3.6.2 \(\int \frac {1}{x^{5/2} (a+b x^2)^2 (c+d x^2)^3} \, dx\) [502]

Optimal. Leaf size=805 \[ \frac {-56 b^3 c^3+96 a b^2 c^2 d-189 a^2 b c d^2+77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} (7 b c-19 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {b^{15/4} (7 b c-19 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^4}+\frac {b^{15/4} (7 b c-19 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {b^{15/4} (7 b c-19 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right ) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^4} \]

[Out]

1/48*(77*a^3*d^3-189*a^2*b*c*d^2+96*a*b^2*c^2*d-56*b^3*c^3)/a^2/c^3/(-a*d+b*c)^3/x^(3/2)+1/4*d*(a*d+2*b*c)/a/c
/(-a*d+b*c)^2/x^(3/2)/(d*x^2+c)^2+1/2*b/a/(-a*d+b*c)/x^(3/2)/(b*x^2+a)/(d*x^2+c)^2+1/16*d*(-11*a^2*d^2+27*a*b*
c*d+8*b^2*c^2)/a/c^2/(-a*d+b*c)^3/x^(3/2)/(d*x^2+c)+1/8*b^(15/4)*(-19*a*d+7*b*c)*arctan(1-b^(1/4)*2^(1/2)*x^(1
/2)/a^(1/4))/a^(11/4)/(-a*d+b*c)^4*2^(1/2)-1/8*b^(15/4)*(-19*a*d+7*b*c)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/
4))/a^(11/4)/(-a*d+b*c)^4*2^(1/2)+1/64*d^(11/4)*(77*a^2*d^2-266*a*b*c*d+285*b^2*c^2)*arctan(1-d^(1/4)*2^(1/2)*
x^(1/2)/c^(1/4))/c^(15/4)/(-a*d+b*c)^4*2^(1/2)-1/64*d^(11/4)*(77*a^2*d^2-266*a*b*c*d+285*b^2*c^2)*arctan(1+d^(
1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(15/4)/(-a*d+b*c)^4*2^(1/2)+1/16*b^(15/4)*(-19*a*d+7*b*c)*ln(a^(1/2)+x*b^(1/2)
-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/(-a*d+b*c)^4*2^(1/2)-1/16*b^(15/4)*(-19*a*d+7*b*c)*ln(a^(1/2)+x*b^(
1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(11/4)/(-a*d+b*c)^4*2^(1/2)+1/128*d^(11/4)*(77*a^2*d^2-266*a*b*c*d+285
*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^(15/4)/(-a*d+b*c)^4*2^(1/2)-1/128*d^(11/4)*(
77*a^2*d^2-266*a*b*c*d+285*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^(15/4)/(-a*d+b*c)^
4*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.90, antiderivative size = 805, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {477, 483, 593, 597, 536, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {(7 b c-19 a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ) b^{15/4}}{4 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {(7 b c-19 a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right ) b^{15/4}}{4 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {(7 b c-19 a d) \log \left (\sqrt {b} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}\right ) b^{15/4}}{8 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {(7 b c-19 a d) \log \left (\sqrt {b} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}\right ) b^{15/4}}{8 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {b}{2 a (b c-a d) x^{3/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt {2} c^{15/4} (b c-a d)^4}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \log \left (\sqrt {d} x-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \log \left (\sqrt {d} x+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}\right )}{64 \sqrt {2} c^{15/4} (b c-a d)^4}+\frac {d \left (8 b^2 c^2+27 a b d c-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (d x^2+c\right )}-\frac {56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (d x^2+c\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

