (3 d^{3/2} \left (55 b^2 c^2-30 a b c d+7 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)^4}-\frac {\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}-\frac {\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{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+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}+\frac {\left (3 b^{11/4} (b c-5 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^{7/4} (b c-a d)^4}-\frac {\left (3 b^{11/4} (b c-5 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^{7/4} (b c-a d)^4}+\frac {\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}-\frac {\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}\\ &=\frac {d (2 b c+a d) \sqrt {x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac {b \sqrt {x}}{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+23 a b c d-7 a^2 d^2\right ) \sqrt {x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac {3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}-\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}+\frac {3 b^{11/4} (b c-5 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{7/4} (b c-a d)^4}-\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}+\frac {3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 3.20, size = 450, normalized size = 0.61 \begin {gather*} \frac {1}{64} \left (-\frac {4 \sqrt {x} \left (8 b^3 c^2 \left (c+d x^2\right )^2-a^3 d^3 \left (11 c+7 d x^2\right )+a b^2 c d^2 x^2 \left (27 c+23 d x^2\right )+a^2 b d^2 \left (27 c^2+12 c d x^2-7 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 {24 \sqrt {2} b^{11/4} (-b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{7/4} (b c-a d)^4}-\frac {3 \sqrt {2} d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}+\frac {24 \sqrt {2} b^{11/4} (b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{7/4} (b c-a d)^4}+\frac {3 \sqrt {2} d^{7/4} \left (55 b^2 c^2-30 a b c d+7 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)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*Sqrt[x]*(8*b^3*c^2*(c + d*x^2)^2 - a^3*d^3*(11*c + 7*d*x^2) + a*b^2*c*d^2*x^2*(27*c + 23*d*x^2) + a^2*b*d
^2*(27*c^2 + 12*c*d*x^2 - 7*d^2*x^4)))/(a*c^2*(-(b*c) + a*d)^3*(a + b*x^2)*(c + d*x^2)^2) + (24*Sqrt[2]*b^(11/
4)*(-(b*c) + 5*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(7/4)*(b*c - a*d)^4) -
(3*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*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)^4) + (24*Sqrt[2]*b^(11/4)*(b*c - 5*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(7/4)*(b*c - a*d)^4) + (3*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*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)^4))/64

________________________________________________________________________________________

Maple [A]
time = 0.19, size = 375, normalized size = 0.51

method	result	size
derivativedivides	\(-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 a \left (b \,x^{2}+a \right )}+\frac {3 \left (5 a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{2} \left (\frac {\frac {d \left (7 a^{2} d^{2}-30 a b c d +23 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-38 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (7 a^{2} d^{2}-30 a b c d +55 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{4}}\)	\(375\)
default	\(-\frac {2 b^{3} \left (\frac {\left (a d -b c \right ) \sqrt {x}}{4 a \left (b \,x^{2}+a \right )}+\frac {3 \left (5 a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{2} \left (\frac {\frac {d \left (7 a^{2} d^{2}-30 a b c d +23 b^{2} c^{2}\right ) x^{\frac {5}{2}}}{32 c^{2}}+\frac {\left (11 a^{2} d^{2}-38 a b c d +27 b^{2} c^{2}\right ) \sqrt {x}}{32 c}}{\left (d \,x^{2}+c \right )^{2}}+\frac {3 \left (7 a^{2} d^{2}-30 a b c d +55 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c^{3}}\right )}{\left (a d -b c \right )^{4}}\)	\(375\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^3/(a*d-b*c)^4*(1/4*(a*d-b*c)/a*x^(1/2)/(b*x^2+a)+3/32*(5*a*d-b*c)/a^2*(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*(7*a^2*d^2-30*a*b*c*d+23*b^2*c^2)
/c^2*x^(5/2)+1/32*(11*a^2*d^2-38*a*b*c*d+27*b^2*c^2)/c*x^(1/2))/(d*x^2+c)^2+3/256*(7*a^2*d^2-30*a*b*c*d+55*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)))

