3.48 \(\int e^{-b x} x^{13/2} \, dx\)

Optimal. Leaf size=151 \[ \frac{135135 \sqrt{\pi } \text{Erf}\left (\sqrt{b} \sqrt{x}\right )}{128 b^{15/2}}-\frac{135135 \sqrt{x} e^{-b x}}{64 b^7}-\frac{45045 x^{3/2} e^{-b x}}{32 b^6}-\frac{9009 x^{5/2} e^{-b x}}{16 b^5}-\frac{1287 x^{7/2} e^{-b x}}{8 b^4}-\frac{143 x^{9/2} e^{-b x}}{4 b^3}-\frac{13 x^{11/2} e^{-b x}}{2 b^2}-\frac{x^{13/2} e^{-b x}}{b} \]

[Out]

(-135135*Sqrt[x])/(64*b^7*E^(b*x)) - (45045*x^(3/2))/(32*b^6*E^(b*x)) - (9009*x^
(5/2))/(16*b^5*E^(b*x)) - (1287*x^(7/2))/(8*b^4*E^(b*x)) - (143*x^(9/2))/(4*b^3*
E^(b*x)) - (13*x^(11/2))/(2*b^2*E^(b*x)) - x^(13/2)/(b*E^(b*x)) + (135135*Sqrt[P
i]*Erf[Sqrt[b]*Sqrt[x]])/(128*b^(15/2))

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Rubi [A]  time = 0.24062, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{135135 \sqrt{\pi } \text{Erf}\left (\sqrt{b} \sqrt{x}\right )}{128 b^{15/2}}-\frac{135135 \sqrt{x} e^{-b x}}{64 b^7}-\frac{45045 x^{3/2} e^{-b x}}{32 b^6}-\frac{9009 x^{5/2} e^{-b x}}{16 b^5}-\frac{1287 x^{7/2} e^{-b x}}{8 b^4}-\frac{143 x^{9/2} e^{-b x}}{4 b^3}-\frac{13 x^{11/2} e^{-b x}}{2 b^2}-\frac{x^{13/2} e^{-b x}}{b} \]

Antiderivative was successfully verified.

[In]  Int[x^(13/2)/E^(b*x),x]

[Out]

(-135135*Sqrt[x])/(64*b^7*E^(b*x)) - (45045*x^(3/2))/(32*b^6*E^(b*x)) - (9009*x^
(5/2))/(16*b^5*E^(b*x)) - (1287*x^(7/2))/(8*b^4*E^(b*x)) - (143*x^(9/2))/(4*b^3*
E^(b*x)) - (13*x^(11/2))/(2*b^2*E^(b*x)) - x^(13/2)/(b*E^(b*x)) + (135135*Sqrt[P
i]*Erf[Sqrt[b]*Sqrt[x]])/(128*b^(15/2))

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Rubi in Sympy [A]  time = 30.5925, size = 138, normalized size = 0.91 \[ - \frac{x^{\frac{13}{2}} e^{- b x}}{b} - \frac{13 x^{\frac{11}{2}} e^{- b x}}{2 b^{2}} - \frac{143 x^{\frac{9}{2}} e^{- b x}}{4 b^{3}} - \frac{1287 x^{\frac{7}{2}} e^{- b x}}{8 b^{4}} - \frac{9009 x^{\frac{5}{2}} e^{- b x}}{16 b^{5}} - \frac{45045 x^{\frac{3}{2}} e^{- b x}}{32 b^{6}} - \frac{135135 \sqrt{x} e^{- b x}}{64 b^{7}} + \frac{135135 \sqrt{\pi } \operatorname{erf}{\left (\sqrt{b} \sqrt{x} \right )}}{128 b^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(13/2)/exp(b*x),x)

[Out]

-x**(13/2)*exp(-b*x)/b - 13*x**(11/2)*exp(-b*x)/(2*b**2) - 143*x**(9/2)*exp(-b*x
)/(4*b**3) - 1287*x**(7/2)*exp(-b*x)/(8*b**4) - 9009*x**(5/2)*exp(-b*x)/(16*b**5
) - 45045*x**(3/2)*exp(-b*x)/(32*b**6) - 135135*sqrt(x)*exp(-b*x)/(64*b**7) + 13
5135*sqrt(pi)*erf(sqrt(b)*sqrt(x))/(128*b**(15/2))

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Mathematica [A]  time = 0.0656602, size = 91, normalized size = 0.6 \[ \frac{135135 \sqrt{\pi } \text{Erf}\left (\sqrt{b} \sqrt{x}\right )}{128 b^{15/2}}-\frac{\sqrt{x} e^{-b x} \left (64 b^6 x^6+416 b^5 x^5+2288 b^4 x^4+10296 b^3 x^3+36036 b^2 x^2+90090 b x+135135\right )}{64 b^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(13/2)/E^(b*x),x]

[Out]

-(Sqrt[x]*(135135 + 90090*b*x + 36036*b^2*x^2 + 10296*b^3*x^3 + 2288*b^4*x^4 + 4
16*b^5*x^5 + 64*b^6*x^6))/(64*b^7*E^(b*x)) + (135135*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x
]])/(128*b^(15/2))