-1/48*(56*b^3*c^3 - 96*a*b^2*c^2*d + 189*a^2*b*c*d^2 - 77*a^3*d^3)/(a^2*c^3*(b*c - a*d)^3*x^(3/2)) + (d*(2*b*c
+ a*d))/(4*a*c*(b*c - a*d)^2*x^(3/2)*(c + d*x^2)^2) + b/(2*a*(b*c - a*d)*x^(3/2)*(a + b*x^2)*(c + d*x^2)^2) +
(d*(8*b^2*c^2 + 27*a*b*c*d - 11*a^2*d^2))/(16*a*c^2*(b*c - a*d)^3*x^(3/2)*(c + d*x^2)) + (b^(15/4)*(7*b*c - 1
9*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) - (b^(15/4)*(7*b*c -
19*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) + (d^(11/4)*(285*b^2
*c^2 - 266*a*b*c*d + 77*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(15/4)*(b*c - a*
d)^4) - (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*
Sqrt[2]*c^(15/4)*(b*c - a*d)^4) + (b^(15/4)*(7*b*c - 19*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + S
qrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) - (b^(15/4)*(7*b*c - 19*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/
4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(11/4)*(b*c - a*d)^4) + (d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^
2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(15/4)*(b*c - a*d)^4) - (d^(11/4)
*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt
[2]*c^(15/4)*(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 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 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 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 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -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}{x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^4 \left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt {x}\right )\\ &=\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {-7 b c+4 a d-15 b d x^4}{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)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac {\text {Subst}\left (\int \frac {-4 \left (14 b^2 c^2-16 a b c d+11 a^2 d^2\right )-44 b d (2 b c+a d) x^4}{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)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {-4 \left (56 b^3 c^3-96 a b^2 c^2 d+189 a^2 b c d^2-77 a^3 d^3\right )-28 b d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right ) x^4}{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 {56 b^3 c^3-96 a b^2 c^2 d+189 a^2 b c d^2-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {-12 \left (56 b^4 c^4-96 a b^3 c^3 d-96 a^2 b^2 c^2 d^2+189 a^3 b c d^3-77 a^4 d^4\right )-12 b d \left (56 b^3 c^3-96 a b^2 c^2 d+189 a^2 b c d^2-77 a^3 d^3\right ) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{192 a^2 c^3 (b c-a d)^3}\\ &=-\frac {56 b^3 c^3-96 a b^2 c^2 d+189 a^2 b c d^2-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}-\frac {\left (b^4 (7 b c-19 a d)\right ) \text {Subst}\left (\int \frac {1}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 a^2 (b c-a d)^4}-\frac {\left (d^3 \left (285 b^2 c^2-266 a b c d+77 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{c+d x^4} \, dx,x,\sqrt {x}\right )}{16 c^3 (b c-a d)^4}\\ &=-\frac {56 b^3 c^3-96 a b^2 c^2 d+189 a^2 b c d^2-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}-\frac {\left (b^4 (7 b c-19 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^{5/2} (b c-a d)^4}-\frac {\left (b^4 (7 b c-19 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^{5/2} (b c-a d)^4}-\frac {\left (d^3 \left (285 b^2 c^2-266 a b c d+77 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^{7/2} (b c-a d)^4}-\frac {\left (d^3 \left (285 b^2 c^2-266 a b c d+77 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^{7/2} (b c-a d)^4}\\ &=-\frac {56 b^3 c^3-96 a b^2 c^2 d+189 a^2 b c d^2-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}-\frac {\left (b^{7/2} (7 b c-19 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^{5/2} (b c-a d)^4}-\frac {\left (b^{7/2} (7 b c-19 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^{5/2} (b c-a d)^4}+\frac {\left (b^{15/4} (7 b c-19 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^{11/4} (b c-a d)^4}+\frac {\left (b^{15/4} (7 b c-19 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^{11/4} (b c-a d)^4}-\frac {\left (d^{5/2} \left (285 b^2 c^2-266 a b c d+77 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^{7/2} (b c-a d)^4}-\frac {\left (d^{5/2} \left (285 b^2 c^2-266 a b c d+77 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^{7/2} (b c-a d)^4}+\frac {\left (d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}+\frac {\left (d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}\\ &=-\frac {56 b^3 c^3-96 a b^2 c^2 d+189 a^2 b c d^2-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} (7 b c-19 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {b^{15/4} (7 b c-19 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}-\frac {\left (b^{15/4} (7 b c-19 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^{11/4} (b c-a d)^4}+\frac {\left (b^{15/4} (7 b c-19 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^{11/4} (b c-a d)^4}-\frac {\left (d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}+\frac {\left (d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}\\ &=-\frac {56 b^3 c^3-96 a b^2 c^2 d+189 a^2 b c d^2-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac {d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (c+d x^2\right )^2}+\frac {b}{2 a (b c-a d) x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {d \left (8 b^2 c^2+27 a b c d-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (c+d x^2\right )}+\frac {b^{15/4} (7 b c-19 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {b^{15/4} (7 b c-19 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}+\frac {b^{15/4} (7 b c-19 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} (b c-a d)^4}-\frac {b^{15/4} (7 b c-19 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{11/4} (b c-a d)^4}+\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}-\frac {d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}\\ \end {align*}