________________________________________________________________________________________

Maxima [A]
time = 0.54, size = 951, normalized size = 1.29 \begin {gather*} \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (b c - 5 \, a d\right )} \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} {\left (b c - 5 \, a d\right )} \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} {\left (b c - 5 \, a d\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b c - 5 \, a d\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )} b^{3}}{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 (8 \, b^{3} c^{2} d^{2} + 23 \, a b^{2} c d^{3} - 7 \, a^{2} b d^{4}\right )} x^{\frac {9}{2}} + {\left (16 \, b^{3} c^{3} d + 27 \, a b^{2} c^{2} d^{2} + 12 \, a^{2} b c d^{3} - 7 \, a^{3} d^{4}\right )} x^{\frac {5}{2}} + {\left (8 \, b^{3} c^{4} + 27 \, a^{2} b c^{2} d^{2} - 11 \, a^{3} c d^{3}\right )} \sqrt {x}}{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 )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} {\left (55 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 7 \, a^{2} d^{4}\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 (55 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 7 \, a^{2} d^{4}\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 (55 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 7 \, a^{2} d^{4}\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 (55 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 7 \, a^{2} d^{4}\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}}}\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*(2*sqrt(2)*(b*c - 5*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sq
rt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b*c - 5*a*d)*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*c - 5*a*d)*log(sqrt(
2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b*c - 5*a*d)*log(-sqrt(2)*a^(1/
4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))*b^3/(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) + 1/16*((8*b^3*c^2*d^2 + 23*a*b^2*c*d^3 - 7*a^2*b*d^4)*x^(9/2) + (16*b^3*c^3*d
+ 27*a*b^2*c^2*d^2 + 12*a^2*b*c*d^3 - 7*a^3*d^4)*x^(5/2) + (8*b^3*c^4 + 27*a^2*b*c^2*d^2 - 11*a^3*c*d^3)*sqrt(
x))/(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) + 3/
128*(2*sqrt(2)*(55*b^2*c^2*d^2 - 30*a*b*c*d^3 + 7*a^2*d^4)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqr
t(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(c)*sqrt(sqrt(c)*sqrt(d))) + 2*sqrt(2)*(55*b^2*c^2*d^2 - 30*a*b*c*d^
3 + 7*a^2*d^4)*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)*(55*b^2*c^2*d^2 - 30*a*b*c*d^3 + 7*a^2*d^4)*log(sqrt(2)*c^(1/4)*d^(1/4)*sq
rt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(55*b^2*c^2*d^2 - 30*a*b*c*d^3 + 7*a^2*d^4)*log(-sqrt
(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(3/4)*d^(1/4)))/(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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1253 vs. \(2 (583) = 1166\).
time = 2.59, size = 1253, normalized size = 1.70 \begin {gather*} \frac {b^{3} \sqrt {x}}{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 {3 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} 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 {3 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} 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 {3 \, {\left (55 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\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} - 4 \, \sqrt {2} a b^{3} c^{6} d + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{2} - 4 \, \sqrt {2} a^{3} b c^{4} d^{3} + \sqrt {2} a^{4} c^{3} d^{4}\right )}} + \frac {3 \, {\left (55 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\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} - 4 \, \sqrt {2} a b^{3} c^{6} d + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{2} - 4 \, \sqrt {2} a^{3} b c^{4} d^{3} + \sqrt {2} a^{4} c^{3} d^{4}\right )}} + \frac {3 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} 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 {3 \, {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c - 5 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} 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 {3 \, {\left (55 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\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} - 4 \, \sqrt {2} a b^{3} c^{6} d + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{2} - 4 \, \sqrt {2} a^{3} b c^{4} d^{3} + \sqrt {2} a^{4} c^{3} d^{4}\right )}} - \frac {3 \, {\left (55 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d - 30 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{2} + 7 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{3}\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} - 4 \, \sqrt {2} a b^{3} c^{6} d + 6 \, \sqrt {2} a^{2} b^{2} c^{5} d^{2} - 4 \, \sqrt {2} a^{3} b c^{4} d^{3} + \sqrt {2} a^{4} c^{3} d^{4}\right )}} + \frac {23 \, b c d^{3} x^{\frac {5}{2}} - 7 \, a d^{4} x^{\frac {5}{2}} + 27 \, b c^{2} d^{2} \sqrt {x} - 11 \, a c d^{3} \sqrt {x}}{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(1/(b*x^2+a)^2/(d*x^2+c)^3/x^(1/2),x, algorithm="giac")

[Out]

