3.16.5 \(\int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx\) [1505]

Optimal. Leaf size=131 \[ \frac {10 e^2 (b d-a e)^3 x}{b^5}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e^3 (b d-a e)^2 (a+b x)^2}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^3}{3 b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6} \]

[Out]

10*e^2*(-a*e+b*d)^3*x/b^5-(-a*e+b*d)^5/b^6/(b*x+a)+5*e^3*(-a*e+b*d)^2*(b*x+a)^2/b^6+5/3*e^4*(-a*e+b*d)*(b*x+a)
^3/b^6+1/4*e^5*(b*x+a)^4/b^6+5*e*(-a*e+b*d)^4*ln(b*x+a)/b^6

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Rubi [A]
time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \begin {gather*} \frac {5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac {5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {10 e^2 x (b d-a e)^3}{b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*e^2*(b*d - a*e)^3*x)/b^5 - (b*d - a*e)^5/(b^6*(a + b*x)) + (5*e^3*(b*d - a*e)^2*(a + b*x)^2)/b^6 + (5*e^4*
(b*d - a*e)*(a + b*x)^3)/(3*b^6) + (e^5*(a + b*x)^4)/(4*b^6) + (5*e*(b*d - a*e)^4*Log[a + b*x])/b^6

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(d+e x)^5}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^5}{(a+b x)^2} \, dx\\ &=\int \left (\frac {10 e^2 (b d-a e)^3}{b^5}+\frac {(b d-a e)^5}{b^5 (a+b x)^2}+\frac {5 e (b d-a e)^4}{b^5 (a+b x)}+\frac {10 e^3 (b d-a e)^2 (a+b x)}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^2}{b^5}+\frac {e^5 (a+b x)^3}{b^5}\right ) \, dx\\ &=\frac {10 e^2 (b d-a e)^3 x}{b^5}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e^3 (b d-a e)^2 (a+b x)^2}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^3}{3 b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 230, normalized size = 1.76 \begin {gather*} \frac {12 a^5 e^5-12 a^4 b e^4 (5 d+4 e x)+30 a^3 b^2 e^3 \left (4 d^2+6 d e x-e^2 x^2\right )+10 a^2 b^3 e^2 \left (-12 d^3-24 d^2 e x+12 d e^2 x^2+e^3 x^3\right )-5 a b^4 e \left (-12 d^4-24 d^3 e x+36 d^2 e^2 x^2+8 d e^3 x^3+e^4 x^4\right )+b^5 \left (-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right )+60 e (b d-a e)^4 (a+b x) \log (a+b x)}{12 b^6 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*a^5*e^5 - 12*a^4*b*e^4*(5*d + 4*e*x) + 30*a^3*b^2*e^3*(4*d^2 + 6*d*e*x - e^2*x^2) + 10*a^2*b^3*e^2*(-12*d^
3 - 24*d^2*e*x + 12*d*e^2*x^2 + e^3*x^3) - 5*a*b^4*e*(-12*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 8*d*e^3*x^3 + e^
4*x^4) + b^5*(-12*d^5 + 120*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 20*d*e^4*x^4 + 3*e^5*x^5) + 60*e*(b*d - a*e)^4*(a +
b*x)*Log[a + b*x])/(12*b^6*(a + b*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(259\) vs. \(2(127)=254\).
time = 0.71, size = 260, normalized size = 1.98