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Mathematica [A]
time = 6.12, size = 521, normalized size = 0.65 \begin {gather*} \frac {1}{192} \left (-\frac {4 \left (-56 b^4 c^3 x^2 \left (c+d x^2\right )^2-32 a b^3 c^2 \left (c-3 d x^2\right ) \left (c+d x^2\right )^2+a^4 d^3 \left (32 c^2+121 c d x^2+77 d^2 x^4\right )+3 a^2 b^2 c d \left (32 c^3+32 c^2 d x^2-67 c d^2 x^4-63 d^3 x^6\right )+a^3 b d^2 \left (-96 c^3-265 c^2 d x^2-68 c d^2 x^4+77 d^3 x^6\right )\right )}{a^2 c^3 (-b c+a d)^3 x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac {24 \sqrt {2} b^{15/4} (7 b c-19 a d) \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{11/4} (b c-a d)^4}+\frac {3 \sqrt {2} d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}+\frac {24 \sqrt {2} b^{15/4} (-7 b c+19 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{11/4} (b c-a d)^4}-\frac {3 \sqrt {2} d^{11/4} \left (285 b^2 c^2-266 a b c d+77 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^{15/4} (b c-a d)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3),x]

[Out]

((-4*(-56*b^4*c^3*x^2*(c + d*x^2)^2 - 32*a*b^3*c^2*(c - 3*d*x^2)*(c + d*x^2)^2 + a^4*d^3*(32*c^2 + 121*c*d*x^2
+ 77*d^2*x^4) + 3*a^2*b^2*c*d*(32*c^3 + 32*c^2*d*x^2 - 67*c*d^2*x^4 - 63*d^3*x^6) + a^3*b*d^2*(-96*c^3 - 265*
c^2*d*x^2 - 68*c*d^2*x^4 + 77*d^3*x^6)))/(a^2*c^3*(-(b*c) + a*d)^3*x^(3/2)*(a + b*x^2)*(c + d*x^2)^2) + (24*Sq
rt[2]*b^(15/4)*(7*b*c - 19*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(11/4)*(b*
c - a*d)^4) + (3*Sqrt[2]*d^(11/4)*(285*b^2*c^2 - 266*a*b*c*d + 77*a^2*d^2)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[
2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(15/4)*(b*c - a*d)^4) + (24*Sqrt[2]*b^(15/4)*(-7*b*c + 19*a*d)*ArcTanh[(Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(11/4)*(b*c - a*d)^4) - (3*Sqrt[2]*d^(11/4)*(285*b^2*c^
2 - 266*a*b*c*d + 77*a^2*d^2)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(15/4)*(b*c
- a*d)^4))/192

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Maple [A]
time = 0.35, size = 385, normalized size = 0.48

method	result	size
derivativedivides	\(\frac {2 b^{4} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (19 a d -7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} \left (a d -b c \right )^{4}}-\frac {2 d^{3} \left (\frac {\left (\frac {15}{32} a^{2} d^{3}-\frac {23}{16} a b c \,d^{2}+\frac {31}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-54 a b c d +35 b^{2} c^{2}\right ) \sqrt {x}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-266 a b c d +285 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c}\right )}{c^{3} \left (a d -b c \right )^{4}}-\frac {2}{3 a^{2} c^{3} x^{\frac {3}{2}}}\)	\(385\)
default	\(\frac {2 b^{4} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) \sqrt {x}}{b \,x^{2}+a}+\frac {\left (19 a d -7 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{a^{2} \left (a d -b c \right )^{4}}-\frac {2 d^{3} \left (\frac {\left (\frac {15}{32} a^{2} d^{3}-\frac {23}{16} a b c \,d^{2}+\frac {31}{32} b^{2} c^{2} d \right ) x^{\frac {5}{2}}+\frac {c \left (19 a^{2} d^{2}-54 a b c d +35 b^{2} c^{2}\right ) \sqrt {x}}{32}}{\left (d \,x^{2}+c \right )^{2}}+\frac {\left (77 a^{2} d^{2}-266 a b c d +285 b^{2} c^{2}\right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{256 c}\right )}{c^{3} \left (a d -b c \right )^{4}}-\frac {2}{3 a^{2} c^{3} x^{\frac {3}{2}}}\)	\(385\)
risch	\(\text {Expression too large to display}\)	\(1143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

