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

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

[Out]

(d*(2*b*c + a*d)*Sqrt[x])/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*Sqrt[x])/(2*a*(b*c - a*d)*(a + b*x^2)*(c +
d*x^2)^2) + (d*(8*b^2*c^2 + 23*a*b*c*d - 7*a^2*d^2)*Sqrt[x])/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(11/4
)*(b*c - 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (3*b^(11/4)
*(b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(
55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c
- a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(3
2*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) - (3*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (3*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*L
og[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) + (3*d^(7/4)*(5
5*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^
(11/4)*(b*c - a*d)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.977757, antiderivative size = 739, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {466, 414, 527, 522, 211, 1165, 628, 1162, 617, 204} \[ \frac{d \sqrt{x} \left (-7 a^2 d^2+23 a b c d+8 b^2 c^2\right )}{16 a c^2 \left (c+d x^2\right ) (b c-a d)^3}-\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}-\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^4}-\frac{3 b^{11/4} (b c-5 a d) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{b \sqrt{x}}{2 a \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}+\frac{d \sqrt{x} (a d+2 b c)}{4 a c \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(2*b*c + a*d)*Sqrt[x])/(4*a*c*(b*c - a*d)^2*(c + d*x^2)^2) + (b*Sqrt[x])/(2*a*(b*c - a*d)*(a + b*x^2)*(c +
d*x^2)^2) + (d*(8*b^2*c^2 + 23*a*b*c*d - 7*a^2*d^2)*Sqrt[x])/(16*a*c^2*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(11/4
)*(b*c - 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (3*b^(11/4)
*(b*c - 5*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(
55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(11/4)*(b*c
- a*d)^4) + (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(3
2*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) - (3*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) + (3*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(7/4)*(b*c - a*d)^4) - (3*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*L
og[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(11/4)*(b*c - a*d)^4) + (3*d^(7/4)*(5
5*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^
(11/4)*(b*c - a*d)^4)

Rule 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 414

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

Rule 527

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

Rule 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}{\sqrt{x} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )\\ &=\frac{b \sqrt{x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-3 b c+4 a d-11 b d x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )}{2 a (b c-a d)}\\ &=\frac{d (2 b c+a d) \sqrt{x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{b \sqrt{x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-4 \left (6 b^2 c^2-16 a b c d+7 a^2 d^2\right )-28 b d (2 b c+a d) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )}{16 a c (b c-a d)^2}\\ &=\frac{d (2 b c+a d) \sqrt{x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{b \sqrt{x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt{x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-12 \left (8 b^3 c^3-32 a b^2 c^2 d+23 a^2 b c d^2-7 a^3 d^3\right )-12 b d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) x^4}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{64 a c^2 (b c-a d)^3}\\ &=\frac{d (2 b c+a d) \sqrt{x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{b \sqrt{x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt{x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac{\left (3 b^3 (b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a (b c-a d)^4}+\frac{\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c^2 (b c-a d)^4}\\ &=\frac{d (2 b c+a d) \sqrt{x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{b \sqrt{x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt{x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac{\left (3 b^3 (b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{3/2} (b c-a d)^4}+\frac{\left (3 b^3 (b c-5 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{3/2} (b c-a d)^4}+\frac{\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c^{5/2} (b c-a d)^4}+\frac{\left (3 d^2 \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c^{5/2} (b c-a d)^4}\\ &=\frac{d (2 b c+a d) \sqrt{x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{b \sqrt{x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt{x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}+\frac{\left (3 b^{5/2} (b c-5 a d)\right ) \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^{3/2} (b c-a d)^4}+\frac{\left (3 b^{5/2} (b c-5 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^{3/2} (b c-a d)^4}-\frac{\left (3 b^{11/4} (b c-5 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^{7/4} (b c-a d)^4}-\frac{\left (3 b^{11/4} (b c-5 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^{7/4} (b c-a d)^4}+\frac{\left (3 d^{3/2} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \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^{5/2} (b c-a d)^4}+\frac{\left (3 d^{3/2} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \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^{5/2} (b c-a d)^4}-\frac{\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \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^{11/4} (b c-a d)^4}-\frac{\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \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^{11/4} (b c-a d)^4}\\ &=\frac{d (2 b c+a d) \sqrt{x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{b \sqrt{x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt{x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac{3 b^{11/4} (b c-5 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{\left (3 b^{11/4} (b c-5 a d)\right ) \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^{7/4} (b c-a d)^4}-\frac{\left (3 b^{11/4} (b c-5 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^{7/4} (b c-a d)^4}+\frac{\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \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^{11/4} (b c-a d)^4}-\frac{\left (3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right )\right ) \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^{11/4} (b c-a d)^4}\\ &=\frac{d (2 b c+a d) \sqrt{x}}{4 a c (b c-a d)^2 \left (c+d x^2\right )^2}+\frac{b \sqrt{x}}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (8 b^2 c^2+23 a b c d-7 a^2 d^2\right ) \sqrt{x}}{16 a c^2 (b c-a d)^3 \left (c+d x^2\right )}-\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{11/4} (b c-a d)^4}-\frac{3 b^{11/4} (b c-5 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}+\frac{3 b^{11/4} (b c-5 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{7/4} (b c-a d)^4}-\frac{3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}+\frac{3 d^{7/4} \left (55 b^2 c^2-30 a b c d+7 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{11/4} (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 2.0101, size = 692, normalized size = 0.94 \[ \frac{1}{128} \left (-\frac{3 \sqrt{2} d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (b c-a d)^4}+\frac{3 \sqrt{2} d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{11/4} (b c-a d)^4}-\frac{6 \sqrt{2} d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{11/4} (b c-a d)^4}+\frac{6 \sqrt{2} d^{7/4} \left (7 a^2 d^2-30 a b c d+55 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{11/4} (b c-a d)^4}+\frac{24 \sqrt{2} b^{11/4} (5 a d-b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4} (b c-a d)^4}+\frac{24 \sqrt{2} b^{11/4} (b c-5 a d) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4} (b c-a d)^4}+\frac{48 \sqrt{2} b^{11/4} (5 a d-b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4} (b c-a d)^4}+\frac{48 \sqrt{2} b^{11/4} (b c-5 a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4} (b c-a d)^4}-\frac{64 b^3 \sqrt{x}}{a \left (a+b x^2\right ) (a d-b c)^3}+\frac{8 d^2 \sqrt{x} (23 b c-7 a d)}{c^2 \left (c+d x^2\right ) (b c-a d)^3}+\frac{32 d^2 \sqrt{x}}{c \left (c+d x^2\right )^2 (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-64*b^3*Sqrt[x])/(a*(-(b*c) + a*d)^3*(a + b*x^2)) + (32*d^2*Sqrt[x])/(c*(b*c - a*d)^2*(c + d*x^2)^2) + (8*d^
2*(23*b*c - 7*a*d)*Sqrt[x])/(c^2*(b*c - a*d)^3*(c + d*x^2)) + (48*Sqrt[2]*b^(11/4)*(-(b*c) + 5*a*d)*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*(b*c - a*d)^4) + (48*Sqrt[2]*b^(11/4)*(b*c - 5*a*d)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*(b*c - a*d)^4) - (6*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a
^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^4) + (6*Sqrt[2]*d^(7/4)*(55*b^2*c
^2 - 30*a*b*c*d + 7*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(c^(11/4)*(b*c - a*d)^4) + (24*Sqr
t[2]*b^(11/4)*(-(b*c) + 5*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)*(b*c - a*d
)^4) + (24*Sqrt[2]*b^(11/4)*(b*c - 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)
*(b*c - a*d)^4) - (3*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/
4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(b*c - a*d)^4) + (3*Sqrt[2]*d^(7/4)*(55*b^2*c^2 - 30*a*b*c*d + 7*a^2*d^2)*L
og[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(c^(11/4)*(b*c - a*d)^4))/128

