3.5.79 \(\int \frac {1}{x^{7/2} (a+b x^2) (c+d x^2)^2} \, dx\) [479]

Optimal. Leaf size=618 \[ -\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {b^{13/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2} \]

[Out]

1/10*(9*a*d-4*b*c)/a/c^2/(-a*d+b*c)/x^(5/2)-1/2*d/c/(-a*d+b*c)/x^(5/2)/(d*x^2+c)-1/2*b^(13/4)*arctan(1-b^(1/4)
*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a*d+b*c)^2*2^(1/2)+1/2*b^(13/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/
a^(9/4)/(-a*d+b*c)^2*2^(1/2)+1/8*d^(9/4)*(-9*a*d+13*b*c)*arctan(1-d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-
a*d+b*c)^2*2^(1/2)-1/8*d^(9/4)*(-9*a*d+13*b*c)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(13/4)/(-a*d+b*c)^2
*2^(1/2)+1/4*b^(13/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)^2*2^(1/2)-1/4*b
^(13/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)^2*2^(1/2)-1/16*d^(9/4)*(-9*a*
d+13*b*c)*ln(c^(1/2)+x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(13/4)/(-a*d+b*c)^2*2^(1/2)+1/16*d^(9/4)*(-9
*a*d+13*b*c)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(13/4)/(-a*d+b*c)^2*2^(1/2)+1/2*(-9*a^2*d
^2+4*a*b*c*d+4*b^2*c^2)/a^2/c^3/(-a*d+b*c)/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.65, antiderivative size = 618, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {477, 483, 597, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {b^{13/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}+\frac {-9 a^2 d^2+4 a b c d+4 b^2 c^2}{2 a^2 c^3 \sqrt {x} (b c-a d)}+\frac {d^{9/4} (13 b c-9 a d) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {4 b c-9 a d}{10 a c^2 x^{5/2} (b c-a d)}-\frac {d}{2 c x^{5/2} \left (c+d x^2\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

-1/10*(4*b*c - 9*a*d)/(a*c^2*(b*c - a*d)*x^(5/2)) + (4*b^2*c^2 + 4*a*b*c*d - 9*a^2*d^2)/(2*a^2*c^3*(b*c - a*d)
*Sqrt[x]) - d/(2*c*(b*c - a*d)*x^(5/2)*(c + d*x^2)) - (b^(13/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])
/(Sqrt[2]*a^(9/4)*(b*c - a*d)^2) + (b^(13/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(
b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(
b*c - a*d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(13/4)*(
b*c - a*d)^2) + (b^(13/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c
- a*d)^2) - (b^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*
d)^2) - (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(13
/4)*(b*c - a*d)^2) + (d^(9/4)*(13*b*c - 9*a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*
Sqrt[2]*c^(13/4)*(b*c - a*d)^2)

Rule 210

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

Rule 303

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

Rule 477

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

Rule 483

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

Rule 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 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 631