2*b^4/a^2/(a*d-b*c)^4*((1/4*a*d-1/4*b*c)*x^(1/2)/(b*x^2+a)+1/32*(19*a*d-7*b*c)*(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^3/c^3/(a*d-b*c)^4*(((15/32*a^2*d^3-23/16*a*b*c*d^2
+31/32*b^2*c^2*d)*x^(5/2)+1/32*c*(19*a^2*d^2-54*a*b*c*d+35*b^2*c^2)*x^(1/2))/(d*x^2+c)^2+1/256*(77*a^2*d^2-266
*a*b*c*d+285*b^2*c^2)*(c/d)^(1/4)/c*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)))-2
/3/a^2/c^3/x^(3/2)

________________________________________________________________________________________

Maxima [A]
time = 0.54, size = 1064, normalized size = 1.32 \begin {gather*} -\frac {{\left (\frac {2 \, \sqrt {2} {\left (7 \, b c - 19 \, 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 (7 \, b c - 19 \, 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 (7 \, b c - 19 \, 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 (7 \, b c - 19 \, 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^{4}}{16 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4}\right )}} - \frac {32 \, a b^{3} c^{5} - 96 \, a^{2} b^{2} c^{4} d + 96 \, a^{3} b c^{3} d^{2} - 32 \, a^{4} c^{2} d^{3} + {\left (56 \, b^{4} c^{3} d^{2} - 96 \, a b^{3} c^{2} d^{3} + 189 \, a^{2} b^{2} c d^{4} - 77 \, a^{3} b d^{5}\right )} x^{6} + {\left (112 \, b^{4} c^{4} d - 160 \, a b^{3} c^{3} d^{2} + 201 \, a^{2} b^{2} c^{2} d^{3} + 68 \, a^{3} b c d^{4} - 77 \, a^{4} d^{5}\right )} x^{4} + {\left (56 \, b^{4} c^{5} - 32 \, a b^{3} c^{4} d - 96 \, a^{2} b^{2} c^{3} d^{2} + 265 \, a^{3} b c^{2} d^{3} - 121 \, a^{4} c d^{4}\right )} x^{2}}{48 \, {\left ({\left (a^{2} b^{4} c^{6} d^{2} - 3 \, a^{3} b^{3} c^{5} d^{3} + 3 \, a^{4} b^{2} c^{4} d^{4} - a^{5} b c^{3} d^{5}\right )} x^{\frac {15}{2}} + {\left (2 \, a^{2} b^{4} c^{7} d - 5 \, a^{3} b^{3} c^{6} d^{2} + 3 \, a^{4} b^{2} c^{5} d^{3} + a^{5} b c^{4} d^{4} - a^{6} c^{3} d^{5}\right )} x^{\frac {11}{2}} + {\left (a^{2} b^{4} c^{8} - a^{3} b^{3} c^{7} d - 3 \, a^{4} b^{2} c^{6} d^{2} + 5 \, a^{5} b c^{5} d^{3} - 2 \, a^{6} c^{4} d^{4}\right )} x^{\frac {7}{2}} + {\left (a^{3} b^{3} c^{8} - 3 \, a^{4} b^{2} c^{7} d + 3 \, a^{5} b c^{6} d^{2} - a^{6} c^{5} d^{3}\right )} x^{\frac {3}{2}}\right )}} - \frac {\frac {2 \, \sqrt {2} {\left (285 \, b^{2} c^{2} d^{3} - 266 \, a b c d^{4} + 77 \, a^{2} d^{5}\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 (285 \, b^{2} c^{2} d^{3} - 266 \, a b c d^{4} + 77 \, a^{2} d^{5}\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 (285 \, b^{2} c^{2} d^{3} - 266 \, a b c d^{4} + 77 \, a^{2} d^{5}\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 (285 \, b^{2} c^{2} d^{3} - 266 \, a b c d^{4} + 77 \, a^{2} d^{5}\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^{4} c^{7} - 4 \, a b^{3} c^{6} d + 6 \, a^{2} b^{2} c^{5} d^{2} - 4 \, a^{3} b c^{4} d^{3} + a^{4} c^{3} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="maxima")

[Out]