method	result	size
norman	\(\frac {\frac {e^{5} x^{5}}{4 b}-\frac {5 e^{2} \left (e^{3} a^{3}-4 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -4 b^{3} d^{3}\right ) x^{2}}{2 b^{4}}+\frac {5 e^{3} \left (a^{2} e^{2}-4 a b d e +6 b^{2} d^{2}\right ) x^{3}}{6 b^{3}}-\frac {5 e^{4} \left (a e -4 b d \right ) x^{4}}{12 b^{2}}-\frac {\left (5 a^{5} e^{5}-20 a^{4} b d \,e^{4}+30 a^{3} b^{2} d^{2} e^{3}-20 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) x}{a \,b^{5}}}{b x +a}+\frac {5 e \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(256\)
default	\(-\frac {e^{2} \left (-\frac {1}{4} b^{3} x^{4} e^{3}+\frac {2}{3} a \,b^{2} e^{3} x^{3}-\frac {5}{3} b^{3} d \,e^{2} x^{3}-\frac {3}{2} a^{2} b \,e^{3} x^{2}+5 a \,b^{2} d \,e^{2} x^{2}-5 b^{3} d^{2} e \,x^{2}+4 e^{3} a^{3} x -15 a^{2} b d \,e^{2} x +20 a \,b^{2} d^{2} e x -10 b^{3} d^{3} x \right )}{b^{5}}-\frac {-a^{5} e^{5}+5 a^{4} b d \,e^{4}-10 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}-5 a \,b^{4} d^{4} e +b^{5} d^{5}}{b^{6} \left (b x +a \right )}+\frac {5 e \left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(260\)
risch	\(\frac {e^{5} x^{4}}{4 b^{2}}-\frac {2 e^{5} a \,x^{3}}{3 b^{3}}+\frac {5 e^{4} d \,x^{3}}{3 b^{2}}+\frac {3 e^{5} a^{2} x^{2}}{2 b^{4}}-\frac {5 e^{4} a d \,x^{2}}{b^{3}}+\frac {5 e^{3} d^{2} x^{2}}{b^{2}}-\frac {4 e^{5} a^{3} x}{b^{5}}+\frac {15 e^{4} a^{2} d x}{b^{4}}-\frac {20 e^{3} a \,d^{2} x}{b^{3}}+\frac {10 e^{2} d^{3} x}{b^{2}}+\frac {a^{5} e^{5}}{b^{6} \left (b x +a \right )}-\frac {5 a^{4} d \,e^{4}}{b^{5} \left (b x +a \right )}+\frac {10 a^{3} d^{2} e^{3}}{b^{4} \left (b x +a \right )}-\frac {10 a^{2} d^{3} e^{2}}{b^{3} \left (b x +a \right )}+\frac {5 a \,d^{4} e}{b^{2} \left (b x +a \right )}-\frac {d^{5}}{b \left (b x +a \right )}+\frac {5 e^{5} \ln \left (b x +a \right ) a^{4}}{b^{6}}-\frac {20 e^{4} \ln \left (b x +a \right ) a^{3} d}{b^{5}}+\frac {30 e^{3} \ln \left (b x +a \right ) a^{2} d^{2}}{b^{4}}-\frac {20 e^{2} \ln \left (b x +a \right ) a \,d^{3}}{b^{3}}+\frac {5 e \ln \left (b x +a \right ) d^{4}}{b^{2}}\)	\(326\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

-e^2/b^5*(-1/4*b^3*x^4*e^3+2/3*a*b^2*e^3*x^3-5/3*b^3*d*e^2*x^3-3/2*a^2*b*e^3*x^2+5*a*b^2*d*e^2*x^2-5*b^3*d^2*e
*x^2+4*e^3*a^3*x-15*a^2*b*d*e^2*x+20*a*b^2*d^2*e*x-10*b^3*d^3*x)-1/b^6*(-a^5*e^5+5*a^4*b*d*e^4-10*a^3*b^2*d^2*
e^3+10*a^2*b^3*d^3*e^2-5*a*b^4*d^4*e+b^5*d^5)/(b*x+a)+5/b^6*e*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3
*d^3*e+b^4*d^4)*ln(b*x+a)

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Maxima [A]
time = 0.27, size = 249, normalized size = 1.90 \begin {gather*} -\frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} x^{4} e^{5} + 4 \, {\left (5 \, b^{3} d e^{4} - 2 \, a b^{2} e^{5}\right )} x^{3} + 6 \, {\left (10 \, b^{3} d^{2} e^{3} - 10 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{2} + 12 \, {\left (10 \, b^{3} d^{3} e^{2} - 20 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} - 4 \, a^{3} e^{5}\right )} x}{12 \, b^{5}} + \frac {5 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left (b x + a\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

