3.470 \(\int \frac{x^{11/2}}{(a+b x^2) (c+d x^2)^2} \, dx\)

Optimal. Leaf size=570 \[ \frac{a^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}+\frac{a^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{5/4} (b c-a d)^2}+\frac{c^{5/4} (5 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{\sqrt{x} (5 b c-4 a d)}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.84591, antiderivative size = 570, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {466, 470, 582, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac{a^{9/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}+\frac{a^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{5/4} (b c-a d)^2}+\frac{c^{5/4} (5 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{\sqrt{x} (5 b c-4 a d)}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d \left (c+d x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[x^(11/2)/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

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

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 470

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

Rule 582

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*d*(m + n*(p + q
+ 1) + 1)), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 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{x^{11/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^{12}}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (5 a c+(5 b c-4 a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{2 d (b c-a d)}\\ &=\frac{(5 b c-4 a d) \sqrt{x}}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{a c (5 b c-4 a d)+\left (5 b^2 c^2-4 a b c d-4 a^2 d^2\right ) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{2 b d^2 (b c-a d)}\\ &=\frac{(5 b c-4 a d) \sqrt{x}}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b (b c-a d)^2}-\frac{\left (c^2 (5 b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{2 d^2 (b c-a d)^2}\\ &=\frac{(5 b c-4 a d) \sqrt{x}}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac{a^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b (b c-a d)^2}-\frac{a^{5/2} \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b (b c-a d)^2}-\frac{\left (c^{3/2} (5 b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 d^2 (b c-a d)^2}-\frac{\left (c^{3/2} (5 b c-9 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{4 d^2 (b c-a d)^2}\\ &=\frac{(5 b c-4 a d) \sqrt{x}}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}-\frac{a^{5/2} \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 )}{2 b^{3/2} (b c-a d)^2}-\frac{a^{5/2} \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 )}{2 b^{3/2} (b c-a d)^2}+\frac{a^{9/4} \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 )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}+\frac{a^{9/4} \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 )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}-\frac{\left (c^{3/2} (5 b c-9 a d)\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 )}{8 d^{5/2} (b c-a d)^2}-\frac{\left (c^{3/2} (5 b c-9 a d)\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 )}{8 d^{5/2} (b c-a d)^2}+\frac{\left (c^{5/4} (5 b c-9 a d)\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 )}{8 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{\left (c^{5/4} (5 b c-9 a d)\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 )}{8 \sqrt{2} d^{9/4} (b c-a d)^2}\\ &=\frac{(5 b c-4 a d) \sqrt{x}}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}+\frac{a^{9/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}+\frac{c^{5/4} (5 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} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 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} d^{9/4} (b c-a d)^2}-\frac{a^{9/4} \operatorname{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} b^{5/4} (b c-a d)^2}+\frac{a^{9/4} \operatorname{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} b^{5/4} (b c-a d)^2}-\frac{\left (c^{5/4} (5 b c-9 a d)\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 )}{4 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{\left (c^{5/4} (5 b c-9 a d)\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 )}{4 \sqrt{2} d^{9/4} (b c-a d)^2}\\ &=\frac{(5 b c-4 a d) \sqrt{x}}{2 b d^2 (b c-a d)}-\frac{c x^{5/2}}{2 d (b c-a d) \left (c+d x^2\right )}+\frac{a^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{5/4} (b c-a d)^2}+\frac{c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 b c-9 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{9/4} (b c-a d)^2}+\frac{a^{9/4} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}-\frac{a^{9/4} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} b^{5/4} (b c-a d)^2}+\frac{c^{5/4} (5 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} d^{9/4} (b c-a d)^2}-\frac{c^{5/4} (5 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} d^{9/4} (b c-a d)^2}\\ \end{align*}

Mathematica [A]  time = 0.375928, size = 563, normalized size = 0.99 \[ \frac{4 \sqrt{2} a^{9/4} d^{9/4} \left (c+d x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-4 \sqrt{2} a^{9/4} d^{9/4} \left (c+d x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+8 \sqrt{2} a^{9/4} d^{9/4} \left (c+d x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-8 \sqrt{2} a^{9/4} d^{9/4} \left (c+d x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+\sqrt{2} b^{5/4} c^{5/4} \left (c+d x^2\right ) (5 b c-9 a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-\sqrt{2} b^{5/4} c^{5/4} \left (c+d x^2\right ) (5 b c-9 a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+2 \sqrt{2} b^{5/4} c^{5/4} \left (c+d x^2\right ) (5 b c-9 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )-2 \sqrt{2} b^{5/4} c^{5/4} \left (c+d x^2\right ) (5 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )+8 b^{5/4} c^2 \sqrt [4]{d} \sqrt{x} (b c-a d)+32 \sqrt [4]{b} \sqrt [4]{d} \sqrt{x} \left (c+d x^2\right ) (b c-a d)^2}{16 b^{5/4} d^{9/4} \left (c+d x^2\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(11/2)/((a + b*x^2)*(c + d*x^2)^2),x]

[Out]

