3.1726 \(\int \frac{(A+B x) (d+e x)^{5/2}}{a+b x} \, dx\)

Optimal. Leaf size=164 \[ -\frac{2 (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)^2}{b^4}+\frac{2 (d+e x)^{3/2} (A b-a B) (b d-a e)}{3 b^3}+\frac{2 (d+e x)^{5/2} (A b-a B)}{5 b^2}+\frac{2 B (d+e x)^{7/2}}{7 b e} \]

[Out]

(2*(A*b - a*B)*(b*d - a*e)^2*Sqrt[d + e*x])/b^4 + (2*(A*b - a*B)*(b*d - a*e)*(d
+ e*x)^(3/2))/(3*b^3) + (2*(A*b - a*B)*(d + e*x)^(5/2))/(5*b^2) + (2*B*(d + e*x)
^(7/2))/(7*b*e) - (2*(A*b - a*B)*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/b^(9/2)

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Rubi [A]  time = 0.276734, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}}+\frac{2 \sqrt{d+e x} (A b-a B) (b d-a e)^2}{b^4}+\frac{2 (d+e x)^{3/2} (A b-a B) (b d-a e)}{3 b^3}+\frac{2 (d+e x)^{5/2} (A b-a B)}{5 b^2}+\frac{2 B (d+e x)^{7/2}}{7 b e} \]

Antiderivative was successfully verified.

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

[Out]

(2*(A*b - a*B)*(b*d - a*e)^2*Sqrt[d + e*x])/b^4 + (2*(A*b - a*B)*(b*d - a*e)*(d
+ e*x)^(3/2))/(3*b^3) + (2*(A*b - a*B)*(d + e*x)^(5/2))/(5*b^2) + (2*B*(d + e*x)
^(7/2))/(7*b*e) - (2*(A*b - a*B)*(b*d - a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x
])/Sqrt[b*d - a*e]])/b^(9/2)

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Rubi in Sympy [A]  time = 33.6048, size = 144, normalized size = 0.88 \[ \frac{2 B \left (d + e x\right )^{\frac{7}{2}}}{7 b e} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{2}} - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b - B a\right ) \left (a e - b d\right )}{3 b^{3}} + \frac{2 \sqrt{d + e x} \left (A b - B a\right ) \left (a e - b d\right )^{2}}{b^{4}} - \frac{2 \left (A b - B a\right ) \left (a e - b d\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(d + e*x)**(7/2)/(7*b*e) + 2*(d + e*x)**(5/2)*(A*b - B*a)/(5*b**2) - 2*(d +
e*x)**(3/2)*(A*b - B*a)*(a*e - b*d)/(3*b**3) + 2*sqrt(d + e*x)*(A*b - B*a)*(a*e
- b*d)**2/b**4 - 2*(A*b - B*a)*(a*e - b*d)**(5/2)*atan(sqrt(b)*sqrt(d + e*x)/sqr
t(a*e - b*d))/b**(9/2)

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Mathematica [A]  time = 0.319855, size = 185, normalized size = 1.13 \[ \frac{2 \sqrt{d+e x} \left (-105 a^3 B e^3+35 a^2 b e^2 (3 A e+7 B d+B e x)-7 a b^2 e \left (5 A e (7 d+e x)+B \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )+b^3 \left (7 A e \left (23 d^2+11 d e x+3 e^2 x^2\right )+15 B (d+e x)^3\right )\right )}{105 b^4 e}-\frac{2 (A b-a B) (b d-a e)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(-105*a^3*B*e^3 + 35*a^2*b*e^2*(7*B*d + 3*A*e + B*e*x) - 7*a*b^
2*e*(5*A*e*(7*d + e*x) + B*(23*d^2 + 11*d*e*x + 3*e^2*x^2)) + b^3*(15*B*(d + e*x
)^3 + 7*A*e*(23*d^2 + 11*d*e*x + 3*e^2*x^2))))/(105*b^4*e) - (2*(A*b - a*B)*(b*d
- a*e)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(9/2)

