3.2014 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx\)

Optimal. Leaf size=362 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{14 e^7 (a+b x) (d+e x)^{14}}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{15}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{16 e^7 (a+b x) (d+e x)^{16}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}+\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{13 e^7 (a+b x) (d+e x)^{13}} \]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(16*e^7*(a + b*x)*(d + e*x)^16) +
(2*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^15
) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14*e^7*(a + b*x)*(d +
e*x)^14) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x
)*(d + e*x)^13) - (5*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a
+ b*x)*(d + e*x)^12) + (6*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7
*(a + b*x)*(d + e*x)^11) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^7*(a + b*x)
*(d + e*x)^10)

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Rubi [A]  time = 0.610363, antiderivative size = 362, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{15 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{14 e^7 (a+b x) (d+e x)^{14}}+\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{15}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{16 e^7 (a+b x) (d+e x)^{16}}-\frac{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}+\frac{6 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^7 (a+b x) (d+e x)^{12}}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{13 e^7 (a+b x) (d+e x)^{13}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^17,x]

[Out]

-((b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(16*e^7*(a + b*x)*(d + e*x)^16) +
(2*b*(b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)*(d + e*x)^15
) - (15*b^2*(b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(14*e^7*(a + b*x)*(d +
e*x)^14) + (20*b^3*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x
)*(d + e*x)^13) - (5*b^4*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a
+ b*x)*(d + e*x)^12) + (6*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7
*(a + b*x)*(d + e*x)^11) - (b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^7*(a + b*x)
*(d + e*x)^10)

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Rubi in Sympy [A]  time = 52.3533, size = 275, normalized size = 0.76 \[ - \frac{b^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{7280 e^{6} \left (d + e x\right )^{11}} + \frac{b^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{80080 e^{7} \left (a + b x\right ) \left (d + e x\right )^{11}} - \frac{b^{4} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4368 e^{5} \left (d + e x\right )^{12}} - \frac{b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{364 e^{4} \left (d + e x\right )^{13}} - \frac{b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{560 e^{3} \left (d + e x\right )^{14}} - \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{40 e^{2} \left (d + e x\right )^{15}} - \frac{\left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{16 e \left (d + e x\right )^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**17,x)

[Out]

-b**5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(7280*e**6*(d + e*x)**11) + b**5*(a*e - b
*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(80080*e**7*(a + b*x)*(d + e*x)**11) - b**4
*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(4368*e**5*(d + e*x)**12) - b**3
*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(364*e**4*(d + e*x)**13) - b**2*(5*a + 5*b*
x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(560*e**3*(d + e*x)**14) - b*(a**2 + 2*a*
b*x + b**2*x**2)**(5/2)/(40*e**2*(d + e*x)**15) - (a + b*x)*(a**2 + 2*a*b*x + b*
*2*x**2)**(5/2)/(16*e*(d + e*x)**16)

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Mathematica [A]  time = 0.21807, size = 295, normalized size = 0.81 \[ -\frac{\sqrt{(a+b x)^2} \left (5005 a^6 e^6+2002 a^5 b e^5 (d+16 e x)+715 a^4 b^2 e^4 \left (d^2+16 d e x+120 e^2 x^2\right )+220 a^3 b^3 e^3 \left (d^3+16 d^2 e x+120 d e^2 x^2+560 e^3 x^3\right )+55 a^2 b^4 e^2 \left (d^4+16 d^3 e x+120 d^2 e^2 x^2+560 d e^3 x^3+1820 e^4 x^4\right )+10 a b^5 e \left (d^5+16 d^4 e x+120 d^3 e^2 x^2+560 d^2 e^3 x^3+1820 d e^4 x^4+4368 e^5 x^5\right )+b^6 \left (d^6+16 d^5 e x+120 d^4 e^2 x^2+560 d^3 e^3 x^3+1820 d^2 e^4 x^4+4368 d e^5 x^5+8008 e^6 x^6\right )\right )}{80080 e^7 (a+b x) (d+e x)^{16}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^17,x]

[Out]

