3.3.87 \(\int \frac {a+b x^2+c x^4}{x^8 (d+e x^2)^2} \, dx\) [287]

Optimal. Leaf size=167 \[ -\frac {a}{7 d^2 x^7}-\frac {b d-2 a e}{5 d^3 x^5}-\frac {c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac {e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}+\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {e^{3/2} \left (5 c d^2-e (7 b d-9 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{11/2}} \]

[Out]

-1/7*a/d^2/x^7+1/5*(2*a*e-b*d)/d^3/x^5+1/3*(-c*d^2+e*(-3*a*e+2*b*d))/d^4/x^3+e*(2*c*d^2-e*(-4*a*e+3*b*d))/d^5/
x+1/2*e^2*(a*e^2-b*d*e+c*d^2)*x/d^5/(e*x^2+d)+1/2*e^(3/2)*(5*c*d^2-e*(-9*a*e+7*b*d))*arctan(x*e^(1/2)/d^(1/2))
/d^(11/2)

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Rubi [A]
time = 0.21, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1273, 1816, 211} \begin {gather*} \frac {e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 c d^2-e (7 b d-9 a e)\right )}{2 d^{11/2}}+\frac {e^2 x \left (a e^2-b d e+c d^2\right )}{2 d^5 \left (d+e x^2\right )}+\frac {e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}-\frac {c d^2-e (2 b d-3 a e)}{3 d^4 x^3}-\frac {b d-2 a e}{5 d^3 x^5}-\frac {a}{7 d^2 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(x^8*(d + e*x^2)^2),x]

[Out]

-1/7*a/(d^2*x^7) - (b*d - 2*a*e)/(5*d^3*x^5) - (c*d^2 - e*(2*b*d - 3*a*e))/(3*d^4*x^3) + (e*(2*c*d^2 - e*(3*b*
d - 4*a*e)))/(d^5*x) + (e^2*(c*d^2 - b*d*e + a*e^2)*x)/(2*d^5*(d + e*x^2)) + (e^(3/2)*(5*c*d^2 - e*(7*b*d - 9*
a*e))*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(11/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1273

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(-d)^(m
/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*
p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x],
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {a+b x^2+c x^4}{x^8 \left (d+e x^2\right )^2} \, dx &=\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {\int \frac {2 a d^4 e^2+2 d^3 e^2 (b d-a e) x^2+2 d^2 e^2 \left (c d^2-b d e+a e^2\right ) x^4-2 d e^3 \left (c d^2-b d e+a e^2\right ) x^6+e^4 \left (c d^2-b d e+a e^2\right ) x^8}{x^8 \left (d+e x^2\right )} \, dx}{2 d^5 e^2}\\ &=\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {\int \left (\frac {2 a d^3 e^2}{x^8}+\frac {2 d^2 e^2 (b d-2 a e)}{x^6}+\frac {2 d e^2 \left (c d^2-e (2 b d-3 a e)\right )}{x^4}+\frac {2 e^3 \left (-2 c d^2+e (3 b d-4 a e)\right )}{x^2}+\frac {e^4 \left (5 c d^2-e (7 b d-9 a e)\right )}{d+e x^2}\right ) \, dx}{2 d^5 e^2}\\ &=-\frac {a}{7 d^2 x^7}-\frac {b d-2 a e}{5 d^3 x^5}-\frac {c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac {e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}+\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {\left (e^2 \left (5 c d^2-e (7 b d-9 a e)\right )\right ) \int \frac {1}{d+e x^2} \, dx}{2 d^5}\\ &=-\frac {a}{7 d^2 x^7}-\frac {b d-2 a e}{5 d^3 x^5}-\frac {c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac {e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}+\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {e^{3/2} \left (5 c d^2-e (7 b d-9 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 166, normalized size = 0.99 \begin {gather*} -\frac {a}{7 d^2 x^7}+\frac {-b d+2 a e}{5 d^3 x^5}+\frac {-c d^2+2 b d e-3 a e^2}{3 d^4 x^3}+\frac {e \left (2 c d^2-3 b d e+4 a e^2\right )}{d^5 x}+\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {e^{3/2} \left (5 c d^2-7 b d e+9 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^8*(d + e*x^2)^2),x]

[Out]

-1/7*a/(d^2*x^7) + (-(b*d) + 2*a*e)/(5*d^3*x^5) + (-(c*d^2) + 2*b*d*e - 3*a*e^2)/(3*d^4*x^3) + (e*(2*c*d^2 - 3
*b*d*e + 4*a*e^2))/(d^5*x) + (e^2*(c*d^2 - b*d*e + a*e^2)*x)/(2*d^5*(d + e*x^2)) + (e^(3/2)*(5*c*d^2 - 7*b*d*e
+ 9*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(11/2))