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

Rule 642

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

Rule 1176

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

Rule 1179

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

Rubi steps

\begin {align*} \int \frac {1}{x^{7/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=2 \text {Subst}\left (\int \frac {1}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {4 b c-9 a d-9 b d x^4}{x^6 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{2 c (b c-a d)}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {\text {Subst}\left (\int \frac {5 \left (4 b^2 c^2+4 a b c d-9 a^2 d^2\right )+5 b d (4 b c-9 a d) x^4}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{10 a c^2 (b c-a d)}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \frac {x^2 \left (5 \left (4 b^3 c^3+4 a b^2 c^2 d+4 a^2 b c d^2-9 a^3 d^3\right )+5 b d \left (4 b^2 c^2+4 a b c d-9 a^2 d^2\right ) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{10 a^2 c^3 (b c-a d)}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {\text {Subst}\left (\int \left (\frac {20 b^4 c^3 x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {5 a^2 d^3 (-13 b c+9 a d) x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{10 a^2 c^3 (b c-a d)}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^2}-\frac {\left (d^3 (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{2 c^3 (b c-a d)^2}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {b^{7/2} \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^2}+\frac {b^{7/2} \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a^2 (b c-a d)^2}+\frac {\left (d^{5/2} (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^3 (b c-a d)^2}-\frac {\left (d^{5/2} (13 b c-9 a d)\right ) \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{4 c^3 (b c-a d)^2}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {b^3 \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 a^2 (b c-a d)^2}+\frac {b^3 \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 a^2 (b c-a d)^2}+\frac {b^{13/4} \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^{9/4} (b c-a d)^2}+\frac {b^{13/4} \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^{9/4} (b c-a d)^2}-\frac {\left (d^2 (13 b c-9 a d)\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 )}{8 c^3 (b c-a d)^2}-\frac {\left (d^2 (13 b c-9 a d)\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 )}{8 c^3 (b c-a d)^2}-\frac {\left (d^{9/4} (13 b c-9 a d)\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 )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {\left (d^{9/4} (13 b c-9 a d)\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 )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}+\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {b^{13/4} \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^{9/4} (b c-a d)^2}-\frac {b^{13/4} \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^{9/4} (b c-a d)^2}-\frac {\left (d^{9/4} (13 b c-9 a d)\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 )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {\left (d^{9/4} (13 b c-9 a d)\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 )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}\\ &=-\frac {4 b c-9 a d}{10 a c^2 (b c-a d) x^{5/2}}+\frac {4 b^2 c^2+4 a b c d-9 a^2 d^2}{2 a^2 c^3 (b c-a d) \sqrt {x}}-\frac {d}{2 c (b c-a d) x^{5/2} \left (c+d x^2\right )}-\frac {b^{13/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {b^{13/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{4 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {b^{13/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {b^{13/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)^2}-\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}+\frac {d^{9/4} (13 b c-9 a d) \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{8 \sqrt {2} c^{13/4} (b c-a d)^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.28, size = 378, normalized size = 0.61 \begin {gather*} \frac {1}{40} \left (-\frac {4 \left (20 b^2 c^2 x^2 \left (c+d x^2\right )+a^2 d \left (4 c^2-36 c d x^2-45 d^2 x^4\right )-4 a b c \left (c^2-4 c d x^2-5 d^2 x^4\right )\right )}{a^2 c^3 (-b c+a d) x^{5/2} \left (c+d x^2\right )}-\frac {20 \sqrt {2} b^{13/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{a^{9/4} (b c-a d)^2}+\frac {5 \sqrt {2} d^{9/4} (13 b c-9 a d) \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{c^{13/4} (b c-a d)^2}-\frac {20 \sqrt {2} b^{13/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{a^{9/4} (b c-a d)^2}+\frac {5 \sqrt {2} d^{9/4} (13 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{c^{13/4} (b c-a d)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^2),x]

[Out]

((-4*(20*b^2*c^2*x^2*(c + d*x^2) + a^2*d*(4*c^2 - 36*c*d*x^2 - 45*d^2*x^4) - 4*a*b*c*(c^2 - 4*c*d*x^2 - 5*d^2*
x^4)))/(a^2*c^3*(-(b*c) + a*d)*x^(5/2)*(c + d*x^2)) - (20*Sqrt[2]*b^(13/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(a^(9/4)*(b*c - a*d)^2) + (5*Sqrt[2]*d^(9/4)*(13*b*c - 9*a*d)*ArcTan[(Sqrt[c] -
Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/(c^(13/4)*(b*c - a*d)^2) - (20*Sqrt[2]*b^(13/4)*ArcTanh[(Sqrt[2
]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(a^(9/4)*(b*c - a*d)^2) + (5*Sqrt[2]*d^(9/4)*(13*b*c - 9*a*
d)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/(c^(13/4)*(b*c - a*d)^2))/40