-(Sqrt[(a + b*x)^2]*(5005*a^6*e^6 + 2002*a^5*b*e^5*(d + 16*e*x) + 715*a^4*b^2*e^
4*(d^2 + 16*d*e*x + 120*e^2*x^2) + 220*a^3*b^3*e^3*(d^3 + 16*d^2*e*x + 120*d*e^2
*x^2 + 560*e^3*x^3) + 55*a^2*b^4*e^2*(d^4 + 16*d^3*e*x + 120*d^2*e^2*x^2 + 560*d
*e^3*x^3 + 1820*e^4*x^4) + 10*a*b^5*e*(d^5 + 16*d^4*e*x + 120*d^3*e^2*x^2 + 560*
d^2*e^3*x^3 + 1820*d*e^4*x^4 + 4368*e^5*x^5) + b^6*(d^6 + 16*d^5*e*x + 120*d^4*e
^2*x^2 + 560*d^3*e^3*x^3 + 1820*d^2*e^4*x^4 + 4368*d*e^5*x^5 + 8008*e^6*x^6)))/(
80080*e^7*(a + b*x)*(d + e*x)^16)

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Maple [A]  time = 0.019, size = 392, normalized size = 1.1 \[ -{\frac{8008\,{x}^{6}{b}^{6}{e}^{6}+43680\,{x}^{5}a{b}^{5}{e}^{6}+4368\,{x}^{5}{b}^{6}d{e}^{5}+100100\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+18200\,{x}^{4}a{b}^{5}d{e}^{5}+1820\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+123200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+30800\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+5600\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+560\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+85800\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+26400\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+6600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+1200\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+120\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+32032\,x{a}^{5}b{e}^{6}+11440\,x{a}^{4}{b}^{2}d{e}^{5}+3520\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+880\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+160\,xa{b}^{5}{d}^{4}{e}^{2}+16\,x{b}^{6}{d}^{5}e+5005\,{a}^{6}{e}^{6}+2002\,{a}^{5}bd{e}^{5}+715\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+220\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+55\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}+10\,{d}^{5}a{b}^{5}e+{b}^{6}{d}^{6}}{80080\,{e}^{7} \left ( ex+d \right ) ^{16} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^17,x)

[Out]

-1/80080/e^7*(8008*b^6*e^6*x^6+43680*a*b^5*e^6*x^5+4368*b^6*d*e^5*x^5+100100*a^2
*b^4*e^6*x^4+18200*a*b^5*d*e^5*x^4+1820*b^6*d^2*e^4*x^4+123200*a^3*b^3*e^6*x^3+3
0800*a^2*b^4*d*e^5*x^3+5600*a*b^5*d^2*e^4*x^3+560*b^6*d^3*e^3*x^3+85800*a^4*b^2*
e^6*x^2+26400*a^3*b^3*d*e^5*x^2+6600*a^2*b^4*d^2*e^4*x^2+1200*a*b^5*d^3*e^3*x^2+
120*b^6*d^4*e^2*x^2+32032*a^5*b*e^6*x+11440*a^4*b^2*d*e^5*x+3520*a^3*b^3*d^2*e^4
*x+880*a^2*b^4*d^3*e^3*x+160*a*b^5*d^4*e^2*x+16*b^6*d^5*e*x+5005*a^6*e^6+2002*a^
5*b*d*e^5+715*a^4*b^2*d^2*e^4+220*a^3*b^3*d^3*e^3+55*a^2*b^4*d^4*e^2+10*a*b^5*d^
5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^16/(b*x+a)^5