(8*b^(5/4)*c^2*d^(1/4)*(b*c - a*d)*Sqrt[x] + 32*b^(1/4)*d^(1/4)*(b*c - a*d)^2*Sqrt[x]*(c + d*x^2) + 8*Sqrt[2]*
a^(9/4)*d^(9/4)*(c + d*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 8*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x
^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 2*Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*ArcT
an[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 2*Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*ArcTan[1 + (
Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 4*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x] - 4*Sqrt[2]*a^(9/4)*d^(9/4)*(c + d*x^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x] + Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x
] + Sqrt[d]*x] - Sqrt[2]*b^(5/4)*c^(5/4)*(5*b*c - 9*a*d)*(c + d*x^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqr
t[x] + Sqrt[d]*x])/(16*b^(5/4)*d^(9/4)*(b*c - a*d)^2*(c + d*x^2))

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Maple [A]  time = 0.016, size = 582, normalized size = 1. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(11/2)/(b*x^2+a)/(d*x^2+c)^2,x)

[Out]

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

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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(x^(11/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 161.814, size = 7129, normalized size = 12.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(11/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

1/8*(4*(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2
*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*
a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17
))^(1/4)*arctan(((b^6*c^6*d^7 - 6*a*b^5*c^5*d^8 + 15*a^2*b^4*c^4*d^9 - 20*a^3*b^3*c^3*d^10 + 15*a^4*b^2*c^2*d^
11 - 6*a^5*b*c*d^12 + a^6*d^13)*sqrt((25*b^2*c^4 - 90*a*b*c^3*d + 81*a^2*c^2*d^2)*x + (b^4*c^4*d^4 - 4*a*b^3*c
^3*d^5 + 6*a^2*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a^4*d^8)*sqrt(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^
7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3
*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17)))
*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^8*
d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^14
+ 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(3/4) + (5*b^7*c^8*d^7 - 39*a*b^6*c^7*d^8 + 129*a^2*b^5*c
^6*d^9 - 235*a^3*b^4*c^5*d^10 + 255*a^4*b^3*c^4*d^11 - 165*a^5*b^2*c^3*d^12 + 59*a^6*b*c^2*d^13 - 9*a^7*c*d^14
)*sqrt(x)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/
(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3
*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(3/4))/(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a
^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)) - 16*(-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*
c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7
+ a^8*b^5*d^8))^(1/4)*(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2)*arctan(((b^10*c^6 - 6*a*b^9*c^5*d
+ 15*a^2*b^8*c^4*d^2 - 20*a^3*b^7*c^3*d^3 + 15*a^4*b^6*c^2*d^4 - 6*a^5*b^5*c*d^5 + a^6*b^4*d^6)*(-a^9/(b^13*c
^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28
*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(3/4)*sqrt(a^4*x + (b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2
*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*sqrt(-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10
*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))) - (
a^2*b^10*c^6 - 6*a^3*b^9*c^5*d + 15*a^4*b^8*c^4*d^2 - 20*a^5*b^7*c^3*d^3 + 15*a^6*b^6*c^2*d^4 - 6*a^7*b^5*c*d^
5 + a^8*b^4*d^6)*(-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4
*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(3/4)*sqrt(x))/a^9) - 4*(-a^9
/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*
d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(1/4)*(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d
^4)*x^2)*log(a^2*sqrt(x) + (-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a
^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(1/4)*(b^3*c^2 - 2*
a*b^2*c*d + a^2*b*d^2)) + 4*(-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6*d^2 - 56*a^3*b^10*c^5*d^3 + 70*
a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 + a^8*b^5*d^8))^(1/4)*(b^2*c^2*d^2
- a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2)*log(a^2*sqrt(x) - (-a^9/(b^13*c^8 - 8*a*b^12*c^7*d + 28*a^2*b^11*c^6
*d^2 - 56*a^3*b^10*c^5*d^3 + 70*a^4*b^9*c^4*d^4 - 56*a^5*b^8*c^3*d^5 + 28*a^6*b^7*c^2*d^6 - 8*a^7*b^6*c*d^7 +
a^8*b^5*d^8))^(1/4)*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)) + (b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^
2)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^
8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^
14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(1/4)*log(-(5*b*c^2 - 9*a*c*d)*sqrt(x) + (b^2*c^2*d^2 -
2*a*b*c*d^3 + a^2*d^4)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 + 6561
*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^4*d^1
3 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(1/4)) - (b^2*c^2*d^2 - a*b*c*d^3
+ (b^2*c*d^3 - a*b*d^4)*x^2)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 - 14580*a^3*b*c^6*d^3 +
6561*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5*d^12 + 70*a^4*b^4*c^
4*d^13 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(1/4)*log(-(5*b*c^2 - 9*a*c*d
)*sqrt(x) - (b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(-(625*b^4*c^9 - 4500*a*b^3*c^8*d + 12150*a^2*b^2*c^7*d^2 -
14580*a^3*b*c^6*d^3 + 6561*a^4*c^5*d^4)/(b^8*c^8*d^9 - 8*a*b^7*c^7*d^10 + 28*a^2*b^6*c^6*d^11 - 56*a^3*b^5*c^5
*d^12 + 70*a^4*b^4*c^4*d^13 - 56*a^5*b^3*c^3*d^14 + 28*a^6*b^2*c^2*d^15 - 8*a^7*b*c*d^16 + a^8*d^17))^(1/4)) +
4*(5*b*c^2 - 4*a*c*d + 4*(b*c*d - a*d^2)*x^2)*sqrt(x))/(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2)

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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(x**(11/2)/(b*x**2+a)/(d*x**2+c)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.55658, size = 969, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(11/2)/(b*x^2+a)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

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