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 306, normalized size = 0.50

method	result	size
derivativedivides	\(\frac {b^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9 a d}{4}-\frac {13 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3} \left (a d -b c \right )^{2}}-\frac {2}{5 c^{2} a \,x^{\frac {5}{2}}}-\frac {2 \left (-2 a d -b c \right )}{a^{2} c^{3} \sqrt {x}}\)	\(306\)
default	\(\frac {b^{3} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {2 d^{3} \left (\frac {\left (\frac {a d}{4}-\frac {b c}{4}\right ) x^{\frac {3}{2}}}{d \,x^{2}+c}+\frac {\left (\frac {9 a d}{4}-\frac {13 b c}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{8 d \left (\frac {c}{d}\right )^{\frac {1}{4}}}\right )}{c^{3} \left (a d -b c \right )^{2}}-\frac {2}{5 c^{2} a \,x^{\frac {5}{2}}}-\frac {2 \left (-2 a d -b c \right )}{a^{2} c^{3} \sqrt {x}}\)	\(306\)
risch	\(-\frac {2 \left (-10 a d \,x^{2}-5 c \,x^{2} b +a c \right )}{5 a^{2} c^{3} x^{\frac {5}{2}}}+\frac {b^{3} \sqrt {2}\, \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 )}{4 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {b^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {b^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 a^{2} \left (a d -b c \right )^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {a \,d^{4} x^{\frac {3}{2}}}{2 c^{3} \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}-\frac {d^{3} x^{\frac {3}{2}} b}{2 c^{2} \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {9 a \,d^{3} \sqrt {2}\, \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 )}{16 c^{3} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {9 a \,d^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 c^{3} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {9 a \,d^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 c^{3} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {13 d^{2} \sqrt {2}\, b \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 )}{16 c^{2} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {13 d^{2} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{8 c^{2} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}-\frac {13 d^{2} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{8 c^{2} \left (a d -b c \right )^{2} \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(606\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^2,x,method=_RETURNVERBOSE)

[Out]

1/4*b^3/a^2/(a*d-b*c)^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^3/c^3/(a*d-b*c)^2*((1/4*a*d-1/4*b*c)*x^(3/2)/(d*x^2+c)+1/8*(9/4*a*d-13/4*b*c)/d/(c/d)^(1/4)*2^(1/2)*(ln((x-(
c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1
/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)))-2/5/c^2/a/x^(5/2)-2*(-2*a*d-b*c)/a^2/c^3/x^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.52, size = 551, normalized size = 0.89 \begin {gather*} \frac {b^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )}} - \frac {{\left (13 \, b c d^{3} - 9 \, a d^{4}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{16 \, {\left (b^{2} c^{5} - 2 \, a b c^{4} d + a^{2} c^{3} d^{2}\right )}} - \frac {4 \, a b c^{3} - 4 \, a^{2} c^{2} d - 5 \, {\left (4 \, b^{2} c^{2} d + 4 \, a b c d^{2} - 9 \, a^{2} d^{3}\right )} x^{4} - 4 \, {\left (5 \, b^{2} c^{3} + 4 \, a b c^{2} d - 9 \, a^{2} c d^{2}\right )} x^{2}}{10 \, {\left ({\left (a^{2} b c^{4} d - a^{3} c^{3} d^{2}\right )} x^{\frac {9}{2}} + {\left (a^{2} b c^{5} - a^{3} c^{4} d\right )} x^{\frac {5}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