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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((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^17,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300244, size = 699, normalized size = 1.93 \[ -\frac{8008 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 10 \, a b^{5} d^{5} e + 55 \, a^{2} b^{4} d^{4} e^{2} + 220 \, a^{3} b^{3} d^{3} e^{3} + 715 \, a^{4} b^{2} d^{2} e^{4} + 2002 \, a^{5} b d e^{5} + 5005 \, a^{6} e^{6} + 4368 \,{\left (b^{6} d e^{5} + 10 \, a b^{5} e^{6}\right )} x^{5} + 1820 \,{\left (b^{6} d^{2} e^{4} + 10 \, a b^{5} d e^{5} + 55 \, a^{2} b^{4} e^{6}\right )} x^{4} + 560 \,{\left (b^{6} d^{3} e^{3} + 10 \, a b^{5} d^{2} e^{4} + 55 \, a^{2} b^{4} d e^{5} + 220 \, a^{3} b^{3} e^{6}\right )} x^{3} + 120 \,{\left (b^{6} d^{4} e^{2} + 10 \, a b^{5} d^{3} e^{3} + 55 \, a^{2} b^{4} d^{2} e^{4} + 220 \, a^{3} b^{3} d e^{5} + 715 \, a^{4} b^{2} e^{6}\right )} x^{2} + 16 \,{\left (b^{6} d^{5} e + 10 \, a b^{5} d^{4} e^{2} + 55 \, a^{2} b^{4} d^{3} e^{3} + 220 \, a^{3} b^{3} d^{2} e^{4} + 715 \, a^{4} b^{2} d e^{5} + 2002 \, a^{5} b e^{6}\right )} x}{80080 \,{\left (e^{23} x^{16} + 16 \, d e^{22} x^{15} + 120 \, d^{2} e^{21} x^{14} + 560 \, d^{3} e^{20} x^{13} + 1820 \, d^{4} e^{19} x^{12} + 4368 \, d^{5} e^{18} x^{11} + 8008 \, d^{6} e^{17} x^{10} + 11440 \, d^{7} e^{16} x^{9} + 12870 \, d^{8} e^{15} x^{8} + 11440 \, d^{9} e^{14} x^{7} + 8008 \, d^{10} e^{13} x^{6} + 4368 \, d^{11} e^{12} x^{5} + 1820 \, d^{12} e^{11} x^{4} + 560 \, d^{13} e^{10} x^{3} + 120 \, d^{14} e^{9} x^{2} + 16 \, d^{15} e^{8} x + d^{16} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^17,x, algorithm="fricas")

[Out]

-1/80080*(8008*b^6*e^6*x^6 + b^6*d^6 + 10*a*b^5*d^5*e + 55*a^2*b^4*d^4*e^2 + 220
*a^3*b^3*d^3*e^3 + 715*a^4*b^2*d^2*e^4 + 2002*a^5*b*d*e^5 + 5005*a^6*e^6 + 4368*
(b^6*d*e^5 + 10*a*b^5*e^6)*x^5 + 1820*(b^6*d^2*e^4 + 10*a*b^5*d*e^5 + 55*a^2*b^4
*e^6)*x^4 + 560*(b^6*d^3*e^3 + 10*a*b^5*d^2*e^4 + 55*a^2*b^4*d*e^5 + 220*a^3*b^3
*e^6)*x^3 + 120*(b^6*d^4*e^2 + 10*a*b^5*d^3*e^3 + 55*a^2*b^4*d^2*e^4 + 220*a^3*b
^3*d*e^5 + 715*a^4*b^2*e^6)*x^2 + 16*(b^6*d^5*e + 10*a*b^5*d^4*e^2 + 55*a^2*b^4*
d^3*e^3 + 220*a^3*b^3*d^2*e^4 + 715*a^4*b^2*d*e^5 + 2002*a^5*b*e^6)*x)/(e^23*x^1
6 + 16*d*e^22*x^15 + 120*d^2*e^21*x^14 + 560*d^3*e^20*x^13 + 1820*d^4*e^19*x^12
+ 4368*d^5*e^18*x^11 + 8008*d^6*e^17*x^10 + 11440*d^7*e^16*x^9 + 12870*d^8*e^15*
x^8 + 11440*d^9*e^14*x^7 + 8008*d^10*e^13*x^6 + 4368*d^11*e^12*x^5 + 1820*d^12*e
^11*x^4 + 560*d^13*e^10*x^3 + 120*d^14*e^9*x^2 + 16*d^15*e^8*x + d^16*e^7)