-(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/(b^7*x + a*b^6)
+ 1/12*(3*b^3*x^4*e^5 + 4*(5*b^3*d*e^4 - 2*a*b^2*e^5)*x^3 + 6*(10*b^3*d^2*e^3 - 10*a*b^2*d*e^4 + 3*a^2*b*e^5)
*x^2 + 12*(10*b^3*d^3*e^2 - 20*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 - 4*a^3*e^5)*x)/b^5 + 5*(b^4*d^4*e - 4*a*b^3*d^3
*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)*log(b*x + a)/b^6

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (129) = 258\).
time = 2.67, size = 340, normalized size = 2.60 \begin {gather*} -\frac {12 \, b^{5} d^{5} - 60 \, a b^{4} d^{4} e - {\left (3 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} - 30 \, a^{3} b^{2} x^{2} - 48 \, a^{4} b x + 12 \, a^{5}\right )} e^{5} - 20 \, {\left (b^{5} d x^{4} - 2 \, a b^{4} d x^{3} + 6 \, a^{2} b^{3} d x^{2} + 9 \, a^{3} b^{2} d x - 3 \, a^{4} b d\right )} e^{4} - 60 \, {\left (b^{5} d^{2} x^{3} - 3 \, a b^{4} d^{2} x^{2} - 4 \, a^{2} b^{3} d^{2} x + 2 \, a^{3} b^{2} d^{2}\right )} e^{3} - 120 \, {\left (b^{5} d^{3} x^{2} + a b^{4} d^{3} x - a^{2} b^{3} d^{3}\right )} e^{2} - 60 \, {\left ({\left (a^{4} b x + a^{5}\right )} e^{5} - 4 \, {\left (a^{3} b^{2} d x + a^{4} b d\right )} e^{4} + 6 \, {\left (a^{2} b^{3} d^{2} x + a^{3} b^{2} d^{2}\right )} e^{3} - 4 \, {\left (a b^{4} d^{3} x + a^{2} b^{3} d^{3}\right )} e^{2} + {\left (b^{5} d^{4} x + a b^{4} d^{4}\right )} e\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

-1/12*(12*b^5*d^5 - 60*a*b^4*d^4*e - (3*b^5*x^5 - 5*a*b^4*x^4 + 10*a^2*b^3*x^3 - 30*a^3*b^2*x^2 - 48*a^4*b*x +
12*a^5)*e^5 - 20*(b^5*d*x^4 - 2*a*b^4*d*x^3 + 6*a^2*b^3*d*x^2 + 9*a^3*b^2*d*x - 3*a^4*b*d)*e^4 - 60*(b^5*d^2*
x^3 - 3*a*b^4*d^2*x^2 - 4*a^2*b^3*d^2*x + 2*a^3*b^2*d^2)*e^3 - 120*(b^5*d^3*x^2 + a*b^4*d^3*x - a^2*b^3*d^3)*e
^2 - 60*((a^4*b*x + a^5)*e^5 - 4*(a^3*b^2*d*x + a^4*b*d)*e^4 + 6*(a^2*b^3*d^2*x + a^3*b^2*d^2)*e^3 - 4*(a*b^4*
d^3*x + a^2*b^3*d^3)*e^2 + (b^5*d^4*x + a*b^4*d^4)*e)*log(b*x + a))/(b^7*x + a*b^6)

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Sympy [A]
time = 0.53, size = 231, normalized size = 1.76 \begin {gather*} x^{3} \left (- \frac {2 a e^{5}}{3 b^{3}} + \frac {5 d e^{4}}{3 b^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} e^{5}}{2 b^{4}} - \frac {5 a d e^{4}}{b^{3}} + \frac {5 d^{2} e^{3}}{b^{2}}\right ) + x \left (- \frac {4 a^{3} e^{5}}{b^{5}} + \frac {15 a^{2} d e^{4}}{b^{4}} - \frac {20 a d^{2} e^{3}}{b^{3}} + \frac {10 d^{3} e^{2}}{b^{2}}\right ) + \frac {a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}}{a b^{6} + b^{7} x} + \frac {e^{5} x^{4}}{4 b^{2}} + \frac {5 e \left (a e - b d\right )^{4} \log {\left (a + b x \right )}}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

