3.1286 \(\int \frac{(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx\)
Optimal. Leaf size=147 \[ \frac{4}{3} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}+2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+\frac{4}{7} d (b d+2 c d x)^{7/2} \]
[Out]
(4*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(3/2))/3 + (4*d*(b*d + 2*c*d*x)^(7/2))/7 + 2*(b^2 - 4*a*c)^(7/4)*d^(9/2)*
ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 2*(b^2 - 4*a*c)^(7/4)*d^(9/2)*ArcTanh[Sqrt[d*(b +
2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]
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Rubi [A] time = 0.138146, antiderivative size = 147, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used =
{692, 694, 329, 298, 203, 206} \[ \frac{4}{3} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}+2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+\frac{4}{7} d (b d+2 c d x)^{7/2} \]
Antiderivative was successfully verified.
[In]
Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2),x]
[Out]
(4*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(3/2))/3 + (4*d*(b*d + 2*c*d*x)^(7/2))/7 + 2*(b^2 - 4*a*c)^(7/4)*d^(9/2)*
ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 2*(b^2 - 4*a*c)^(7/4)*d^(9/2)*ArcTanh[Sqrt[d*(b +
2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]
Rule 692
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])
Rule 694
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[x^m*(
a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]
Rule 329
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 298
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
& !GtQ[a/b, 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx &=\frac{4}{7} d (b d+2 c d x)^{7/2}+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx\\ &=\frac{4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac{4}{7} d (b d+2 c d x)^{7/2}+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac{\sqrt{b d+2 c d x}}{a+b x+c x^2} \, dx\\ &=\frac{4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac{4}{7} d (b d+2 c d x)^{7/2}+\frac{\left (\left (b^2-4 a c\right )^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{2 c}\\ &=\frac{4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac{4}{7} d (b d+2 c d x)^{7/2}+\frac{\left (\left (b^2-4 a c\right )^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{a-\frac{b^2}{4 c}+\frac{x^4}{4 c d^2}} \, dx,x,\sqrt{d (b+2 c x)}\right )}{c}\\ &=\frac{4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac{4}{7} d (b d+2 c d x)^{7/2}-\left (2 \left (b^2-4 a c\right )^2 d^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c} d-x^2} \, dx,x,\sqrt{d (b+2 c x)}\right )+\left (2 \left (b^2-4 a c\right )^2 d^5\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c} d+x^2} \, dx,x,\sqrt{d (b+2 c x)}\right )\\ &=\frac{4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac{4}{7} d (b d+2 c d x)^{7/2}+2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )-2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\\ \end{align*}
Mathematica [A] time = 0.145131, size = 131, normalized size = 0.89 \[ \frac{4 (d (b+2 c x))^{9/2} \left (\frac{1}{7} (b+2 c x)^{7/2}-\frac{1}{6} \left (4 a c-b^2\right ) \left (3 \left (b^2-4 a c\right )^{3/4} \left (\tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-\tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )+2 (b+2 c x)^{3/2}\right )\right )}{(b+2 c x)^{9/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2),x]
[Out]
(4*(d*(b + 2*c*x))^(9/2)*((b + 2*c*x)^(7/2)/7 - ((-b^2 + 4*a*c)*(2*(b + 2*c*x)^(3/2) + 3*(b^2 - 4*a*c)^(3/4)*(
ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)] - ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])))/6))/(b + 2*c*x)^
(9/2)
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Maple [B] time = 0.194, size = 922, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x)
[Out]
4/7*d*(2*c*d*x+b*d)^(7/2)-16/3*(2*c*d*x+b*d)^(3/2)*a*c*d^3+4/3*(2*c*d*x+b*d)^(3/2)*b^2*d^3-16*d^5/(4*a*c*d^2-b
^2*d^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*a^2*c^2+8*d^5/(4*a*c*d^
2-b^2*d^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*a*b^2*c-d^5/(4*a*c*d
^2-b^2*d^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*b^4+8*d^5/(4*a*c*d^
2-b^2*d^2)^(1/4)*2^(1/2)*ln((2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*
d^2)^(1/2))/(2*c*d*x+b*d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2)))*a^2
*c^2-4*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*ln((2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(
1/2)+(4*a*c*d^2-b^2*d^2)^(1/2))/(2*c*d*x+b*d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-
b^2*d^2)^(1/2)))*a*b^2*c+1/2*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*ln((2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(
2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2))/(2*c*d*x+b*d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/
2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2)))*b^4+16*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(4*a*c*d^2-
b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*a^2*c^2-8*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(4*a*c*d^
2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*a*b^2*c+d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(4*a*c*d^
2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*b^4
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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((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.19676, size = 3245, normalized size = 22.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x, algorithm="fricas")
[Out]
8/21*(12*c^3*d^4*x^3 + 18*b*c^2*d^4*x^2 + 4*(4*b^2*c - 7*a*c^2)*d^4*x + (5*b^3 - 14*a*b*c)*d^4)*sqrt(2*c*d*x +
b*d) - 4*((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 +
28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(1/4)*arctan(-(((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8
*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(1/4)*(b^10 - 20*a*b^8*
c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(2*c*d*x + b*d)*d^13 + sqrt(2*(b^
20*c - 40*a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*a^5*b^10*c^6 + 86016
0*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 - 2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)*d^27*x +
(b^21 - 40*a*b^19*c + 720*a^2*b^17*c^2 - 7680*a^3*b^15*c^3 + 53760*a^4*b^13*c^4 - 258048*a^5*b^11*c^5 + 86016
0*a^6*b^9*c^6 - 1966080*a^7*b^7*c^7 + 2949120*a^8*b^5*c^8 - 2621440*a^9*b^3*c^9 + 1048576*a^10*b*c^10)*d^27 +
sqrt((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*
a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)*(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^
4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)*((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 224
0*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(1/4))/((b^14
- 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6
- 16384*a^7*c^7)*d^18)) + ((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 215
04*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(1/4)*log(-(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 64
0*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(2*c*d*x + b*d)*d^13 + ((b^14 - 28*a*b^12*c + 336*a^2*b^1
0*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(3/
4)) - ((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 2867
2*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(1/4)*log(-(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*
a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(2*c*d*x + b*d)*d^13 - ((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*
c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(3/4))
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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((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a),x)
[Out]
Timed out
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Giac [B] time = 1.25914, size = 612, normalized size = 4.16 \begin{align*} \frac{4}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} d^{3} - \frac{16}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c d^{3} + \frac{4}{7} \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} d + \frac{1}{2} \, \sqrt{2}{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \log \left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac{1}{2} \, \sqrt{2}{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \log \left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) -{\left (\sqrt{2} b^{2} d^{3} - 4 \, \sqrt{2} a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) -{\left (\sqrt{2} b^{2} d^{3} - 4 \, \sqrt{2} a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x, algorithm="giac")
[Out]
4/3*(2*c*d*x + b*d)^(3/2)*b^2*d^3 - 16/3*(2*c*d*x + b*d)^(3/2)*a*c*d^3 + 4/7*(2*c*d*x + b*d)^(7/2)*d + 1/2*sqr
t(2)*(b^2*d^3 - 4*a*c*d^3)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*log(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/
4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2)) - 1/2*sqrt(2)*(b^2*d^3 - 4*a*c*d^3)*(-b^2*d^2 + 4*a*c*d^2
)^(3/4)*log(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d
^2)) - (sqrt(2)*b^2*d^3 - 4*sqrt(2)*a*c*d^3)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b^2*d^
2 + 4*a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4)) - (sqrt(2)*b^2*d^3 - 4*sqrt(2)*a*c
*d^3)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x
+ b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))