\(\int e^{-b x} x^{13/2} \, dx\) [48]

Optimal result

Integrand size = 12, antiderivative size = 151 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {135135 e^{-b x} \sqrt {x}}{64 b^7}-\frac {45045 e^{-b x} x^{3/2}}{32 b^6}-\frac {9009 e^{-b x} x^{5/2}}{16 b^5}-\frac {1287 e^{-b x} x^{7/2}}{8 b^4}-\frac {143 e^{-b x} x^{9/2}}{4 b^3}-\frac {13 e^{-b x} x^{11/2}}{2 b^2}-\frac {e^{-b x} x^{13/2}}{b}+\frac {135135 \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 b^{15/2}} \]

[Out]

-45045/32*x^(3/2)/b^6/exp(b*x)-9009/16*x^(5/2)/b^5/exp(b*x)-1287/8*x^(7/2)/b^4/exp(b*x)-143/4*x^(9/2)/b^3/exp(
b*x)-13/2*x^(11/2)/b^2/exp(b*x)-x^(13/2)/b/exp(b*x)+135135/128*erf(b^(1/2)*x^(1/2))*Pi^(1/2)/b^(15/2)-135135/6
4*x^(1/2)/b^7/exp(b*x)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2207, 2211, 2236} \[ \int e^{-b x} x^{13/2} \, dx=\frac {135135 \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 b^{15/2}}-\frac {135135 \sqrt {x} e^{-b x}}{64 b^7}-\frac {45045 x^{3/2} e^{-b x}}{32 b^6}-\frac {9009 x^{5/2} e^{-b x}}{16 b^5}-\frac {1287 x^{7/2} e^{-b x}}{8 b^4}-\frac {143 x^{9/2} e^{-b x}}{4 b^3}-\frac {13 x^{11/2} e^{-b x}}{2 b^2}-\frac {x^{13/2} e^{-b x}}{b} \]

[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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2236

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*} \text {integral}& = -\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 \text {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] (verified)

Time = 0.61 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.16 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {\sqrt {b x} \Gamma \left (\frac {15}{2},b x\right )}{b^8 \sqrt {x}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.52

method	result	size
meijerg	\(\frac {-\frac {\sqrt {x}\, \sqrt {b}\, \left (960 b^{6} x^{6}+6240 b^{5} x^{5}+34320 b^{4} x^{4}+154440 b^{3} x^{3}+540540 b^{2} x^{2}+1351350 b x +2027025\right ) {\mathrm e}^{-b x}}{960}+\frac {135135 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{128}}{b^{\frac {15}{2}}}\)	\(78\)
derivativedivides	\(-\frac {x^{\frac {13}{2}} {\mathrm e}^{-b x}}{b}+\frac {-\frac {13 x^{\frac {11}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {13 \left (-\frac {11 x^{\frac {9}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {11 \left (-\frac {9 x^{\frac {7}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {9 \left (-\frac {7 x^{\frac {5}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {7 \left (-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{b}}{b}\)	\(145\)
default	\(-\frac {x^{\frac {13}{2}} {\mathrm e}^{-b x}}{b}+\frac {-\frac {13 x^{\frac {11}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {13 \left (-\frac {11 x^{\frac {9}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {11 \left (-\frac {9 x^{\frac {7}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {9 \left (-\frac {7 x^{\frac {5}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {7 \left (-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{b}}{b}\)	\(145\)

[In]

int(x^(13/2)/exp(b*x),x,method=_RETURNVERBOSE)

[Out]

1/b^(15/2)*(-1/960*x^(1/2)*b^(1/2)*(960*b^6*x^6+6240*b^5*x^5+34320*b^4*x^4+154440*b^3*x^3+540540*b^2*x^2+13513
50*b*x+2027025)*exp(-b*x)+135135/128*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.54 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {2 \, {\left (64 \, b^{7} x^{6} + 416 \, b^{6} x^{5} + 2288 \, b^{5} x^{4} + 10296 \, b^{4} x^{3} + 36036 \, b^{3} x^{2} + 90090 \, b^{2} x + 135135 \, b\right )} \sqrt {x} e^{\left (-b x\right )} - 135135 \, \sqrt {\pi } \sqrt {b} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 \, b^{8}} \]

[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

Sympy [A] (verification not implemented)

Time = 127.00 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int e^{-b x} x^{13/2} \, dx=- \frac {x^{\frac {13}{2}} e^{- b x}}{b} - \frac {13 x^{\frac {11}{2}} e^{- b x}}{2 b^{2}} - \frac {143 x^{\frac {9}{2}} e^{- b x}}{4 b^{3}} - \frac {1287 x^{\frac {7}{2}} e^{- b x}}{8 b^{4}} - \frac {9009 x^{\frac {5}{2}} e^{- b x}}{16 b^{5}} - \frac {45045 x^{\frac {3}{2}} e^{- b x}}{32 b^{6}} - \frac {135135 \sqrt {x} e^{- b x}}{64 b^{7}} + \frac {135135 \sqrt {\pi } \operatorname {erf}{\left (\sqrt {b} \sqrt {x} \right )}}{128 b^{\frac {15}{2}}} \]

[In]

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) + 135135*sqrt(pi)*erf(sqrt(b)*sqrt(x))/(128*b**(15/2))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {{\left (64 \, b^{6} x^{\frac {13}{2}} + 416 \, b^{5} x^{\frac {11}{2}} + 2288 \, b^{4} x^{\frac {9}{2}} + 10296 \, b^{3} x^{\frac {7}{2}} + 36036 \, b^{2} x^{\frac {5}{2}} + 90090 \, b x^{\frac {3}{2}} + 135135 \, \sqrt {x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} + \frac {135135 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 \, b^{\frac {15}{2}}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.53 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {{\left (64 \, b^{6} x^{\frac {13}{2}} + 416 \, b^{5} x^{\frac {11}{2}} + 2288 \, b^{4} x^{\frac {9}{2}} + 10296 \, b^{3} x^{\frac {7}{2}} + 36036 \, b^{2} x^{\frac {5}{2}} + 90090 \, b x^{\frac {3}{2}} + 135135 \, \sqrt {x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} - \frac {135135 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} \sqrt {x}\right )}{128 \, b^{\frac {15}{2}}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.59 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {135135\,x^{13/2}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{128\,b\,{\left (b\,x\right )}^{13/2}}-\frac {x^{13/2}\,{\mathrm {e}}^{-b\,x}\,\left (\frac {135135\,\sqrt {b\,x}}{64}+\frac {45045\,{\left (b\,x\right )}^{3/2}}{32}+\frac {9009\,{\left (b\,x\right )}^{5/2}}{16}+\frac {1287\,{\left (b\,x\right )}^{7/2}}{8}+\frac {143\,{\left (b\,x\right )}^{9/2}}{4}+\frac {13\,{\left (b\,x\right )}^{11/2}}{2}+{\left (b\,x\right )}^{13/2}\right )}{b\,{\left (b\,x\right )}^{13/2}} \]

[In]

int(x^(13/2)*exp(-b*x),x)

[Out]

- (135135*x^(13/2)*pi^(1/2)*erfc((b*x)^(1/2)))/(128*b*(b*x)^(13/2)) - (x^(13/2)*exp(-b*x)*((135135*(b*x)^(1/2)
)/64 + (45045*(b*x)^(3/2))/32 + (9009*(b*x)^(5/2))/16 + (1287*(b*x)^(7/2))/8 + (143*(b*x)^(9/2))/4 + (13*(b*x)
^(11/2))/2 + (b*x)^(13/2)))/(b*(b*x)^(13/2))