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Maple [B]  time = 0.016, size = 573, normalized size = 3.5 \[{\frac{2\,B}{7\,be} \left ( ex+d \right ) ^{{\frac{7}{2}}}}+{\frac{2\,A}{5\,b} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Ba}{5\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}}-{\frac{2\,Aae}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,Ad}{3\,b} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+{\frac{2\,eB{a}^{2}}{3\,{b}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{2\,Bad}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{{a}^{2}A{e}^{2}\sqrt{ex+d}}{{b}^{3}}}-4\,{\frac{aAde\sqrt{ex+d}}{{b}^{2}}}+2\,{\frac{A{d}^{2}\sqrt{ex+d}}{b}}-2\,{\frac{{e}^{2}B{a}^{3}\sqrt{ex+d}}{{b}^{4}}}+4\,{\frac{eB{a}^{2}d\sqrt{ex+d}}{{b}^{3}}}-2\,{\frac{Ba{d}^{2}\sqrt{ex+d}}{{b}^{2}}}-2\,{\frac{{a}^{3}A{e}^{3}}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+6\,{\frac{{a}^{2}A{e}^{2}d}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-6\,{\frac{aA{d}^{2}e}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{A{d}^{3}}{\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+2\,{\frac{B{a}^{4}{e}^{3}}{{b}^{4}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-6\,{\frac{{e}^{2}B{a}^{3}d}{{b}^{3}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }+6\,{\frac{eB{a}^{2}{d}^{2}}{{b}^{2}\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) }-2\,{\frac{Ba{d}^{3}}{b\sqrt{ \left ( ae-bd \right ) b}}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{ \left ( ae-bd \right ) b}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*B*(e*x+d)^(7/2)/b/e+2/5/b*A*(e*x+d)^(5/2)-2/5/b^2*B*(e*x+d)^(5/2)*a-2/3*e/b^
2*A*(e*x+d)^(3/2)*a+2/3/b*A*(e*x+d)^(3/2)*d+2/3*e/b^3*B*(e*x+d)^(3/2)*a^2-2/3/b^
2*B*(e*x+d)^(3/2)*a*d+2*e^2/b^3*A*a^2*(e*x+d)^(1/2)-4*e/b^2*A*a*d*(e*x+d)^(1/2)+
2/b*A*d^2*(e*x+d)^(1/2)-2*e^2/b^4*B*a^3*(e*x+d)^(1/2)+4*e/b^3*B*a^2*d*(e*x+d)^(1
/2)-2/b^2*B*a*d^2*(e*x+d)^(1/2)-2*e^3/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/
2)*b/((a*e-b*d)*b)^(1/2))*A*a^3+6*e^2/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/
2)*b/((a*e-b*d)*b)^(1/2))*A*a^2*d-6*e/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)
*b/((a*e-b*d)*b)^(1/2))*A*a*d^2+2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a
*e-b*d)*b)^(1/2))*A*d^3+2*e^3/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a
*e-b*d)*b)^(1/2))*B*a^4-6*e^2/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a
*e-b*d)*b)^(1/2))*B*a^3*d+6*e/b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a
*e-b*d)*b)^(1/2))*B*a^2*d^2-2/b/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)*b/((a*e
-b*d)*b)^(1/2))*B*a*d^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241404, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} +{\left (B a^{3} - A a^{2} b\right )} e^{3}\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e + 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (15 \, B b^{3} e^{3} x^{3} + 15 \, B b^{3} d^{3} - 161 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e + 245 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} - 105 \,{\left (B a^{3} - A a^{2} b\right )} e^{3} + 3 \,{\left (15 \, B b^{3} d e^{2} - 7 \,{\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{2} +{\left (45 \, B b^{3} d^{2} e - 77 \,{\left (B a b^{2} - A b^{3}\right )} d e^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, b^{4} e}, \frac{2 \,{\left (105 \,{\left ({\left (B a b^{2} - A b^{3}\right )} d^{2} e - 2 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} +{\left (B a^{3} - A a^{2} b\right )} e^{3}\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) +{\left (15 \, B b^{3} e^{3} x^{3} + 15 \, B b^{3} d^{3} - 161 \,{\left (B a b^{2} - A b^{3}\right )} d^{2} e + 245 \,{\left (B a^{2} b - A a b^{2}\right )} d e^{2} - 105 \,{\left (B a^{3} - A a^{2} b\right )} e^{3} + 3 \,{\left (15 \, B b^{3} d e^{2} - 7 \,{\left (B a b^{2} - A b^{3}\right )} e^{3}\right )} x^{2} +{\left (45 \, B b^{3} d^{2} e - 77 \,{\left (B a b^{2} - A b^{3}\right )} d e^{2} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} e^{3}\right )} x\right )} \sqrt{e x + d}\right )}}{105 \, b^{4} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/105*(105*((B*a*b^2 - A*b^3)*d^2*e - 2*(B*a^2*b - A*a*b^2)*d*e^2 + (B*a^3 - A*
a^2*b)*e^3)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqr
t((b*d - a*e)/b))/(b*x + a)) + 2*(15*B*b^3*e^3*x^3 + 15*B*b^3*d^3 - 161*(B*a*b^2
- A*b^3)*d^2*e + 245*(B*a^2*b - A*a*b^2)*d*e^2 - 105*(B*a^3 - A*a^2*b)*e^3 + 3*
(15*B*b^3*d*e^2 - 7*(B*a*b^2 - A*b^3)*e^3)*x^2 + (45*B*b^3*d^2*e - 77*(B*a*b^2 -
A*b^3)*d*e^2 + 35*(B*a^2*b - A*a*b^2)*e^3)*x)*sqrt(e*x + d))/(b^4*e), 2/105*(10
5*((B*a*b^2 - A*b^3)*d^2*e - 2*(B*a^2*b - A*a*b^2)*d*e^2 + (B*a^3 - A*a^2*b)*e^3
)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) + (15*B*b^3*e^
3*x^3 + 15*B*b^3*d^3 - 161*(B*a*b^2 - A*b^3)*d^2*e + 245*(B*a^2*b - A*a*b^2)*d*e
^2 - 105*(B*a^3 - A*a^2*b)*e^3 + 3*(15*B*b^3*d*e^2 - 7*(B*a*b^2 - A*b^3)*e^3)*x^
2 + (45*B*b^3*d^2*e - 77*(B*a*b^2 - A*b^3)*d*e^2 + 35*(B*a^2*b - A*a*b^2)*e^3)*x
)*sqrt(e*x + d))/(b^4*e)]