-1/16*(2*sqrt(2)*(7*b*c - 19*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))) + 2*sqrt(2)*(7*b*c - 19*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)*(7*b*c - 19*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)*(7*b*c - 19*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^4/(a^2*b^4*c^4 - 4*a^3*b^3*c^3*d +
6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4) - 1/48*(32*a*b^3*c^5 - 96*a^2*b^2*c^4*d + 96*a^3*b*c^3*d^2 - 32*
a^4*c^2*d^3 + (56*b^4*c^3*d^2 - 96*a*b^3*c^2*d^3 + 189*a^2*b^2*c*d^4 - 77*a^3*b*d^5)*x^6 + (112*b^4*c^4*d - 16
0*a*b^3*c^3*d^2 + 201*a^2*b^2*c^2*d^3 + 68*a^3*b*c*d^4 - 77*a^4*d^5)*x^4 + (56*b^4*c^5 - 32*a*b^3*c^4*d - 96*a
^2*b^2*c^3*d^2 + 265*a^3*b*c^2*d^3 - 121*a^4*c*d^4)*x^2)/((a^2*b^4*c^6*d^2 - 3*a^3*b^3*c^5*d^3 + 3*a^4*b^2*c^4
*d^4 - a^5*b*c^3*d^5)*x^(15/2) + (2*a^2*b^4*c^7*d - 5*a^3*b^3*c^6*d^2 + 3*a^4*b^2*c^5*d^3 + a^5*b*c^4*d^4 - a^
6*c^3*d^5)*x^(11/2) + (a^2*b^4*c^8 - a^3*b^3*c^7*d - 3*a^4*b^2*c^6*d^2 + 5*a^5*b*c^5*d^3 - 2*a^6*c^4*d^4)*x^(7
/2) + (a^3*b^3*c^8 - 3*a^4*b^2*c^7*d + 3*a^5*b*c^6*d^2 - a^6*c^5*d^3)*x^(3/2)) - 1/128*(2*sqrt(2)*(285*b^2*c^2
*d^3 - 266*a*b*c*d^4 + 77*a^2*d^5)*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)*(285*b^2*c^2*d^3 - 266*a*b*c*d^4 + 77*a^2*d^5)*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)*(285*b^2*c^2*d^3 - 266*a*b*c*d^4 + 77*a^2*d^5)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x
+ sqrt(c))/(c^(3/4)*d^(1/4)) - sqrt(2)*(285*b^2*c^2*d^3 - 266*a*b*c*d^4 + 77*a^2*d^5)*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^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*
c^4*d^3 + a^4*c^3*d^4)

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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(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,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/x**(5/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [A]
time = 1.80, size = 1278, normalized size = 1.59 \begin {gather*} -\frac {b^{4} \sqrt {x}}{2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} {\left (b x^{2} + a\right )}} - \frac {{\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{4} c - 19 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{3} 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^{3} b^{4} c^{4} - 4 \, \sqrt {2} a^{4} b^{3} c^{3} d + 6 \, \sqrt {2} a^{5} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{6} b c d^{3} + \sqrt {2} a^{7} d^{4}\right )}} - \frac {{\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{4} c - 19 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{3} 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^{3} b^{4} c^{4} - 4 \, \sqrt {2} a^{4} b^{3} c^{3} d + 6 \, \sqrt {2} a^{5} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{6} b c d^{3} + \sqrt {2} a^{7} d^{4}\right )}} - \frac {{\left (285 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d^{2} - 266 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{3} + 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{4}\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^{8} - 4 \, \sqrt {2} a b^{3} c^{7} d + 6 \, \sqrt {2} a^{2} b^{2} c^{6} d^{2} - 4 \, \sqrt {2} a^{3} b c^{5} d^{3} + \sqrt {2} a^{4} c^{4} d^{4}\right )}} - \frac {{\left (285 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d^{2} - 266 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{3} + 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{4}\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^{8} - 4 \, \sqrt {2} a b^{3} c^{7} d + 6 \, \sqrt {2} a^{2} b^{2} c^{6} d^{2} - 4 \, \sqrt {2} a^{3} b c^{5} d^{3} + \sqrt {2} a^{4} c^{4} d^{4}\right )}} - \frac {{\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{4} c - 19 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{3} 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^{3} b^{4} c^{4} - 4 \, \sqrt {2} a^{4} b^{3} c^{3} d + 6 \, \sqrt {2} a^{5} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{6} b c d^{3} + \sqrt {2} a^{7} d^{4}\right )}} + \frac {{\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{4} c - 19 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{3} 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^{3} b^{4} c^{4} - 4 \, \sqrt {2} a^{4} b^{3} c^{3} d + 6 \, \sqrt {2} a^{5} b^{2} c^{2} d^{2} - 4 \, \sqrt {2} a^{6} b c d^{3} + \sqrt {2} a^{7} d^{4}\right )}} - \frac {{\left (285 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d^{2} - 266 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{3} + 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{4}\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^{8} - 4 \, \sqrt {2} a b^{3} c^{7} d + 6 \, \sqrt {2} a^{2} b^{2} c^{6} d^{2} - 4 \, \sqrt {2} a^{3} b c^{5} d^{3} + \sqrt {2} a^{4} c^{4} d^{4}\right )}} + \frac {{\left (285 \, \left (c d^{3}\right )^{\frac {1}{4}} b^{2} c^{2} d^{2} - 266 \, \left (c d^{3}\right )^{\frac {1}{4}} a b c d^{3} + 77 \, \left (c d^{3}\right )^{\frac {1}{4}} a^{2} d^{4}\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^{8} - 4 \, \sqrt {2} a b^{3} c^{7} d + 6 \, \sqrt {2} a^{2} b^{2} c^{6} d^{2} - 4 \, \sqrt {2} a^{3} b c^{5} d^{3} + \sqrt {2} a^{4} c^{4} d^{4}\right )}} - \frac {31 \, b c d^{4} x^{\frac {5}{2}} - 15 \, a d^{5} x^{\frac {5}{2}} + 35 \, b c^{2} d^{3} \sqrt {x} - 19 \, a c d^{4} \sqrt {x}}{16 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{2}} - \frac {2}{3 \, a^{2} c^{3} x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x, algorithm="giac")