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Maple [A]
time = 0.13, size = 149, normalized size = 0.89

method	result	size
default	\(\frac {e^{2} \left (\frac {\left (\frac {1}{2} a \,e^{2}-\frac {1}{2} d e b +\frac {1}{2} c \,d^{2}\right ) x}{e \,x^{2}+d}+\frac {\left (9 a \,e^{2}-7 d e b +5 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{d^{5}}-\frac {a}{7 x^{7} d^{2}}-\frac {-2 a e +b d}{5 d^{3} x^{5}}-\frac {3 a \,e^{2}-2 d e b +c \,d^{2}}{3 d^{4} x^{3}}+\frac {e \left (4 a \,e^{2}-3 d e b +2 c \,d^{2}\right )}{d^{5} x}\)	\(149\)
risch	\(\frac {\frac {e^{2} \left (9 a \,e^{2}-7 d e b +5 c \,d^{2}\right ) x^{8}}{2 d^{5}}+\frac {e \left (9 a \,e^{2}-7 d e b +5 c \,d^{2}\right ) x^{6}}{3 d^{4}}-\frac {\left (9 a \,e^{2}-7 d e b +5 c \,d^{2}\right ) x^{4}}{15 d^{3}}+\frac {\left (9 a e -7 b d \right ) x^{2}}{35 d^{2}}-\frac {a}{7 d}}{x^{7} \left (e \,x^{2}+d \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (d^{11} \textit {\_Z}^{2}+81 a^{2} e^{7}-126 a b d \,e^{6}+90 a c \,d^{2} e^{5}+49 b^{2} d^{2} e^{5}-70 b c \,d^{3} e^{4}+25 c^{2} d^{4} e^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} d^{11}+162 a^{2} e^{7}-252 a b d \,e^{6}+180 a c \,d^{2} e^{5}+98 b^{2} d^{2} e^{5}-140 b c \,d^{3} e^{4}+50 c^{2} d^{4} e^{3}\right ) x +\left (-9 a \,d^{6} e^{3}+7 b \,d^{7} e^{2}-5 c \,d^{8} e \right ) \textit {\_R} \right )\right )}{4}\)	\(294\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^8/(e*x^2+d)^2,x,method=_RETURNVERBOSE)

[Out]

e^2/d^5*((1/2*a*e^2-1/2*d*e*b+1/2*c*d^2)*x/(e*x^2+d)+1/2*(9*a*e^2-7*b*d*e+5*c*d^2)/(d*e)^(1/2)*arctan(e*x/(d*e
)^(1/2)))-1/7*a/x^7/d^2-1/5*(-2*a*e+b*d)/d^3/x^5-1/3*(3*a*e^2-2*b*d*e+c*d^2)/d^4/x^3+e*(4*a*e^2-3*b*d*e+2*c*d^
2)/d^5/x

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Maxima [A]
time = 0.52, size = 165, normalized size = 0.99 \begin {gather*} \frac {105 \, {\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{8} + 70 \, {\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{6} - 30 \, a d^{4} - 14 \, {\left (5 \, c d^{4} - 7 \, b d^{3} e + 9 \, a d^{2} e^{2}\right )} x^{4} - 6 \, {\left (7 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2}}{210 \, {\left (d^{5} x^{9} e + d^{6} x^{7}\right )}} + \frac {{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

1/210*(105*(5*c*d^2*e^2 - 7*b*d*e^3 + 9*a*e^4)*x^8 + 70*(5*c*d^3*e - 7*b*d^2*e^2 + 9*a*d*e^3)*x^6 - 30*a*d^4 -
14*(5*c*d^4 - 7*b*d^3*e + 9*a*d^2*e^2)*x^4 - 6*(7*b*d^4 - 9*a*d^3*e)*x^2)/(d^5*x^9*e + d^6*x^7) + 1/2*(5*c*d^
2*e^2 - 7*b*d*e^3 + 9*a*e^4)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(11/2)