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Sympy [A]  time = 75.4846, size = 340, normalized size = 2.07 \[ \frac{2 B \left (d + e x\right )^{\frac{7}{2}}}{7 b e} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 A b - 2 B a\right )}{5 b^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (- 2 A a b e + 2 A b^{2} d + 2 B a^{2} e - 2 B a b d\right )}{3 b^{3}} + \frac{\sqrt{d + e x} \left (2 A a^{2} b e^{2} - 4 A a b^{2} d e + 2 A b^{3} d^{2} - 2 B a^{3} e^{2} + 4 B a^{2} b d e - 2 B a b^{2} d^{2}\right )}{b^{4}} + \frac{2 \left (- A b + B a\right ) \left (a e - b d\right )^{3} \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e - b d}{b}}} \right )}}{b \sqrt{\frac{a e - b d}{b}}} & \text{for}\: \frac{a e - b d}{b} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: d + e x > \frac{- a e + b d}{b} \wedge \frac{a e - b d}{b} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{- a e + b d}{b}}} \right )}}{b \sqrt{\frac{- a e + b d}{b}}} & \text{for}\: \frac{a e - b d}{b} < 0 \wedge d + e x < \frac{- a e + b d}{b} \end{cases}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(d + e*x)**(7/2)/(7*b*e) + (d + e*x)**(5/2)*(2*A*b - 2*B*a)/(5*b**2) + (d +
e*x)**(3/2)*(-2*A*a*b*e + 2*A*b**2*d + 2*B*a**2*e - 2*B*a*b*d)/(3*b**3) + sqrt(d
+ e*x)*(2*A*a**2*b*e**2 - 4*A*a*b**2*d*e + 2*A*b**3*d**2 - 2*B*a**3*e**2 + 4*B*
a**2*b*d*e - 2*B*a*b**2*d**2)/b**4 + 2*(-A*b + B*a)*(a*e - b*d)**3*Piecewise((at
an(sqrt(d + e*x)/sqrt((a*e - b*d)/b))/(b*sqrt((a*e - b*d)/b)), (a*e - b*d)/b > 0
), (-acoth(sqrt(d + e*x)/sqrt((-a*e + b*d)/b))/(b*sqrt((-a*e + b*d)/b)), ((a*e -
b*d)/b < 0) & (d + e*x > (-a*e + b*d)/b)), (-atanh(sqrt(d + e*x)/sqrt((-a*e + b
*d)/b))/(b*sqrt((-a*e + b*d)/b)), ((a*e - b*d)/b < 0) & (d + e*x < (-a*e + b*d)/
b)))/b**4