[Out]

-1/2*b^4*sqrt(x)/((a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*(b*x^2 + a)) - 1/4*(7*(a*b^3)^(1/4
)*b^4*c - 19*(a*b^3)^(1/4)*a*b^3*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)
*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 - 4*sqrt(2)*a^6*b*c*d^3 + sqrt(2)*a^7*d^4)
- 1/4*(7*(a*b^3)^(1/4)*b^4*c - 19*(a*b^3)^(1/4)*a*b^3*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))
/(a/b)^(1/4))/(sqrt(2)*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 - 4*sqrt(2)*a^6*b*c*d
^3 + sqrt(2)*a^7*d^4) - 1/32*(285*(c*d^3)^(1/4)*b^2*c^2*d^2 - 266*(c*d^3)^(1/4)*a*b*c*d^3 + 77*(c*d^3)^(1/4)*a
^2*d^4)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^8 - 4*sqrt(2)*a*b^3*c
^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*a^3*b*c^5*d^3 + sqrt(2)*a^4*c^4*d^4) - 1/32*(285*(c*d^3)^(1/4)*b^
2*c^2*d^2 - 266*(c*d^3)^(1/4)*a*b*c*d^3 + 77*(c*d^3)^(1/4)*a^2*d^4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) -
2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^8 - 4*sqrt(2)*a*b^3*c^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*a^3*
b*c^5*d^3 + sqrt(2)*a^4*c^4*d^4) - 1/8*(7*(a*b^3)^(1/4)*b^4*c - 19*(a*b^3)^(1/4)*a*b^3*d)*log(sqrt(2)*sqrt(x)*
(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 - 4*sq
rt(2)*a^6*b*c*d^3 + sqrt(2)*a^7*d^4) + 1/8*(7*(a*b^3)^(1/4)*b^4*c - 19*(a*b^3)^(1/4)*a*b^3*d)*log(-sqrt(2)*sqr
t(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^4*c^4 - 4*sqrt(2)*a^4*b^3*c^3*d + 6*sqrt(2)*a^5*b^2*c^2*d^2 -
4*sqrt(2)*a^6*b*c*d^3 + sqrt(2)*a^7*d^4) - 1/64*(285*(c*d^3)^(1/4)*b^2*c^2*d^2 - 266*(c*d^3)^(1/4)*a*b*c*d^3
+ 77*(c*d^3)^(1/4)*a^2*d^4)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^8 - 4*sqrt(2)*a*b^
3*c^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*a^3*b*c^5*d^3 + sqrt(2)*a^4*c^4*d^4) + 1/64*(285*(c*d^3)^(1/4)
*b^2*c^2*d^2 - 266*(c*d^3)^(1/4)*a*b*c*d^3 + 77*(c*d^3)^(1/4)*a^2*d^4)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x +
sqrt(c/d))/(sqrt(2)*b^4*c^8 - 4*sqrt(2)*a*b^3*c^7*d + 6*sqrt(2)*a^2*b^2*c^6*d^2 - 4*sqrt(2)*a^3*b*c^5*d^3 + sq
rt(2)*a^4*c^4*d^4) - 1/16*(31*b*c*d^4*x^(5/2) - 15*a*d^5*x^(5/2) + 35*b*c^2*d^3*sqrt(x) - 19*a*c*d^4*sqrt(x))/
((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*(d*x^2 + c)^2) - 2/3/(a^2*c^3*x^(3/2))

