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

Optimal. Leaf size=703 \[ -\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} (b c-a d)^4}-\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 d x^{3/2} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 d x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]

[Out]

(-3*d*x^(3/2))/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x^(3/2)/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (3*d*(7*b
*c + a*d)*x^(3/2))/(16*c*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[
x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (S
qrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a
^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c +
3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) - (3*b^(5
/4)*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4
) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])
/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(
1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4)

________________________________________________________________________________________

Rubi [A]  time = 1.02006, antiderivative size = 703, normalized size of antiderivative = 1., 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 = {466, 471, 579, 584, 297, 1162, 617, 204, 1165, 628} \[ -\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} (b c-a d)^4}-\frac{3 \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}+\frac{3 b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 d x^{3/2} (a d+7 b c)}{16 c \left (c+d x^2\right ) (b c-a d)^3}-\frac{x^{3/2}}{2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 d x^{3/2}}{4 \left (c+d x^2\right )^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*d*x^(3/2))/(4*(b*c - a*d)^2*(c + d*x^2)^2) - x^(3/2)/(2*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^2) - (3*d*(7*b
*c + a*d)*x^(3/2))/(16*c*(b*c - a*d)^3*(c + d*x^2)) - (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[
x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (S
qrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a
^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) + (3*b^(5/4)*(b*c +
3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4) - (3*b^(5
/4)*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*(b*c - a*d)^4
) - (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])
/(64*Sqrt[2]*c^(5/4)*(b*c - a*d)^4) + (3*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(
1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*(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 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 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] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && 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 584

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 297

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 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])

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]

Rubi steps

\begin{align*} \int \frac{x^{5/2}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^6}{\left (a+b x^4\right )^2 \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 c-9 d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^3} \, dx,x,\sqrt{x}\right )}{2 (b c-a d)}\\ &=-\frac{3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (12 c (2 b c+a d)-60 b c d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )^2} \, dx,x,\sqrt{x}\right )}{16 c (b c-a d)^2}\\ &=-\frac{3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (12 c \left (8 b^2 c^2+17 a b c d-a^2 d^2\right )-12 b c d (7 b c+a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt{x}\right )}{64 c^2 (b c-a d)^3}\\ &=-\frac{3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{96 b^2 c^2 (b c+3 a d) x^2}{(b c-a d) \left (a+b x^4\right )}-\frac{12 c d \left (15 b^2 c^2+18 a b c d-a^2 d^2\right ) x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{64 c^2 (b c-a d)^3}\\ &=-\frac{3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac{\left (3 b^2 (b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 (b c-a d)^4}-\frac{\left (3 d \left (15 b^2 c^2+18 a b c d-a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c (b c-a d)^4}\\ &=-\frac{3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac{\left (3 b^{3/2} (b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 (b c-a d)^4}+\frac{\left (3 b^{3/2} (b c+3 a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 (b c-a d)^4}+\frac{\left (3 \sqrt{d} \left (15 b^2 c^2+18 a b c d-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 (b c-a d)^4}-\frac{\left (3 \sqrt{d} \left (15 b^2 c^2+18 a b c d-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 (b c-a d)^4}\\ &=-\frac{3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac{(3 b (b c+3 a d)) \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 (b c-a d)^4}+\frac{(3 b (b c+3 a d)) \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 (b c-a d)^4}+\frac{\left (3 b^{5/4} (b c+3 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} \sqrt [4]{a} (b c-a d)^4}+\frac{\left (3 b^{5/4} (b c+3 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} \sqrt [4]{a} (b c-a d)^4}-\frac{\left (3 \left (15 b^2 c^2+18 a b c d-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 (b c-a d)^4}-\frac{\left (3 \left (15 b^2 c^2+18 a b c d-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 (b c-a d)^4}-\frac{\left (3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}-\frac{\left (3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}\\ &=-\frac{3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}+\frac{3 b^{5/4} (b c+3 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (b c+3 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}+\frac{\left (3 b^{5/4} (b c+3 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} \sqrt [4]{a} (b c-a d)^4}-\frac{\left (3 b^{5/4} (b c+3 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} \sqrt [4]{a} (b c-a d)^4}-\frac{\left (3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}+\frac{\left (3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}\\ &=-\frac{3 d x^{3/2}}{4 (b c-a d)^2 \left (c+d x^2\right )^2}-\frac{x^{3/2}}{2 (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2}-\frac{3 d (7 b c+a d) x^{3/2}}{16 c (b c-a d)^3 \left (c+d x^2\right )}-\frac{3 b^{5/4} (b c+3 a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}+\frac{3 b^{5/4} (b c+3 a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}-\frac{3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}+\frac{3 b^{5/4} (b c+3 a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 b^{5/4} (b c+3 a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} (b c-a d)^4}-\frac{3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}+\frac{3 \sqrt [4]{d} \left (15 b^2 c^2+18 a b c d-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^{5/4} (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 1.82837, size = 604, normalized size = 0.86 \[ \frac{\frac{3 \sqrt{2} \sqrt [4]{d} \left (a^2 d^2-18 a b c d-15 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{3 \sqrt{2} \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{6 \sqrt{2} \sqrt [4]{d} \left (-a^2 d^2+18 a b c d+15 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4}}+\frac{6 \sqrt{2} \sqrt [4]{d} \left (a^2 d^2-18 a b c d-15 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac{64 b^2 x^{3/2} (b c-a d)}{a+b x^2}+\frac{24 \sqrt{2} b^{5/4} (3 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{24 \sqrt{2} b^{5/4} (3 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{48 \sqrt{2} b^{5/4} (3 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{48 \sqrt{2} b^{5/4} (3 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}+\frac{8 d x^{3/2} (a d-b c) (3 a d+13 b c)}{c \left (c+d x^2\right )}-\frac{32 d x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}}{128 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-64*b^2*(b*c - a*d)*x^(3/2))/(a + b*x^2) - (32*d*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (8*d*(-(b*c) + a*d)*
(13*b*c + 3*a*d)*x^(3/2))/(c*(c + d*x^2)) - (48*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/a^(1/4) + (48*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1
/4) + (6*Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^
(5/4) + (6*Sqrt[2]*d^(1/4)*(-15*b^2*c^2 - 18*a*b*c*d + a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])
/c^(5/4) + (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/
4) - (24*Sqrt[2]*b^(5/4)*(b*c + 3*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4) + (
3*Sqrt[2]*d^(1/4)*(-15*b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]
*x])/c^(5/4) + (3*Sqrt[2]*d^(1/4)*(15*b^2*c^2 + 18*a*b*c*d - a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sq
rt[x] + Sqrt[d]*x])/c^(5/4))/(128*(b*c - a*d)^4)