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Fricas [A]
time = 0.37, size = 458, normalized size = 2.74 \begin {gather*} \left [\frac {1890 \, a x^{8} e^{4} - 140 \, c d^{4} x^{4} - 84 \, b d^{4} x^{2} - 60 \, a d^{4} + 105 \, {\left (5 \, c d^{3} x^{7} e + 9 \, a x^{9} e^{4} - {\left (7 \, b d x^{9} - 9 \, a d x^{7}\right )} e^{3} + {\left (5 \, c d^{2} x^{9} - 7 \, b d^{2} x^{7}\right )} e^{2}\right )} \sqrt {-\frac {e}{d}} \log \left (\frac {x^{2} e + 2 \, d x \sqrt {-\frac {e}{d}} - d}{x^{2} e + d}\right ) - 210 \, {\left (7 \, b d x^{8} - 6 \, a d x^{6}\right )} e^{3} + 14 \, {\left (75 \, c d^{2} x^{8} - 70 \, b d^{2} x^{6} - 18 \, a d^{2} x^{4}\right )} e^{2} + 4 \, {\left (175 \, c d^{3} x^{6} + 49 \, b d^{3} x^{4} + 27 \, a d^{3} x^{2}\right )} e}{420 \, {\left (d^{5} x^{9} e + d^{6} x^{7}\right )}}, \frac {945 \, a x^{8} e^{4} - 70 \, c d^{4} x^{4} - 42 \, b d^{4} x^{2} - 30 \, a d^{4} + \frac {105 \, {\left (5 \, c d^{3} x^{7} e + 9 \, a x^{9} e^{4} - {\left (7 \, b d x^{9} - 9 \, a d x^{7}\right )} e^{3} + {\left (5 \, c d^{2} x^{9} - 7 \, b d^{2} x^{7}\right )} e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}}}{\sqrt {d}} - 105 \, {\left (7 \, b d x^{8} - 6 \, a d x^{6}\right )} e^{3} + 7 \, {\left (75 \, c d^{2} x^{8} - 70 \, b d^{2} x^{6} - 18 \, a d^{2} x^{4}\right )} e^{2} + 2 \, {\left (175 \, c d^{3} x^{6} + 49 \, b d^{3} x^{4} + 27 \, a d^{3} x^{2}\right )} e}{210 \, {\left (d^{5} x^{9} e + d^{6} x^{7}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

[1/420*(1890*a*x^8*e^4 - 140*c*d^4*x^4 - 84*b*d^4*x^2 - 60*a*d^4 + 105*(5*c*d^3*x^7*e + 9*a*x^9*e^4 - (7*b*d*x
^9 - 9*a*d*x^7)*e^3 + (5*c*d^2*x^9 - 7*b*d^2*x^7)*e^2)*sqrt(-e/d)*log((x^2*e + 2*d*x*sqrt(-e/d) - d)/(x^2*e +
d)) - 210*(7*b*d*x^8 - 6*a*d*x^6)*e^3 + 14*(75*c*d^2*x^8 - 70*b*d^2*x^6 - 18*a*d^2*x^4)*e^2 + 4*(175*c*d^3*x^6
+ 49*b*d^3*x^4 + 27*a*d^3*x^2)*e)/(d^5*x^9*e + d^6*x^7), 1/210*(945*a*x^8*e^4 - 70*c*d^4*x^4 - 42*b*d^4*x^2 -
30*a*d^4 + 105*(5*c*d^3*x^7*e + 9*a*x^9*e^4 - (7*b*d*x^9 - 9*a*d*x^7)*e^3 + (5*c*d^2*x^9 - 7*b*d^2*x^7)*e^2)*
arctan(x*e^(1/2)/sqrt(d))*e^(1/2)/sqrt(d) - 105*(7*b*d*x^8 - 6*a*d*x^6)*e^3 + 7*(75*c*d^2*x^8 - 70*b*d^2*x^6 -
18*a*d^2*x^4)*e^2 + 2*(175*c*d^3*x^6 + 49*b*d^3*x^4 + 27*a*d^3*x^2)*e)/(d^5*x^9*e + d^6*x^7)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (156) = 312\).
time = 1.35, size = 328, normalized size = 1.96 \begin {gather*} - \frac {\sqrt {- \frac {e^{3}}{d^{11}}} \cdot \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \log {\left (- \frac {d^{6} \sqrt {- \frac {e^{3}}{d^{11}}} \cdot \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right )}{9 a e^{4} - 7 b d e^{3} + 5 c d^{2} e^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {e^{3}}{d^{11}}} \cdot \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \log {\left (\frac {d^{6} \sqrt {- \frac {e^{3}}{d^{11}}} \cdot \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right )}{9 a e^{4} - 7 b d e^{3} + 5 c d^{2} e^{2}} + x \right )}}{4} + \frac {- 30 a d^{4} + x^{8} \cdot \left (945 a e^{4} - 735 b d e^{3} + 525 c d^{2} e^{2}\right ) + x^{6} \cdot \left (630 a d e^{3} - 490 b d^{2} e^{2} + 350 c d^{3} e\right ) + x^{4} \left (- 126 a d^{2} e^{2} + 98 b d^{3} e - 70 c d^{4}\right ) + x^{2} \cdot \left (54 a d^{3} e - 42 b d^{4}\right )}{210 d^{6} x^{7} + 210 d^{5} e x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**8/(e*x**2+d)**2,x)

