3.1924 \(\int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=305 \[ -\frac{5 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}}+\frac{5 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^3}-\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac{\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e} \]

[Out]

(5*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(512*c^3*d^3*e^3) - (5*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*
x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^2*d^2*e^2) + ((c*d^2 +
a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(12*c*d*e) - (
5*(c*d^2 - a*e^2)^6*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[
e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(1024*c^(7/2)*d^(7/2)*e^(7/2))

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Rubi [A]  time = 0.352036, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{5 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{7/2} d^{7/2} e^{7/2}}+\frac{5 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^3 d^3 e^3}-\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{192 c^2 d^2 e^2}+\frac{\left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{12 c d e} \]

Antiderivative was successfully verified.

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

[Out]

(5*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(512*c^3*d^3*e^3) - (5*(c*d^2 - a*e^2)^2*(c*d^2 + a*e^2 + 2*c*d*e*
x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(192*c^2*d^2*e^2) + ((c*d^2 +
a*e^2 + 2*c*d*e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(12*c*d*e) - (
5*(c*d^2 - a*e^2)^6*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[
e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(1024*c^(7/2)*d^(7/2)*e^(7/2))

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Rubi in Sympy [A]  time = 54.2038, size = 296, normalized size = 0.97 \[ \frac{\left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{12 c d e} - \frac{5 \left (a e^{2} - c d^{2}\right )^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{192 c^{2} d^{2} e^{2}} + \frac{5 \left (a e^{2} - c d^{2}\right )^{4} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{512 c^{3} d^{3} e^{3}} - \frac{5 \left (a e^{2} - c d^{2}\right )^{6} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{1024 c^{\frac{7}{2}} d^{\frac{7}{2}} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

(a*e**2 + c*d**2 + 2*c*d*e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(5/2)/
(12*c*d*e) - 5*(a*e**2 - c*d**2)**2*(a*e**2 + c*d**2 + 2*c*d*e*x)*(a*d*e + c*d*e
*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(192*c**2*d**2*e**2) + 5*(a*e**2 - c*d**2)**
4*(a*e**2 + c*d**2 + 2*c*d*e*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(
512*c**3*d**3*e**3) - 5*(a*e**2 - c*d**2)**6*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)
/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(10
24*c**(7/2)*d**(7/2)*e**(7/2))

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Mathematica [A]  time = 1.126, size = 374, normalized size = 1.23 \[ \frac{((d+e x) (a e+c d x))^{5/2} \left (\frac{2 \left (15 a^5 e^{10}-5 a^4 c d e^8 (17 d+2 e x)+2 a^3 c^2 d^2 e^6 \left (99 d^2+28 d e x+4 e^2 x^2\right )+6 a^2 c^3 d^3 e^4 \left (33 d^3+198 d^2 e x+212 d e^2 x^2+72 e^3 x^3\right )+a c^4 d^4 e^2 \left (-85 d^4+56 d^3 e x+1272 d^2 e^2 x^2+1696 d e^3 x^3+640 e^4 x^4\right )+c^5 d^5 \left (15 d^5-10 d^4 e x+8 d^3 e^2 x^2+432 d^2 e^3 x^3+640 d e^4 x^4+256 e^5 x^5\right )\right )}{3 c^3 d^3 e^3 (d+e x)^2 (a e+c d x)^2}-\frac{5 \left (c d^2-a e^2\right )^6 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} e^{7/2} (d+e x)^{5/2} (a e+c d x)^{5/2}}\right )}{1024} \]

Antiderivative was successfully verified.

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

[Out]

(((a*e + c*d*x)*(d + e*x))^(5/2)*((2*(15*a^5*e^10 - 5*a^4*c*d*e^8*(17*d + 2*e*x)
+ 2*a^3*c^2*d^2*e^6*(99*d^2 + 28*d*e*x + 4*e^2*x^2) + 6*a^2*c^3*d^3*e^4*(33*d^3
+ 198*d^2*e*x + 212*d*e^2*x^2 + 72*e^3*x^3) + a*c^4*d^4*e^2*(-85*d^4 + 56*d^3*e
*x + 1272*d^2*e^2*x^2 + 1696*d*e^3*x^3 + 640*e^4*x^4) + c^5*d^5*(15*d^5 - 10*d^4
*e*x + 8*d^3*e^2*x^2 + 432*d^2*e^3*x^3 + 640*d*e^4*x^4 + 256*e^5*x^5)))/(3*c^3*d
^3*e^3*(a*e + c*d*x)^2*(d + e*x)^2) - (5*(c*d^2 - a*e^2)^6*Log[a*e^2 + 2*Sqrt[c]
*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(c^(7/2)*d^
(7/2)*e^(7/2)*(a*e + c*d*x)^(5/2)*(d + e*x)^(5/2))))/1024

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Maple [B]  time = 0.009, size = 1247, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)

[Out]

-5/64*d^4*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-5/1024/d^3*e^9/c^3*ln((1
/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
))/(d*e*c)^(1/2)*a^6+15/512/d*e^7/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(
1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^5-75/1024*d*e^5/c*
ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2))/(d*e*c)^(1/2)*a^4-5/96/d*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x
*a^2-5/64*e^4/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+5/192*e^2/c*(a*e*d
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+5/512*d^7/e^3*c^2*(a*e*d+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)+5/256*d^3*e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-5/192*d^
4/e^2*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-5/192/d^2*e^4/c^2*(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)*a^3+1/12*(2*c*d*e*x+a*e^2+c*d^2)*(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(5/2)/c/d/e+5/512/d^3*e^7/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2)*a^5-5/96*d^3/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-5/1024*d^9/e^3*c^3
*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(1/2))/(d*e*c)^(1/2)+25/256*d^3*e^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(
1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^3+5/48*d*e*(a*e*d+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a-15/512/d*e^5/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)*a^4+5/256*d*e^3/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-15/5
12*d^5/e*c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+5/192*d^2*(a*e*d+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(3/2)*a+15/128*d^2*e^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*
x*a^2+5/256*d^6/e^2*c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-75/1024*d^5*e*
c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2))/(d*e*c)^(1/2)*a^2+5/256/d^2*e^6/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*x*a^4+15/512*d^7/e*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e
*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303086, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="fricas")