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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((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**17,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.296047, size = 702, normalized size = 1.94 \[ -\frac{{\left (8008 \, b^{6} x^{6} e^{6}{\rm sign}\left (b x + a\right ) + 4368 \, b^{6} d x^{5} e^{5}{\rm sign}\left (b x + a\right ) + 1820 \, b^{6} d^{2} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 560 \, b^{6} d^{3} x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 120 \, b^{6} d^{4} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 16 \, b^{6} d^{5} x e{\rm sign}\left (b x + a\right ) + b^{6} d^{6}{\rm sign}\left (b x + a\right ) + 43680 \, a b^{5} x^{5} e^{6}{\rm sign}\left (b x + a\right ) + 18200 \, a b^{5} d x^{4} e^{5}{\rm sign}\left (b x + a\right ) + 5600 \, a b^{5} d^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 1200 \, a b^{5} d^{3} x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 160 \, a b^{5} d^{4} x e^{2}{\rm sign}\left (b x + a\right ) + 10 \, a b^{5} d^{5} e{\rm sign}\left (b x + a\right ) + 100100 \, a^{2} b^{4} x^{4} e^{6}{\rm sign}\left (b x + a\right ) + 30800 \, a^{2} b^{4} d x^{3} e^{5}{\rm sign}\left (b x + a\right ) + 6600 \, a^{2} b^{4} d^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 880 \, a^{2} b^{4} d^{3} x e^{3}{\rm sign}\left (b x + a\right ) + 55 \, a^{2} b^{4} d^{4} e^{2}{\rm sign}\left (b x + a\right ) + 123200 \, a^{3} b^{3} x^{3} e^{6}{\rm sign}\left (b x + a\right ) + 26400 \, a^{3} b^{3} d x^{2} e^{5}{\rm sign}\left (b x + a\right ) + 3520 \, a^{3} b^{3} d^{2} x e^{4}{\rm sign}\left (b x + a\right ) + 220 \, a^{3} b^{3} d^{3} e^{3}{\rm sign}\left (b x + a\right ) + 85800 \, a^{4} b^{2} x^{2} e^{6}{\rm sign}\left (b x + a\right ) + 11440 \, a^{4} b^{2} d x e^{5}{\rm sign}\left (b x + a\right ) + 715 \, a^{4} b^{2} d^{2} e^{4}{\rm sign}\left (b x + a\right ) + 32032 \, a^{5} b x e^{6}{\rm sign}\left (b x + a\right ) + 2002 \, a^{5} b d e^{5}{\rm sign}\left (b x + a\right ) + 5005 \, a^{6} e^{6}{\rm sign}\left (b x + a\right )\right )} e^{\left (-7\right )}}{80080 \,{\left (x e + d\right )}^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)/(e*x + d)^17,x, algorithm="giac")

[Out]

-1/80080*(8008*b^6*x^6*e^6*sign(b*x + a) + 4368*b^6*d*x^5*e^5*sign(b*x + a) + 18
20*b^6*d^2*x^4*e^4*sign(b*x + a) + 560*b^6*d^3*x^3*e^3*sign(b*x + a) + 120*b^6*d
^4*x^2*e^2*sign(b*x + a) + 16*b^6*d^5*x*e*sign(b*x + a) + b^6*d^6*sign(b*x + a)
+ 43680*a*b^5*x^5*e^6*sign(b*x + a) + 18200*a*b^5*d*x^4*e^5*sign(b*x + a) + 5600
*a*b^5*d^2*x^3*e^4*sign(b*x + a) + 1200*a*b^5*d^3*x^2*e^3*sign(b*x + a) + 160*a*
b^5*d^4*x*e^2*sign(b*x + a) + 10*a*b^5*d^5*e*sign(b*x + a) + 100100*a^2*b^4*x^4*
e^6*sign(b*x + a) + 30800*a^2*b^4*d*x^3*e^5*sign(b*x + a) + 6600*a^2*b^4*d^2*x^2
*e^4*sign(b*x + a) + 880*a^2*b^4*d^3*x*e^3*sign(b*x + a) + 55*a^2*b^4*d^4*e^2*si
gn(b*x + a) + 123200*a^3*b^3*x^3*e^6*sign(b*x + a) + 26400*a^3*b^3*d*x^2*e^5*sig
n(b*x + a) + 3520*a^3*b^3*d^2*x*e^4*sign(b*x + a) + 220*a^3*b^3*d^3*e^3*sign(b*x
+ a) + 85800*a^4*b^2*x^2*e^6*sign(b*x + a) + 11440*a^4*b^2*d*x*e^5*sign(b*x + a
) + 715*a^4*b^2*d^2*e^4*sign(b*x + a) + 32032*a^5*b*x*e^6*sign(b*x + a) + 2002*a
^5*b*d*e^5*sign(b*x + a) + 5005*a^6*e^6*sign(b*x + a))*e^(-7)/(x*e + d)^16