1/4*b^4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sq
rt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqr
t(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x
+ sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^
(3/4)))/(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2) - 1/16*(13*b*c*d^3 - 9*a*d^4)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqr
t(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*a
rctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))
*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log
(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/(b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d
^2) - 1/10*(4*a*b*c^3 - 4*a^2*c^2*d - 5*(4*b^2*c^2*d + 4*a*b*c*d^2 - 9*a^2*d^3)*x^4 - 4*(5*b^2*c^3 + 4*a*b*c^2
*d - 9*a^2*c*d^2)*x^2)/((a^2*b*c^4*d - a^3*c^3*d^2)*x^(9/2) + (a^2*b*c^5 - a^3*c^4*d)*x^(5/2))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3728 vs. \(2 (467) = 934\).
time = 100.56, size = 3728, normalized size = 6.03 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/40*(20*((a^2*b*c^4*d - a^3*c^3*d^2)*x^5 + (a^2*b*c^5 - a^3*c^4*d)*x^3)*(-(28561*b^4*c^4*d^9 - 79092*a*b^3*c
^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b^6
*c^19*d^2 - 56*a^3*b^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^15*d^6 - 8*a^7*b*c^
14*d^7 + a^8*c^13*d^8))^(1/4)*arctan(((b^2*c^5 - 2*a*b*c^4*d + a^2*c^3*d^2)*sqrt((4826809*b^6*c^6*d^14 - 20049
822*a*b^5*c^5*d^15 + 34701615*a^2*b^4*c^4*d^16 - 32032260*a^3*b^3*c^3*d^17 + 16632135*a^4*b^2*c^2*d^18 - 46058
22*a^5*b*c*d^19 + 531441*a^6*d^20)*x - (28561*b^8*c^15*d^9 - 193336*a*b^7*c^14*d^10 + 569868*a^2*b^6*c^13*d^11
- 955240*a^3*b^5*c^12*d^12 + 995926*a^4*b^4*c^11*d^13 - 661320*a^5*b^3*c^10*d^14 + 273132*a^6*b^2*c^9*d^15 -
64152*a^7*b*c^8*d^16 + 6561*a^8*c^7*d^17)*sqrt(-(28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*
d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a^3*b^5*c^18*
d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^15*d^6 - 8*a^7*b*c^14*d^7 + a^8*c^13*d^8)))*(-(
28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)/(b^8*c
^21 - 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a^3*b^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 +
28*a^6*b^2*c^15*d^6 - 8*a^7*b*c^14*d^7 + a^8*c^13*d^8))^(1/4) + (2197*b^5*c^8*d^7 - 8957*a*b^4*c^7*d^8 + 1448
2*a^2*b^3*c^6*d^9 - 11610*a^3*b^2*c^5*d^10 + 4617*a^4*b*c^4*d^11 - 729*a^5*c^3*d^12)*sqrt(x)*(-(28561*b^4*c^4*
d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)/(b^8*c^21 - 8*a*b^7*
c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a^3*b^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^
15*d^6 - 8*a^7*b*c^14*d^7 + a^8*c^13*d^8))^(1/4))/(28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^