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Maple [A]  time = 0.023, size = 1067, 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(x^(5/2)/(b*x^2+a)^2/(d*x^2+c)^3,x)

[Out]

3/16*d^4/(a*d-b*c)^4/(d*x^2+c)^2/c*x^(7/2)*a^2+5/8*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(7/2)*a*b-13/16*d^2/(a*d-b*c)
^4/(d*x^2+c)^2*c*x^(7/2)*b^2-1/16*d^3/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*a^2+9/8*d^2/(a*d-b*c)^4/(d*x^2+c)^2*x^(3
/2)*c*a*b-17/16*d/(a*d-b*c)^4/(d*x^2+c)^2*x^(3/2)*b^2*c^2+3/64*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)*a^2-27/32*d/(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-45/64/(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+3/64*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)*a^2-27/32*d/(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-45/64/(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+3/128*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)))*a^2-27/64*d/(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-45/128/(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+1/2*b^2/(a*d-b*c)^4*x
^(3/2)/(b*x^2+a)*a*d-1/2*b^3/(a*d-b*c)^4*x^(3/2)/(b*x^2+a)*c+9/16*b/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*a*d*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)))+9/8*b/(a*d-b*c
)^4/(1/b*a)^(1/4)*2^(1/2)*a*d*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)+9/8*b/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*
a*d*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)+3/16*b^2/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*c*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)))+3/8*b^2/(a*d-b*c)^4/(1/b*a)^(1/
4)*2^(1/2)*c*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)+3/8*b^2/(a*d-b*c)^4/(1/b*a)^(1/4)*2^(1/2)*c*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^(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(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(x**(5/2)/(b*x**2+a)**2/(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [B]  time = 2.66757, size = 1671, normalized size = 2.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^2*x^(3/2)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(b*x^2 + a)) + 3/4*((a*b^3)^(3/4)*b*c +
3*(a*b^3)^(3/4)*a*d)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a*b^5*c^4 - 4*
sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4) + 3/4*((a*b^3
)^(3/4)*b*c + 3*(a*b^3)^(3/4)*a*d)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)
*a*b^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4
) - 3/32*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqr
t(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^4*c^6*d^2 - 4*sqrt(2)*a*b^3*c^5*d^3 + 6*sqrt(2)*a^2*b^2*
c^4*d^4 - 4*sqrt(2)*a^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^6) - 3/32*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a
*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b^
4*c^6*d^2 - 4*sqrt(2)*a*b^3*c^5*d^3 + 6*sqrt(2)*a^2*b^2*c^4*d^4 - 4*sqrt(2)*a^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^
6) - 3/8*((a*b^3)^(3/4)*b*c + 3*(a*b^3)^(3/4)*a*d)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a
*b^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4)
+ 3/8*((a*b^3)^(3/4)*b*c + 3*(a*b^3)^(3/4)*a*d)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a*b
^5*c^4 - 4*sqrt(2)*a^2*b^4*c^3*d + 6*sqrt(2)*a^3*b^3*c^2*d^2 - 4*sqrt(2)*a^4*b^2*c*d^3 + sqrt(2)*a^5*b*d^4) +
3/64*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(
1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^6*d^2 - 4*sqrt(2)*a*b^3*c^5*d^3 + 6*sqrt(2)*a^2*b^2*c^4*d^4 - 4*sqrt(2)*a
^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^6) - 3/64*(15*(c*d^3)^(3/4)*b^2*c^2 + 18*(c*d^3)^(3/4)*a*b*c*d - (c*d^3)^(3/4
)*a^2*d^2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b^4*c^6*d^2 - 4*sqrt(2)*a*b^3*c^5*d^3 +
6*sqrt(2)*a^2*b^2*c^4*d^4 - 4*sqrt(2)*a^3*b*c^3*d^5 + sqrt(2)*a^4*c^2*d^6) - 1/16*(13*b*c*d^2*x^(7/2) + 3*a*d^
3*x^(7/2) + 17*b*c^2*d*x^(3/2) - a*c*d^2*x^(3/2))/((b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b*c^2*d^2 - a^3*c*d^3)*(d*
x^2 + c)^2)