x**3*(-2*a*e**5/(3*b**3) + 5*d*e**4/(3*b**2)) + x**2*(3*a**2*e**5/(2*b**4) - 5*a*d*e**4/b**3 + 5*d**2*e**3/b**
2) + x*(-4*a**3*e**5/b**5 + 15*a**2*d*e**4/b**4 - 20*a*d**2*e**3/b**3 + 10*d**3*e**2/b**2) + (a**5*e**5 - 5*a*
*4*b*d*e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d**4*e - b**5*d**5)/(a*b**6 + b**7*x)
+ e**5*x**4/(4*b**2) + 5*e*(a*e - b*d)**4*log(a + b*x)/b**6

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Giac [A]
time = 1.42, size = 257, normalized size = 1.96 \begin {gather*} \frac {5 \, {\left (b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{{\left (b x + a\right )} b^{6}} + \frac {3 \, b^{6} x^{4} e^{5} + 20 \, b^{6} d x^{3} e^{4} + 60 \, b^{6} d^{2} x^{2} e^{3} + 120 \, b^{6} d^{3} x e^{2} - 8 \, a b^{5} x^{3} e^{5} - 60 \, a b^{5} d x^{2} e^{4} - 240 \, a b^{5} d^{2} x e^{3} + 18 \, a^{2} b^{4} x^{2} e^{5} + 180 \, a^{2} b^{4} d x e^{4} - 48 \, a^{3} b^{3} x e^{5}}{12 \, b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*(b^4*d^4*e - 4*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)*log(abs(b*x + a))/b^6 - (b^5*d^5
- 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/((b*x + a)*b^6) + 1/12*(
3*b^6*x^4*e^5 + 20*b^6*d*x^3*e^4 + 60*b^6*d^2*x^2*e^3 + 120*b^6*d^3*x*e^2 - 8*a*b^5*x^3*e^5 - 60*a*b^5*d*x^2*e
^4 - 240*a*b^5*d^2*x*e^3 + 18*a^2*b^4*x^2*e^5 + 180*a^2*b^4*d*x*e^4 - 48*a^3*b^3*x*e^5)/b^8

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Mupad [B]
time = 0.55, size = 326, normalized size = 2.49 \begin {gather*} x\,\left (\frac {10\,d^3\,e^2}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b}-\frac {a^2\,e^5}{b^4}+\frac {10\,d^2\,e^3}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b^2}\right )-x^3\,\left (\frac {2\,a\,e^5}{3\,b^3}-\frac {5\,d\,e^4}{3\,b^2}\right )+x^2\,\left (\frac {a\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b}-\frac {a^2\,e^5}{2\,b^4}+\frac {5\,d^2\,e^3}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,e^5-20\,a^3\,b\,d\,e^4+30\,a^2\,b^2\,d^2\,e^3-20\,a\,b^3\,d^3\,e^2+5\,b^4\,d^4\,e\right )}{b^6}+\frac {a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {e^5\,x^4}{4\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

x*((10*d^3*e^2)/b^2 - (2*a*((2*a*((2*a*e^5)/b^3 - (5*d*e^4)/b^2))/b - (a^2*e^5)/b^4 + (10*d^2*e^3)/b^2))/b + (
a^2*((2*a*e^5)/b^3 - (5*d*e^4)/b^2))/b^2) - x^3*((2*a*e^5)/(3*b^3) - (5*d*e^4)/(3*b^2)) + x^2*((a*((2*a*e^5)/b
^3 - (5*d*e^4)/b^2))/b - (a^2*e^5)/(2*b^4) + (5*d^2*e^3)/b^2) + (log(a + b*x)*(5*a^4*e^5 + 5*b^4*d^4*e - 20*a*
b^3*d^3*e^2 + 30*a^2*b^2*d^2*e^3 - 20*a^3*b*d*e^4))/b^6 + (a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2
*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)/(b*(a*b^5 + b^6*x)) + (e^5*x^4)/(4*b^2)

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