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

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

[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)) - (128
7*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[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(128*b^(15/2))

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Rubi [A]  time = 0.154831, 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, Rules used = {2176, 2180, 2205} \[ \frac{135135 \sqrt{\pi } \text{Erf}\left (\sqrt{b} \sqrt{x}\right )}{128 b^{15/2}}-\frac{13 x^{11/2} e^{-b x}}{2 b^2}-\frac{143 x^{9/2} e^{-b x}}{4 b^3}-\frac{1287 x^{7/2} e^{-b x}}{8 b^4}-\frac{9009 x^{5/2} e^{-b x}}{16 b^5}-\frac{45045 x^{3/2} e^{-b x}}{32 b^6}-\frac{135135 \sqrt{x} e^{-b x}}{64 b^7}-\frac{x^{13/2} e^{-b x}}{b} \]

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)) - (128
7*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[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(128*b^(15/2))

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int e^{-b x} x^{13/2} \, dx &=-\frac{e^{-b x} x^{13/2}}{b}+\frac{13 \int e^{-b x} x^{11/2} \, dx}{2 b}\\ &=-\frac{13 e^{-b x} x^{11/2}}{2 b^2}-\frac{e^{-b x} x^{13/2}}{b}+\frac{143 \int e^{-b x} x^{9/2} \, dx}{4 b^2}\\ &=-\frac{143 e^{-b x} x^{9/2}}{4 b^3}-\frac{13 e^{-b x} x^{11/2}}{2 b^2}-\frac{e^{-b x} x^{13/2}}{b}+\frac{1287 \int e^{-b x} x^{7/2} \, dx}{8 b^3}\\ &=-\frac{1287 e^{-b x} x^{7/2}}{8 b^4}-\frac{143 e^{-b x} x^{9/2}}{4 b^3}-\frac{13 e^{-b x} x^{11/2}}{2 b^2}-\frac{e^{-b x} x^{13/2}}{b}+\frac{9009 \int e^{-b x} x^{5/2} \, dx}{16 b^4}\\ &=-\frac{9009 e^{-b x} x^{5/2}}{16 b^5}-\frac{1287 e^{-b x} x^{7/2}}{8 b^4}-\frac{143 e^{-b x} x^{9/2}}{4 b^3}-\frac{13 e^{-b x} x^{11/2}}{2 b^2}-\frac{e^{-b x} x^{13/2}}{b}+\frac{45045 \int e^{-b x} x^{3/2} \, dx}{32 b^5}\\ &=-\frac{45045 e^{-b x} x^{3/2}}{32 b^6}-\frac{9009 e^{-b x} x^{5/2}}{16 b^5}-\frac{1287 e^{-b x} x^{7/2}}{8 b^4}-\frac{143 e^{-b x} x^{9/2}}{4 b^3}-\frac{13 e^{-b x} x^{11/2}}{2 b^2}-\frac{e^{-b x} x^{13/2}}{b}+\frac{135135 \int e^{-b x} \sqrt{x} \, dx}{64 b^6}\\ &=-\frac{135135 e^{-b x} \sqrt{x}}{64 b^7}-\frac{45045 e^{-b x} x^{3/2}}{32 b^6}-\frac{9009 e^{-b x} x^{5/2}}{16 b^5}-\frac{1287 e^{-b x} x^{7/2}}{8 b^4}-\frac{143 e^{-b x} x^{9/2}}{4 b^3}-\frac{13 e^{-b x} x^{11/2}}{2 b^2}-\frac{e^{-b x} x^{13/2}}{b}+\frac{135135 \int \frac{e^{-b x}}{\sqrt{x}} \, dx}{128 b^7}\\ &=-\frac{135135 e^{-b x} \sqrt{x}}{64 b^7}-\frac{45045 e^{-b x} x^{3/2}}{32 b^6}-\frac{9009 e^{-b x} x^{5/2}}{16 b^5}-\frac{1287 e^{-b x} x^{7/2}}{8 b^4}-\frac{143 e^{-b x} x^{9/2}}{4 b^3}-\frac{13 e^{-b x} x^{11/2}}{2 b^2}-\frac{e^{-b x} x^{13/2}}{b}+\frac{135135 \operatorname{Subst}\left (\int e^{-b x^2} \, dx,x,\sqrt{x}\right )}{64 b^7}\\ &=-\frac{135135 e^{-b x} \sqrt{x}}{64 b^7}-\frac{45045 e^{-b x} x^{3/2}}{32 b^6}-\frac{9009 e^{-b x} x^{5/2}}{16 b^5}-\frac{1287 e^{-b x} x^{7/2}}{8 b^4}-\frac{143 e^{-b x} x^{9/2}}{4 b^3}-\frac{13 e^{-b x} x^{11/2}}{2 b^2}-\frac{e^{-b x} x^{13/2}}{b}+\frac{135135 \sqrt{\pi } \text{erf}\left (\sqrt{b} \sqrt{x}\right )}{128 b^{15/2}}\\ \end{align*}

Mathematica [A]  time = 0.0041943, size = 24, normalized size = 0.16 \[ -\frac{\sqrt{b x} \text{Gamma}\left (\frac{15}{2},b x\right )}{b^8 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(13/2)/E^(b*x),x]

[Out]

-((Sqrt[b*x]*Gamma[15/2, b*x])/(b^8*Sqrt[x]))

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Maple [A]  time = 0.135, size = 145, normalized size = 1. \begin{align*} -{\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 ) } \end{align*}

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)*exp(-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)*erf(b^(1/2)*x^(1/2))))))))

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Maxima [A]  time = 1.07258, size = 107, normalized size = 0.71 \begin{align*} -\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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/exp(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 + 135135/128*sqrt(pi)*erf(sqrt(b)*sqrt(x))/b^(15/2)

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Fricas [A]  time = 1.51932, size = 242, normalized size = 1.6 \begin{align*} -\frac{2 \,{\left (64 \, b^{7} x^{6} + 416 \, b^{6} x^{5} + 2288 \, b^{5} x^{4} + 10296 \, b^{4} x^{3} + 36036 \, b^{3} x^{2} + 90090 \, b^{2} x + 135135 \, b\right )} \sqrt{x} e^{\left (-b x\right )} - 135135 \, \sqrt{\pi } \sqrt{b} \operatorname{erf}\left (\sqrt{b} \sqrt{x}\right )}{128 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/exp(b*x),x, algorithm="fricas")

[Out]

-1/128*(2*(64*b^7*x^6 + 416*b^6*x^5 + 2288*b^5*x^4 + 10296*b^4*x^3 + 36036*b^3*x^2 + 90090*b^2*x + 135135*b)*s
qrt(x)*e^(-b*x) - 135135*sqrt(pi)*sqrt(b)*erf(sqrt(b)*sqrt(x)))/b^8

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

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 [A]  time = 1.1608, size = 108, normalized size = 0.72 \begin{align*} -\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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/exp(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 - 135135/128*sqrt(pi)*erf(-sqrt(b)*sqrt(x))/b^(15/2)