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Maple [A]  time = 0.022, size = 1124, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/16*d^5/(a*d-b*c)^4/(d*x^2+c)^2/c^2*x^(5/2)*a^2-15/8*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^(5/2)*a*b+23/16*d^3/(a*d
-b*c)^4/(d*x^2+c)^2*x^(5/2)*b^2+11/16*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^(1/2)*a^2-19/8*d^3/(a*d-b*c)^4/(d*x^2+c)
^2*x^(1/2)*a*b+27/16*d^2/(a*d-b*c)^4/(d*x^2+c)^2*c*x^(1/2)*b^2+21/64*d^4/(a*d-b*c)^4/c^3*(c/d)^(1/4)*2^(1/2)*a
rctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2-45/32*d^3/(a*d-b*c)^4/c^2*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1
/4)*x^(1/2)-1)*a*b+165/64*d^2/(a*d-b*c)^4/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+21/1
28*d^4/(a*d-b*c)^4/c^3*(c/d)^(1/4)*2^(1/2)*ln((x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x-(c/d)^(1/4)*x^(1/
2)*2^(1/2)+(c/d)^(1/2)))*a^2-45/64*d^3/(a*d-b*c)^4/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*b+165/128*d^2/(a*d-b*c)^4/c*(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+21/64*d^4/(a*d-b*
c)^4/c^3*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2-45/32*d^3/(a*d-b*c)^4/c^2*(c/d)^(1/4)*2
^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b+165/64*d^2/(a*d-b*c)^4/c*(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
c/d)^(1/4)*x^(1/2)+1)*b^2-1/2*b^3/(a*d-b*c)^4*x^(1/2)/(b*x^2+a)*d+1/2*b^4/(a*d-b*c)^4/a*x^(1/2)/(b*x^2+a)*c-15
/8*b^3/(a*d-b*c)^4/a*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*d+3/8*b^4/(a*d-b*c)^4/a^2*(
1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c-15/8*b^3/(a*d-b*c)^4/a*(1/b*a)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*d+3/8*b^4/(a*d-b*c)^4/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(
1/4)*x^(1/2)-1)*c-15/16*b^3/(a*d-b*c)^4/a*(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+3/16*b^4/(a*d-b*c)^4/a^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)))*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/(b*x^2+a)^2/(d*x^2+c)^3/x^(1/2),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/(b*x^2+a)^2/(d*x^2+c)^3/x^(1/2),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/(b*x**2+a)**2/(d*x**2+c)**3/x**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.20438, size = 1692, normalized size = 2.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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