[Out]

[1/6144*(4*(256*c^5*d^5*e^5*x^5 + 15*c^5*d^10 - 85*a*c^4*d^8*e^2 + 198*a^2*c^3*d
^6*e^4 + 198*a^3*c^2*d^4*e^6 - 85*a^4*c*d^2*e^8 + 15*a^5*e^10 + 640*(c^5*d^6*e^4
+ a*c^4*d^4*e^6)*x^4 + 16*(27*c^5*d^7*e^3 + 106*a*c^4*d^5*e^5 + 27*a^2*c^3*d^3*
e^7)*x^3 + 8*(c^5*d^8*e^2 + 159*a*c^4*d^6*e^4 + 159*a^2*c^3*d^4*e^6 + a^3*c^2*d^
2*e^8)*x^2 - 2*(5*c^5*d^9*e - 28*a*c^4*d^7*e^3 - 594*a^2*c^3*d^5*e^5 - 28*a^3*c^
2*d^3*e^7 + 5*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c
*d*e) + 15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^
6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*log(-4*(2*c^2*d^2*e^2*x +
c^2*d^3*e + a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*c^2*d^2*
e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x)*sqrt(
c*d*e)))/(sqrt(c*d*e)*c^3*d^3*e^3), 1/3072*(2*(256*c^5*d^5*e^5*x^5 + 15*c^5*d^10
- 85*a*c^4*d^8*e^2 + 198*a^2*c^3*d^6*e^4 + 198*a^3*c^2*d^4*e^6 - 85*a^4*c*d^2*e
^8 + 15*a^5*e^10 + 640*(c^5*d^6*e^4 + a*c^4*d^4*e^6)*x^4 + 16*(27*c^5*d^7*e^3 +
106*a*c^4*d^5*e^5 + 27*a^2*c^3*d^3*e^7)*x^3 + 8*(c^5*d^8*e^2 + 159*a*c^4*d^6*e^4
+ 159*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8)*x^2 - 2*(5*c^5*d^9*e - 28*a*c^4*d^7*e^
3 - 594*a^2*c^3*d^5*e^5 - 28*a^3*c^2*d^3*e^7 + 5*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) - 15*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15
*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 +
a^6*e^12)*arctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 +
a*d*e + (c*d^2 + a*e^2)*x)*c*d*e)))/(sqrt(-c*d*e)*c^3*d^3*e^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.254144, size = 678, normalized size = 2.22 \[ \frac{1}{1536} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \, c^{2} d^{2} x e^{2} + \frac{5 \,{\left (c^{7} d^{8} e^{6} + a c^{6} d^{6} e^{8}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (27 \, c^{7} d^{9} e^{5} + 106 \, a c^{6} d^{7} e^{7} + 27 \, a^{2} c^{5} d^{5} e^{9}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (c^{7} d^{10} e^{4} + 159 \, a c^{6} d^{8} e^{6} + 159 \, a^{2} c^{5} d^{6} e^{8} + a^{3} c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x - \frac{{\left (5 \, c^{7} d^{11} e^{3} - 28 \, a c^{6} d^{9} e^{5} - 594 \, a^{2} c^{5} d^{7} e^{7} - 28 \, a^{3} c^{4} d^{5} e^{9} + 5 \, a^{4} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac{{\left (15 \, c^{7} d^{12} e^{2} - 85 \, a c^{6} d^{10} e^{4} + 198 \, a^{2} c^{5} d^{8} e^{6} + 198 \, a^{3} c^{4} d^{6} e^{8} - 85 \, a^{4} c^{3} d^{4} e^{10} + 15 \, a^{5} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac{5 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{c d} e^{\left (-\frac{7}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{1024 \, c^{4} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2),x, algorithm="giac")

[Out]

1/1536*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(2*c^2*d^2*x*e^2
+ 5*(c^7*d^8*e^6 + a*c^6*d^6*e^8)*e^(-5)/(c^5*d^5))*x + (27*c^7*d^9*e^5 + 106*a*
c^6*d^7*e^7 + 27*a^2*c^5*d^5*e^9)*e^(-5)/(c^5*d^5))*x + (c^7*d^10*e^4 + 159*a*c^
6*d^8*e^6 + 159*a^2*c^5*d^6*e^8 + a^3*c^4*d^4*e^10)*e^(-5)/(c^5*d^5))*x - (5*c^7
*d^11*e^3 - 28*a*c^6*d^9*e^5 - 594*a^2*c^5*d^7*e^7 - 28*a^3*c^4*d^5*e^9 + 5*a^4*
c^3*d^3*e^11)*e^(-5)/(c^5*d^5))*x + (15*c^7*d^12*e^2 - 85*a*c^6*d^10*e^4 + 198*a
^2*c^5*d^8*e^6 + 198*a^3*c^4*d^6*e^8 - 85*a^4*c^3*d^4*e^10 + 15*a^5*c^2*d^2*e^12
)*e^(-5)/(c^5*d^5)) + 5/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 -
20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*sqrt(c*d
)*e^(-7/2)*ln(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x
^2*e + c*d^2*x + a*x*e^2 + a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^4*d^4)