[Out]

-sqrt(-e**3/d**11)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)*log(-d**6*sqrt(-e**3/d**11)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)
/(9*a*e**4 - 7*b*d*e**3 + 5*c*d**2*e**2) + x)/4 + sqrt(-e**3/d**11)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)*log(d**6*s
qrt(-e**3/d**11)*(9*a*e**2 - 7*b*d*e + 5*c*d**2)/(9*a*e**4 - 7*b*d*e**3 + 5*c*d**2*e**2) + x)/4 + (-30*a*d**4
+ x**8*(945*a*e**4 - 735*b*d*e**3 + 525*c*d**2*e**2) + x**6*(630*a*d*e**3 - 490*b*d**2*e**2 + 350*c*d**3*e) +
x**4*(-126*a*d**2*e**2 + 98*b*d**3*e - 70*c*d**4) + x**2*(54*a*d**3*e - 42*b*d**4))/(210*d**6*x**7 + 210*d**5*
e*x**9)

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Giac [A]
time = 4.41, size = 164, normalized size = 0.98 \begin {gather*} \frac {{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d^{\frac {11}{2}}} + \frac {c d^{2} x e^{2} - b d x e^{3} + a x e^{4}}{2 \, {\left (x^{2} e + d\right )} d^{5}} + \frac {210 \, c d^{2} x^{6} e - 315 \, b d x^{6} e^{2} - 35 \, c d^{3} x^{4} + 420 \, a x^{6} e^{3} + 70 \, b d^{2} x^{4} e - 105 \, a d x^{4} e^{2} - 21 \, b d^{3} x^{2} + 42 \, a d^{2} x^{2} e - 15 \, a d^{3}}{105 \, d^{5} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^8/(e*x^2+d)^2,x, algorithm="giac")

[Out]

1/2*(5*c*d^2*e^2 - 7*b*d*e^3 + 9*a*e^4)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(11/2) + 1/2*(c*d^2*x*e^2 - b*d*x
*e^3 + a*x*e^4)/((x^2*e + d)*d^5) + 1/105*(210*c*d^2*x^6*e - 315*b*d*x^6*e^2 - 35*c*d^3*x^4 + 420*a*x^6*e^3 +
70*b*d^2*x^4*e - 105*a*d*x^4*e^2 - 21*b*d^3*x^2 + 42*a*d^2*x^2*e - 15*a*d^3)/(d^5*x^7)

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Mupad [B]
time = 0.40, size = 156, normalized size = 0.93 \begin {gather*} \frac {\frac {x^2\,\left (9\,a\,e-7\,b\,d\right )}{35\,d^2}-\frac {a}{7\,d}-\frac {x^4\,\left (5\,c\,d^2-7\,b\,d\,e+9\,a\,e^2\right )}{15\,d^3}+\frac {e\,x^6\,\left (5\,c\,d^2-7\,b\,d\,e+9\,a\,e^2\right )}{3\,d^4}+\frac {e^2\,x^8\,\left (5\,c\,d^2-7\,b\,d\,e+9\,a\,e^2\right )}{2\,d^5}}{e\,x^9+d\,x^7}+\frac {e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (5\,c\,d^2-7\,b\,d\,e+9\,a\,e^2\right )}{2\,d^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/(x^8*(d + e*x^2)^2),x)

[Out]

((x^2*(9*a*e - 7*b*d))/(35*d^2) - a/(7*d) - (x^4*(9*a*e^2 + 5*c*d^2 - 7*b*d*e))/(15*d^3) + (e*x^6*(9*a*e^2 + 5
*c*d^2 - 7*b*d*e))/(3*d^4) + (e^2*x^8*(9*a*e^2 + 5*c*d^2 - 7*b*d*e))/(2*d^5))/(d*x^7 + e*x^9) + (e^(3/2)*atan(
(e^(1/2)*x)/d^(1/2))*(9*a*e^2 + 5*c*d^2 - 7*b*d*e))/(2*d^(11/2))

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