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Mupad [B]
time = 8.73, size = 2500, normalized size = 3.11 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b*x^2)^2*(c + d*x^2)^3),x)

[Out]

atan(((x^(1/2)*(857712418202478182400*a^18*b^48*c^62*d^11 - 28925330217666430894080*a^19*b^47*c^61*d^12 + 4658
08355868544602210304*a^20*b^46*c^60*d^13 - 4772189938359453553262592*a^21*b^45*c^59*d^14 + 3498207652982623340
1212928*a^22*b^44*c^58*d^15 - 195811106815542077297786880*a^23*b^43*c^57*d^16 + 873231122236416493313064960*a^
24*b^42*c^56*d^17 - 3201588318340888739356606464*a^25*b^41*c^55*d^18 + 9904866981547362725832687616*a^26*b^40*
c^54*d^19 - 26475613142538536817178705920*a^27*b^39*c^53*d^20 + 62528004036875405150857986048*a^28*b^38*c^52*d
^21 - 133143680796215491474489344000*a^29*b^37*c^51*d^22 + 259595474982835164713400139776*a^30*b^36*c^50*d^23
- 467106577738876991145070559232*a^31*b^35*c^49*d^24 + 775321096823109302674935250944*a^32*b^34*c^48*d^25 - 11
79424943892680059222782640128*a^33*b^33*c^47*d^26 + 1629690593600095833823295569920*a^34*b^32*c^46*d^27 - 2028
143345719314676074795761664*a^35*b^31*c^45*d^28 + 2257905973104023956972306956288*a^36*b^30*c^44*d^29 - 223744
9183565830435563494178816*a^37*b^29*c^43*d^30 + 1966204854457469918399988498432*a^38*b^28*c^42*d^31 - 15276494
06048366621262568488960*a^39*b^27*c^41*d^32 + 1046409458758522347995126562816*a^40*b^26*c^40*d^33 - 6299565235
92774331698776113152*a^41*b^25*c^39*d^34 + 332065764335584004230153764864*a^42*b^24*c^38*d^35 - 15254319696813
3650922715742208*a^43*b^23*c^37*d^36 + 60699171433471101739298979840*a^44*b^22*c^36*d^37 - 2075743669977239574
9793333248*a^45*b^21*c^35*d^38 + 6037825951797032255320227840*a^46*b^20*c^34*d^39 - 14734496390827154795124490
24*a^47*b^19*c^33*d^40 + 296084339424033093684559872*a^48*b^18*c^32*d^41 - 47717950421254308290887680*a^49*b^1
7*c^31*d^42 + 5931528400797457427988480*a^50*b^16*c^30*d^43 - 534037861185724002336768*a^51*b^15*c^29*d^44 + 3
1006369751209579905024*a^52*b^14*c^28*d^45 - 872067188534894657536*a^53*b^13*c^27*d^46) + (-(((143986855936*a^
35*d^35 + 40282095616*b^35*c^35 + 13612059983872*a^2*b^33*c^33*d^2 - 106752016121856*a^3*b^32*c^32*d^3 + 58564
4510281728*a^4*b^31*c^31*d^4 - 2390715430600704*a^5*b^30*c^30*d^5 + 7540414907154432*a^6*b^29*c^29*d^6 - 18829
534178574336*a^7*b^28*c^28*d^7 + 37834420899545088*a^8*b^27*c^27*d^8 - 61812801970110464*a^9*b^26*c^26*d^9 + 8
2612272492445696*a^10*b^25*c^25*d^10 - 90502742771167232*a^11*b^24*c^24*d^11 + 80709771031904256*a^12*b^23*c^2
3*d^12 - 54384137459908608*a^13*b^22*c^22*d^13 + 4937158577455104*a^14*b^21*c^21*d^14 + 112491276045524992*a^1