2*d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)) + 80*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^
2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a
^17*d^8))^(1/4)*((a^2*b*c^4*d - a^3*c^3*d^2)*x^5 + (a^2*b*c^5 - a^3*c^4*d)*x^3)*arctan((sqrt(b^20*x - (a^5*b^1
7*c^4 - 4*a^6*b^16*c^3*d + 6*a^7*b^15*c^2*d^2 - 4*a^8*b^14*c*d^3 + a^9*b^13*d^4)*sqrt(-b^13/(a^9*b^8*c^8 - 8*a
^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^1
5*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8)))*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56
*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^
8))^(1/4)*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2) - (a^2*b^12*c^2 - 2*a^3*b^11*c*d + a^4*b^10*d^2)*(-b^13/(a^9*b
^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*
d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(1/4)*sqrt(x))/b^13) - 20*(-b^13/(a^9*b^8*c^8 - 8*a^10
*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*b
^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(1/4)*((a^2*b*c^4*d - a^3*c^3*d^2)*x^5 + (a^2*b*c^5 - a^3*c^4*d)*x^3)
*log(b^10*sqrt(x) + (a^7*b^6*c^6 - 6*a^8*b^5*c^5*d + 15*a^9*b^4*c^4*d^2 - 20*a^10*b^3*c^3*d^3 + 15*a^11*b^2*c^
2*d^4 - 6*a^12*b*c*d^5 + a^13*d^6)*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*
c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(3/4))
+ 20*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d + 28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4
- 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 - 8*a^16*b*c*d^7 + a^17*d^8))^(1/4)*((a^2*b*c^4*d - a^3*c^3*d^2)*
x^5 + (a^2*b*c^5 - a^3*c^4*d)*x^3)*log(b^10*sqrt(x) - (a^7*b^6*c^6 - 6*a^8*b^5*c^5*d + 15*a^9*b^4*c^4*d^2 - 20
*a^10*b^3*c^3*d^3 + 15*a^11*b^2*c^2*d^4 - 6*a^12*b*c*d^5 + a^13*d^6)*(-b^13/(a^9*b^8*c^8 - 8*a^10*b^7*c^7*d +
28*a^11*b^6*c^6*d^2 - 56*a^12*b^5*c^5*d^3 + 70*a^13*b^4*c^4*d^4 - 56*a^14*b^3*c^3*d^5 + 28*a^15*b^2*c^2*d^6 -
8*a^16*b*c*d^7 + a^17*d^8))^(3/4)) - 5*((a^2*b*c^4*d - a^3*c^3*d^2)*x^5 + (a^2*b*c^5 - a^3*c^4*d)*x^3)*(-(2856
1*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)/(b^8*c^21
- 8*a*b^7*c^20*d + 28*a^2*b^6*c^19*d^2 - 56*a^3*b^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*
a^6*b^2*c^15*d^6 - 8*a^7*b*c^14*d^7 + a^8*c^13*d^8))^(1/4)*log((b^6*c^16 - 6*a*b^5*c^15*d + 15*a^2*b^4*c^14*d^
2 - 20*a^3*b^3*c^13*d^3 + 15*a^4*b^2*c^12*d^4 - 6*a^5*b*c^11*d^5 + a^6*c^10*d^6)*(-(28561*b^4*c^4*d^9 - 79092*
a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12 + 6561*a^4*d^13)/(b^8*c^21 - 8*a*b^7*c^20*d + 28*
a^2*b^6*c^19*d^2 - 56*a^3*b^5*c^18*d^3 + 70*a^4*b^4*c^17*d^4 - 56*a^5*b^3*c^16*d^5 + 28*a^6*b^2*c^15*d^6 - 8*a
^7*b*c^14*d^7 + a^8*c^13*d^8))^(3/4) - (2197*b^3*c^3*d^7 - 4563*a*b^2*c^2*d^8 + 3159*a^2*b*c*d^9 - 729*a^3*d^1
0)*sqrt(x)) + 5*((a^2*b*c^4*d - a^3*c^3*d^2)*x^...