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Maple [A]  time = 0.011, size = 145, normalized size = 1. \[ -{\frac{{{\rm e}^{-bx}}}{b}{x}^{{\frac{13}{2}}}}+13\,{\frac{1}{b} \left ( -1/2\,{\frac{{x}^{11/2}{{\rm e}^{-bx}}}{b}}+11/2\,{\frac{1}{b} \left ( -1/2\,{\frac{{x}^{9/2}{{\rm e}^{-bx}}}{b}}+9/2\,{\frac{1}{b} \left ( -1/2\,{\frac{{x}^{7/2}{{\rm e}^{-bx}}}{b}}+7/2\,{\frac{1}{b} \left ( -1/2\,{\frac{{x}^{5/2}{{\rm e}^{-bx}}}{b}}+5/2\,{\frac{1}{b} \left ( -1/2\,{\frac{{x}^{3/2}{{\rm e}^{-bx}}}{b}}+3/2\,{\frac{1}{b} \left ( -1/2\,{\frac{\sqrt{x}{{\rm e}^{-bx}}}{b}}+1/4\,{\frac{\sqrt{\pi }{\it Erf} \left ( \sqrt{b}\sqrt{x} \right ) }{{b}^{3/2}}} \right ) } \right ) } \right ) } \right ) } \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(13/2)/exp(b*x),x)

[Out]

-1/b*x^(13/2)*exp(-b*x)+13/b*(-1/2/b*x^(11/2)*exp(-b*x)+11/2/b*(-1/2/b*x^(9/2)*e
xp(-b*x)+9/2/b*(-1/2/b*x^(7/2)*exp(-b*x)+7/2/b*(-1/2/b*x^(5/2)*exp(-b*x)+5/2/b*(
-1/2/b*x^(3/2)*exp(-b*x)+3/2/b*(-1/2/b*x^(1/2)*exp(-b*x)+1/4/b^(3/2)*Pi^(1/2)*er
f(b^(1/2)*x^(1/2))))))))

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Maxima [A]  time = 0.803118, size = 107, normalized size = 0.71 \[ -\frac{{\left (64 \, b^{6} x^{\frac{13}{2}} + 416 \, b^{5} x^{\frac{11}{2}} + 2288 \, b^{4} x^{\frac{9}{2}} + 10296 \, b^{3} x^{\frac{7}{2}} + 36036 \, b^{2} x^{\frac{5}{2}} + 90090 \, b x^{\frac{3}{2}} + 135135 \, \sqrt{x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} + \frac{135135 \, \sqrt{\pi } \operatorname{erf}\left (\sqrt{b} \sqrt{x}\right )}{128 \, b^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)*e^(-b*x),x, algorithm="maxima")

[Out]

-1/64*(64*b^6*x^(13/2) + 416*b^5*x^(11/2) + 2288*b^4*x^(9/2) + 10296*b^3*x^(7/2)
+ 36036*b^2*x^(5/2) + 90090*b*x^(3/2) + 135135*sqrt(x))*e^(-b*x)/b^7 + 135135/1
28*sqrt(pi)*erf(sqrt(b)*sqrt(x))/b^(15/2)

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Fricas [A]  time = 0.260485, size = 105, normalized size = 0.7 \[ -\frac{2 \,{\left (64 \, b^{6} x^{6} + 416 \, b^{5} x^{5} + 2288 \, b^{4} x^{4} + 10296 \, b^{3} x^{3} + 36036 \, b^{2} x^{2} + 90090 \, b x + 135135\right )} \sqrt{b} \sqrt{x} e^{\left (-b x\right )} - 135135 \, \sqrt{\pi } \operatorname{erf}\left (\sqrt{b} \sqrt{x}\right )}{128 \, b^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)*e^(-b*x),x, algorithm="fricas")

[Out]

-1/128*(2*(64*b^6*x^6 + 416*b^5*x^5 + 2288*b^4*x^4 + 10296*b^3*x^3 + 36036*b^2*x
^2 + 90090*b*x + 135135)*sqrt(b)*sqrt(x)*e^(-b*x) - 135135*sqrt(pi)*erf(sqrt(b)*
sqrt(x)))/b^(15/2)

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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(x**(13/2)/exp(b*x),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252752, size = 108, normalized size = 0.72 \[ -\frac{{\left (64 \, b^{6} x^{\frac{13}{2}} + 416 \, b^{5} x^{\frac{11}{2}} + 2288 \, b^{4} x^{\frac{9}{2}} + 10296 \, b^{3} x^{\frac{7}{2}} + 36036 \, b^{2} x^{\frac{5}{2}} + 90090 \, b x^{\frac{3}{2}} + 135135 \, \sqrt{x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} - \frac{135135 \, \sqrt{\pi } \operatorname{erf}\left (-\sqrt{b} \sqrt{x}\right )}{128 \, b^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)*e^(-b*x),x, algorithm="giac")

[Out]

-1/64*(64*b^6*x^(13/2) + 416*b^5*x^(11/2) + 2288*b^4*x^(9/2) + 10296*b^3*x^(7/2)
+ 36036*b^2*x^(5/2) + 90090*b*x^(3/2) + 135135*sqrt(x))*e^(-b*x)/b^7 - 135135/1
28*sqrt(pi)*erf(-sqrt(b)*sqrt(x))/b^(15/2)