1/2*b^3*sqrt(x)/((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)) + 3/4*((a*b^3)^(1/4)*b^3
*c - 5*(a*b^3)^(1/4)*a*b^2*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) + 3/4*
((a*b^3)^(1/4)*b^3*c - 5*(a*b^3)^(1/4)*a*b^2*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) + 3/32*(55*(c*d^3)^(1/4)*b^2*c^2*d - 30*(c*d^3)^(1/4)*a*b*c*d^2 + 7*(c*d^3)^(1/4)*a^2*d^3)*arctan
(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^7 - 4*sqrt(2)*a*b^3*c^6*d + 6*sqrt(
2)*a^2*b^2*c^5*d^2 - 4*sqrt(2)*a^3*b*c^4*d^3 + sqrt(2)*a^4*c^3*d^4) + 3/32*(55*(c*d^3)^(1/4)*b^2*c^2*d - 30*(c
*d^3)^(1/4)*a*b*c*d^2 + 7*(c*d^3)^(1/4)*a^2*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(
1/4))/(sqrt(2)*b^4*c^7 - 4*sqrt(2)*a*b^3*c^6*d + 6*sqrt(2)*a^2*b^2*c^5*d^2 - 4*sqrt(2)*a^3*b*c^4*d^3 + sqrt(2)
*a^4*c^3*d^4) + 3/8*((a*b^3)^(1/4)*b^3*c - 5*(a*b^3)^(1/4)*a*b^2*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 + sq
rt(2)*a^6*d^4) - 3/8*((a*b^3)^(1/4)*b^3*c - 5*(a*b^3)^(1/4)*a*b^2*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sq
rt(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) + 3/64*(55*(c*d^3)^(1/4)*b^2*c^2*d - 30*(c*d^3)^(1/4)*a*b*c*d^2 + 7*(c*d^3)^(1/4)*a^2*d^3)*lo
g(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^7 - 4*sqrt(2)*a*b^3*c^6*d + 6*sqrt(2)*a^2*b^2*c^
5*d^2 - 4*sqrt(2)*a^3*b*c^4*d^3 + sqrt(2)*a^4*c^3*d^4) - 3/64*(55*(c*d^3)^(1/4)*b^2*c^2*d - 30*(c*d^3)^(1/4)*a
*b*c*d^2 + 7*(c*d^3)^(1/4)*a^2*d^3)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^7 - 4*sqr
t(2)*a*b^3*c^6*d + 6*sqrt(2)*a^2*b^2*c^5*d^2 - 4*sqrt(2)*a^3*b*c^4*d^3 + sqrt(2)*a^4*c^3*d^4) + 1/16*(23*b*c*d
^3*x^(5/2) - 7*a*d^4*x^(5/2) + 27*b*c^2*d^2*sqrt(x) - 11*a*c*d^3*sqrt(x))/((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)

________________________________________________________________________________________

Mupad [B]
time = 8.10, 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(1/(x^(1/2)*(a + b*x^2)^2*(c + d*x^2)^3),x)

[Out]