________________________________________________________________________________________

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**(7/2)/(b*x**2+a)/(d*x**2+c)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 1.81, size = 715, normalized size = 1.16 \begin {gather*} -\frac {d^{3} x^{\frac {3}{2}}}{2 \, {\left (b c^{4} - a c^{3} d\right )} {\left (d x^{2} + c\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} b \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^{3} b^{2} c^{2} - 2 \, \sqrt {2} a^{4} b c d + \sqrt {2} a^{5} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\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 )}{4 \, {\left (\sqrt {2} b^{2} c^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\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 )}{4 \, {\left (\sqrt {2} b^{2} c^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} + \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} - \frac {{\left (13 \, \left (c d^{3}\right )^{\frac {3}{4}} b c - 9 \, \left (c d^{3}\right )^{\frac {3}{4}} a d\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{8 \, {\left (\sqrt {2} b^{2} c^{6} - 2 \, \sqrt {2} a b c^{5} d + \sqrt {2} a^{2} c^{4} d^{2}\right )}} + \frac {2 \, {\left (5 \, b c x^{2} + 10 \, a d x^{2} - a c\right )}}{5 \, a^{2} c^{3} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

-1/2*d^3*x^(3/2)/((b*c^4 - a*c^3*d)*(d*x^2 + c)) + (a*b^3)^(3/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2
*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) + (a*b^3)^(3/4)*b*arctan(
-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(
2)*a^5*d^2) - 1/2*(a*b^3)^(3/4)*b*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b^2*c^2 - 2*sq
rt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) + 1/2*(a*b^3)^(3/4)*b*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sq
rt(2)*a^3*b^2*c^2 - 2*sqrt(2)*a^4*b*c*d + sqrt(2)*a^5*d^2) - 1/4*(13*(c*d^3)^(3/4)*b*c - 9*(c*d^3)^(3/4)*a*d)*
arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^6 - 2*sqrt(2)*a*b*c^5*d + sqr
t(2)*a^2*c^4*d^2) - 1/4*(13*(c*d^3)^(3/4)*b*c - 9*(c*d^3)^(3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4)
- 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^2*c^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^4*d^2) + 1/8*(13*(c*d^3)^(3/4
)*b*c - 9*(c*d^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^6 - 2*sqrt(2)*a*b
*c^5*d + sqrt(2)*a^2*c^4*d^2) - 1/8*(13*(c*d^3)^(3/4)*b*c - 9*(c*d^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(c/d)^(1
/4) + x + sqrt(c/d))/(sqrt(2)*b^2*c^6 - 2*sqrt(2)*a*b*c^5*d + sqrt(2)*a^2*c^4*d^2) + 2/5*(5*b*c*x^2 + 10*a*d*x
^2 - a*c)/(a^2*c^3*x^(5/2))

________________________________________________________________________________________

Mupad [B]
time = 3.00, size = 2500, normalized size = 4.05 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)^2),x)

[Out]

- (2/(5*a*c) - (2*x^2*(9*a*d + 5*b*c))/(5*a^2*c^2) + (d*x^4*(4*b^2*c^2 - 9*a^2*d^2 + 4*a*b*c*d))/(2*a^2*c^3*(a
*d - b*c)))/(c*x^(5/2) + d*x^(9/2)) - 2*atan((524288*a^3*b^16*c^32*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^
9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 3
2768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5
*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(5/4) + 2654208*a^19*c^16*d^16*x^(1/2)*(-(6561*
a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c
^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^
4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(5/4) + 346112*b^15*
c^18*d^6*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*
a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3
*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20
*d))^(1/4) - 479232*a*b^14*c^17*d^7*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 8213
4*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^
2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*
c^15*d^6 - 32768*a*b^7*c^20*d))^(1/4) - 4194304*a^4*b^15*c^31*d*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 -
79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 3276
8*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^
3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(5/4) - 28901376*a^18*b*c^17*d^15*x^(1/2)*(-(6561*
a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c
^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^
4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(5/4) + 165888*a^2*b
^13*c^16*d^8*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37
908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376
*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*
c^20*d))^(1/4) + 3655808*a^3*b^12*c^15*d^9*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10
+ 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 11
4688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a
^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(1/4) - 10123776*a^4*b^11*c^14*d^10*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^
4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13
*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 22
9376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(1/4) + 10513152*a^5*b^10*c^13*d^11*x^(
1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12
)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^
3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(1/4) -
4852224*a^6*b^9*c^12*d^12*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2
*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^1
9*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6
- 32768*a*b^7*c^20*d))^(1/4) + 839808*a^7*b^8*c^11*d^13*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a
*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8 - 32768*a^7*b*
c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c^17*d^4 - 229376*a^5*b^3*c^16*d
^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(1/4) + 14680064*a^5*b^14*c^30*d^2*x^(1/2)*(-(6561*a^4*d^1
3 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a^3*b*c*d^12)/(4096*b^8*c^21 + 4
096*a^8*c^13*d^8 - 32768*a^7*b*c^14*d^7 + 114688*a^2*b^6*c^19*d^2 - 229376*a^3*b^5*c^18*d^3 + 286720*a^4*b^4*c
^17*d^4 - 229376*a^5*b^3*c^16*d^5 + 114688*a^6*b^2*c^15*d^6 - 32768*a*b^7*c^20*d))^(5/4) - 29360128*a^6*b^13*c
^29*d^3*x^(1/2)*(-(6561*a^4*d^13 + 28561*b^4*c^4*d^9 - 79092*a*b^3*c^3*d^10 + 82134*a^2*b^2*c^2*d^11 - 37908*a
^3*b*c*d^12)/(4096*b^8*c^21 + 4096*a^8*c^13*d^8...

________________________________________________________________________________________