5*b^20*c^20*d^15 - 413241453930905600*a^16*b^19*c^19*d^16 + 1074443231596134400*a^17*b^18*c^18*d^17 - 22365714
58836070400*a^18*b^17*c^17*d^18 + 3832850809857372160*a^19*b^16*c^16*d^19 - 5481339136181731328*a^20*b^15*c^15
*d^20 + 6599213688440389632*a^21*b^14*c^14*d^21 - 6727518677746384896*a^22*b^13*c^13*d^22 + 582709154054548684
8*a^23*b^12*c^12*d^23 - 4293767561145810944*a^24*b^11*c^11*d^24 + 2689585093637472256*a^25*b^10*c^10*d^25 - 14
28045479666450432*a^26*b^9*c^9*d^26 + 639329497516732416*a^27*b^8*c^8*d^27 - 239385911340269568*a^28*b^7*c^7*d
^28 + 74080636676358144*a^29*b^6*c^6*d^29 - 18626082598846464*a^30*b^5*c^5*d^30 + 3711306051231744*a^31*b^4*c^
4*d^31 - 564292849139712*a^32*b^3*c^3*d^32 + 61554295914496*a^33*b^2*c^2*d^33 - 1081861996544*a*b^34*c^34*d -
4293426249728*a^34*b*c*d^34)^2/4 - (4581179456161*a^12*b^15*d^23 + 15840599000625*b^27*c^12*d^11 - 23112188256
1500*a*b^26*c^11*d^12 - 70054782497084*a^11*b^16*c*d^22 + 1442203904732850*a^2*b^25*c^10*d^13 - 50654279047121
40*a^3*b^24*c^9*d^14 + 11150130570636271*a^4*b^23*c^8*d^15 - 16316203958046776*a^5*b^22*c^7*d^16 + 16492413880
109692*a^6*b^21*c^6*d^17 - 11760839441437688*a^7*b^20*c^5*d^18 + 5941572716242975*a^8*b^19*c^4*d^19 - 20942069
29053932*a^9*b^18*c^3*d^20 + 492873253157362*a^10*b^17*c^2*d^21)*(68719476736*a^11*b^32*c^47 + 68719476736*a^4
3*c^15*d^32 - 2199023255552*a^12*b^31*c^46*d - 2199023255552*a^42*b*c^16*d^31 + 34084860461056*a^13*b^30*c^45*
d^2 - 340848604610560*a^14*b^29*c^44*d^3 + 2471152383426560*a^15*b^28*c^43*d^4 - 13838453347188736*a^16*b^27*c
^42*d^5 + 62273040062349312*a^17*b^26*c^41*d^6 - 231299863088726016*a^18*b^25*c^40*d^7 + 722812072152268800*a^
19*b^24*c^39*d^8 - 1927498859072716800*a^20*b^23*c^38*d^9 + 4433247375867248640*a^21*b^22*c^37*d^10 - 88664947
51734497280*a^22*b^21*c^36*d^11 + 15516365815535370240*a^23*b^20*c^35*d^12 - 23871332023900569600*a^24*b^19*c^
34*d^13 + 32396807746722201600*a^25*b^18*c^33*d^14 - 38876169296066641920*a^26*b^17*c^32*d^15 + 41305929877070
807040*a^27*b^16*c^31*d^16 - 38876169296066641920*a^28*b^15*c^30*d^17 + 32396807746722201600*a^29*b^14*c^29*d^
18 - 23871332023900569600*a^30*b^13*c^28*d^19 + 15516365815535370240*a^31*b^12*c^27*d^20 - 8866494751734497280
*a^32*b^11*c^26*d^21 + 4433247375867248640*a^33*b^10*c^25*d^22 - 1927498859072716800*a^34*b^9*c^24*d^23 + 7228
12072152268800*a^35*b^8*c^23*d^24 - 231299863088726016*a^36*b^7*c^22*d^25 + 62273040062349312*a^37*b^6*c^21*d^
26 - 13838453347188736*a^38*b^5*c^20*d^27 + 2471152383426560*a^39*b^4*c^19*d^28 - 340848604610560*a^40*b^3*c^1
8*d^29 + 34084860461056*a^41*b^2*c^17*d^30))^(1...

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