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GIAC/XCAS [A]  time = 0.221615, size = 501, normalized size = 3.05 \[ -\frac{2 \,{\left (B a b^{3} d^{3} - A b^{4} d^{3} - 3 \, B a^{2} b^{2} d^{2} e + 3 \, A a b^{3} d^{2} e + 3 \, B a^{3} b d e^{2} - 3 \, A a^{2} b^{2} d e^{2} - B a^{4} e^{3} + A a^{3} b e^{3}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{4}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B b^{6} e^{6} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B a b^{5} e^{7} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A b^{6} e^{7} - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B a b^{5} d e^{7} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{6} d e^{7} - 105 \, \sqrt{x e + d} B a b^{5} d^{2} e^{7} + 105 \, \sqrt{x e + d} A b^{6} d^{2} e^{7} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B a^{2} b^{4} e^{8} - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A a b^{5} e^{8} + 210 \, \sqrt{x e + d} B a^{2} b^{4} d e^{8} - 210 \, \sqrt{x e + d} A a b^{5} d e^{8} - 105 \, \sqrt{x e + d} B a^{3} b^{3} e^{9} + 105 \, \sqrt{x e + d} A a^{2} b^{4} e^{9}\right )} e^{\left (-7\right )}}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(B*a*b^3*d^3 - A*b^4*d^3 - 3*B*a^2*b^2*d^2*e + 3*A*a*b^3*d^2*e + 3*B*a^3*b*d*
e^2 - 3*A*a^2*b^2*d*e^2 - B*a^4*e^3 + A*a^3*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-
b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) + 2/105*(15*(x*e + d)^(7/2)*B*b^6*e^6
- 21*(x*e + d)^(5/2)*B*a*b^5*e^7 + 21*(x*e + d)^(5/2)*A*b^6*e^7 - 35*(x*e + d)^
(3/2)*B*a*b^5*d*e^7 + 35*(x*e + d)^(3/2)*A*b^6*d*e^7 - 105*sqrt(x*e + d)*B*a*b^5
*d^2*e^7 + 105*sqrt(x*e + d)*A*b^6*d^2*e^7 + 35*(x*e + d)^(3/2)*B*a^2*b^4*e^8 -
35*(x*e + d)^(3/2)*A*a*b^5*e^8 + 210*sqrt(x*e + d)*B*a^2*b^4*d*e^8 - 210*sqrt(x*
e + d)*A*a*b^5*d*e^8 - 105*sqrt(x*e + d)*B*a^3*b^3*e^9 + 105*sqrt(x*e + d)*A*a^2
*b^4*e^9)*e^(-7)/b^7