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

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

[Out]

-(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)) + (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 - 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) - (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*Sq
rt[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] + Sqr
t[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)

________________________________________________________________________________________

Rubi [A]  time = 1.32189, antiderivative size = 805, normalized size of antiderivative = 1., 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 = {466, 472, 579, 583, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac{(7 b c-19 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right ) b^{15/4}}{4 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{(7 b c-19 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right ) b^{15/4}}{4 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{(7 b c-19 a d) \log \left (\sqrt{b} x-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{15/4}}{8 \sqrt{2} a^{11/4} (b c-a d)^4}-\frac{(7 b c-19 a d) \log \left (\sqrt{b} x+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}\right ) b^{15/4}}{8 \sqrt{2} a^{11/4} (b c-a d)^4}+\frac{b}{2 a (b c-a d) x^{3/2} \left (b x^2+a\right ) \left (d x^2+c\right )^2}+\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \log \left (\sqrt{d} x-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^4}-\frac{d^{11/4} \left (285 b^2 c^2-266 a b d c+77 a^2 d^2\right ) \log \left (\sqrt{d} x+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}\right )}{64 \sqrt{2} c^{15/4} (b c-a d)^4}+\frac{d \left (8 b^2 c^2+27 a b d c-11 a^2 d^2\right )}{16 a c^2 (b c-a d)^3 x^{3/2} \left (d x^2+c\right )}-\frac{56 b^3 c^3-96 a b^2 d c^2+189 a^2 b d^2 c-77 a^3 d^3}{48 a^2 c^3 (b c-a d)^3 x^{3/2}}+\frac{d (2 b c+a d)}{4 a c (b c-a d)^2 x^{3/2} \left (d x^2+c\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)) + (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 - 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) - (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*Sq
rt[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] + Sqr
t[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 466

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 472

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 579

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 583

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 522

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 211

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 1165

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]

Rule 628

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 1162

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 617

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 204

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

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 \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 2.12416, size = 707, normalized size = 0.88 \[ \frac{1}{384} \left (\frac{3 \sqrt{2} d^{11/4} \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{15/4} (b c-a d)^4}-\frac{3 \sqrt{2} d^{11/4} \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{15/4} (b c-a d)^4}+\frac{6 \sqrt{2} d^{11/4} \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{15/4} (b c-a d)^4}-\frac{6 \sqrt{2} d^{11/4} \left (77 a^2 d^2-266 a b c d+285 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{15/4} (b c-a d)^4}+\frac{192 b^4 \sqrt{x}}{a^2 \left (a+b x^2\right ) (a d-b c)^3}+\frac{24 \sqrt{2} b^{15/4} (7 b c-19 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} (b c-a d)^4}+\frac{24 \sqrt{2} b^{15/4} (19 a d-7 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{11/4} (b c-a d)^4}+\frac{48 \sqrt{2} b^{15/4} (7 b c-19 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{11/4} (b c-a d)^4}+\frac{48 \sqrt{2} b^{15/4} (19 a d-7 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{11/4} (b c-a d)^4}-\frac{256}{a^2 c^3 x^{3/2}}+\frac{24 d^3 \sqrt{x} (15 a d-31 b c)}{c^3 \left (c+d x^2\right ) (b c-a d)^3}-\frac{96 d^3 \sqrt{x}}{c^2 \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-256/(a^2*c^3*x^(3/2)) + (192*b^4*Sqrt[x])/(a^2*(-(b*c) + a*d)^3*(a + b*x^2)) - (96*d^3*Sqrt[x])/(c^2*(b*c -
a*d)^2*(c + d*x^2)^2) + (24*d^3*(-31*b*c + 15*a*d)*Sqrt[x])/(c^3*(b*c - a*d)^3*(c + d*x^2)) + (48*Sqrt[2]*b^(1
5/4)*(7*b*c - 19*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*(b*c - a*d)^4) + (48*Sqrt[2]*b^
(15/4)*(-7*b*c + 19*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*(b*c - a*d)^4) + (6*Sqrt[2]*
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)])/(c^(15/4)*(b*
c - a*d)^4) - (6*Sqrt[2]*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)])/(c^(15/4)*(b*c - a*d)^4) + (24*Sqrt[2]*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])/(a^(11/4)*(b*c - a*d)^4) + (24*Sqrt[2]*b^(15/4)*(-7*b*c + 19*a*d)*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*b^(1/4)*Sqrt[x] + 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)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(15/4)*(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)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(c^(15/4)*(b*c - a*d)^4))/384

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 1143, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

-15/16*d^6/c^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*a^2+23/8*d^5/c^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*a*b-31/16*d^4/
c/(a*d-b*c)^4/(d*x^2+c)^2*x^(5/2)*b^2-19/16*d^5/c^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*a^2+27/8*d^4/c/(a*d-b*c)^4
/(d*x^2+c)^2*x^(1/2)*a*b-35/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(1/2)*b^2-77/64*d^5/c^4/(a*d-b*c)^4*(c/d)^(1/4)*2
^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2+133/32*d^4/c^3/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)
/(c/d)^(1/4)*x^(1/2)+1)*a*b-285/64*d^3/c^2/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+
1)*b^2-77/64*d^5/c^4/(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2+133/32*d^4/c^3/
(a*d-b*c)^4*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-285/64*d^3/c^2/(a*d-b*c)^4*(c/d)^(1/
4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2-77/128*d^5/c^4/(a*d-b*c)^4*(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)))*a^2+133/64*d^4/c^3/(a*d-b*c)
^4*(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)))*a*b-285/128*d^3/c^2/(a*d-b*c)^4*(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)))*b^2-2/3/a^2/c^3/x^(3/2)+1/2*b^4/a/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*d-1/2*b
^5/a^2/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*c+19/8*b^4/a^2/(a*d-b*c)^4*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(
1/4)*x^(1/2)-1)*d-7/8*b^5/a^3/(a*d-b*c)^4*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c+19/1
6*b^4/a^2/(a*d-b*c)^4*(1/b*a)^(1/4)*2^(1/2)*ln((x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x-(1/b*a)^(1/4
)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))*d-7/16*b^5/a^3/(a*d-b*c)^4*(1/b*a)^(1/4)*2^(1/2)*ln((x+(1/b*a)^(1/4)*x^(1/2)
*2^(1/2)+(1/b*a)^(1/2))/(x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))*c+19/8*b^4/a^2/(a*d-b*c)^4*(1/b*a)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*d-7/8*b^5/a^3/(a*d-b*c)^4*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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

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

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

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 = 2.24992, size = 1725, normalized size = 2.14 \begin{align*} \text{result too large to display} \end{align*}

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))