atan(-((((158640570309279744*a^62*d^62 + 461689330549653504*b^62*c^62 + 1143142782440942075904*a^2*b^60*c^60*d
^2 - 25023561715791219916800*a^3*b^59*c^59*d^3 + 392117365329126217482240*a^4*b^58*c^58*d^4 - 4690198490643886
824751104*a^5*b^57*c^57*d^5 + 44594910394380994297724928*a^6*b^56*c^56*d^6 - 346602278587137521765842944*a^7*b
^55*c^55*d^7 + 2247504424575830750669045760*a^8*b^54*c^54*d^8 - 12350275985199266166472704000*a^9*b^53*c^53*d^
9 + 58231240117103771404688424960*a^10*b^52*c^52*d^10 - 238022522313714176288222085120*a^11*b^51*c^51*d^11 + 8
51128269824272461500629647360*a^12*b^50*c^50*d^12 - 2685471663425998106604003655680*a^13*b^49*c^49*d^13 + 7544
170129817035367585352253440*a^14*b^48*c^48*d^14 - 19068074318507301366835150061568*a^15*b^47*c^47*d^15 + 43925
200681264313454548679131136*a^16*b^46*c^46*d^16 - 93701324613150775962838140715008*a^17*b^45*c^45*d^17 + 18846
4041806198255158575413329920*a^18*b^44*c^44*d^18 - 363482768390639298679139330949120*a^19*b^43*c^43*d^19 + 679
593524406433989867498790453248*a^20*b^42*c^42*d^20 - 1234226492432831870920084030488576*a^21*b^41*c^41*d^21 +
2166299333940469885543144979693568*a^22*b^40*c^40*d^22 - 3649880508285688517650264998543360*a^23*b^39*c^39*d^2
3 + 5882337238786870089625427666534400*a^24*b^38*c^38*d^24 - 9084025233921418993848385529708544*a^25*b^37*c^37
*d^25 + 13517918768320685624871901691117568*a^26*b^36*c^36*d^26 - 19498271125182229871738826673618944*a^27*b^3
5*c^35*d^27 + 27315046443069656705362624071598080*a^28*b^34*c^34*d^28 - 37015781040901615954658395768750080*a^
29*b^33*c^33*d^29 + 48092805215322280459690440055062528*a^30*b^32*c^32*d^30 - 59264887465626927586633770646634
496*a^31*b^31*c^31*d^31 + 68586599768084153161669916447735808*a^32*b^30*c^30*d^32 - 73974197164791541927858637
824327680*a^33*b^29*c^29*d^33 + 73965997892283818508917976575508480*a^34*b^28*c^28*d^34 - 68335704761988738252
796495977775104*a^35*b^27*c^27*d^35 + 58219427824782390172272112611360768*a^36*b^26*c^26*d^36 - 45688108560967
442735282995681296384*a^37*b^25*c^25*d^37 + 33004306099634531959911507013140480*a^38*b^24*c^24*d^38 - 21937255
814019282279521941129789440*a^39*b^23*c^23*d^39 + 13411283618120781029280868454105088*a^40*b^22*c^22*d^40 - 75
37663576430440382672512877592576*a^41*b^21*c^21*d^41 + 3892412049497521843004374964502528*a^42*b^20*c^20*d^42
- 1845284865146033724645937218846720*a^43*b^19*c^19*d^43 + 802242695487291496905120122142720*a^44*b^18*c^18*d^
44 - 319410517078400510775218487164928*a^45*b^17*c^17*d^45 + 116263619225964311813956237787136*a^46*b^16*c^16*
d^46 - 38606608474448543697499060174848*a^47*b^15*c^15*d^47 + 11664498576526727219629743144960*a^48*b^14*c^14*
d^48 - 3196489115423809113423033139200*a^49*b^13*c^13*d^49 + 791409982329733215668467138560*a^50*b^12*c^12*d^5
0 - 176199485733388663821717995520*a^51*b^11*c^11*d^51 + 35073618030151357707960975360*a^52*b^10*c^10*d^52 - 6
197909674539500954745569280*a^53*b^9*c^9*d^53 + 963722299349432543100272640*a^54*b^8*c^8*d^54 - 13038398033557
1997403643904*a^55*b^7*c^7*d^55 + 15126732643705401196412928*a^56*b^6*c^6*d^56 - 1476009532413734912262144*a^5
7*b^5*c^5*d^57 + 117913206827103100600320*a^58*b^4*c^4*d^58 - 7412982469913298862080*a^59*b^3*c^3*d^59 + 34429
5363448368267264*a^60*b^2*c^2*d^60 - 33241631799575052288*a*b^61*c^61*d - 10515603517643685888*a^61*b*c*d^61)^
(1/2) - 398297088*a^31*d^31 - 679477248*b^31*c^31 - 400891576320*a^2*b^29*c^29*d^2 + 3981736673280*a^3*b^28*c^
28*d^3 - 26937875496960*a^4*b^27*c^27*d^4 + 132340424638464*a^5*b^26*c^26*d^5 - 491512097931264*a^6*b^25*c^25*
d^6 + 1416415142246400*a^7*b^24*c^24*d^7 - 3209681400053760*a^8*b^23*c^23*d^8 + 5685622110904320*a^9*b^22*c^22
*d^9 - 7454556262416384*a^10*b^21*c^21*d^10 + 5436179592966144*a^11*b^20*c^20*d^11 + 4665413760860160*a^12*b^1
9*c^19*d^12 - 26292873905971200*a^13*b^18*c^18*d^13 + 58696011926323200*a^14*b^17*c^17*d^14 - 9454494480583680
0*a^15*b^16*c^16*d^15 + 121670839126425600*a^16*b^15*c^15*d^16 - 129462901032960000*a^17*b^14*c^14*d^17 + 1155
61503891947520*a^18*b^13*c^13*d^18 - 87113445112995840*a^19*b^12*c^12*d^19 + 55609782114484224*a^20*b^11*c^11*
d^20 - 30067181023739904*a^21*b^10*c^10*d^21 + 13742000583966720*a^22*b^9*c^9*d^22 - 5286598571980800*a^23*b^8
*c^8*d^23 + 1699967106662400*a^24*b^7*c^7*d^24 - 452124225183744*a^25*b^6*c^6*d^25 + 97916547907584*a^26*b^5*c
^5*d^26 - 16871335464960*a^27*b^4*c^4*d^27 + 2231346216960*a^28*b^3*c^3*d^28 - 213454725120*a^29*b^2*c^2*d^29
+ 24461180928*a*b^30*c^30*d + 13200703488*a^30*b*c*d^30)/(68719476736*a^7*b^32*c^43 + 68719476736*a^39*c^11*d^
32 - 2199023255552*a^8*b^31*c^42*d - 2199023255552*a^38*b*c^12*d^31 + 34084860461056*a^9*b^30*c^41*d^2 - 34084
8604610560*a^10*b^29*c^40*d^3 + 2471152383426560*a^11*b^28*c^39*d^4 - 13838453347188736*a^12*b^27*c^38*d^5 + 6
2273040062349312*a^13*b^26*c^37*d^6 - 231299863088726016*a^14*b^25*c^36*d^7 + 722812072152268800*a^15*b^24*c^3
5*d^8 - 1927498859072716800*a^16*b^23*c^34*d^9 + 4433247375867248640*a^17*b^22*c^33*d^10 - 8866494751734497280
*a^18*b^21*c^32*d^11 + 15516365815535370240*a^1...

________________________________________________________________________________________