\(\int e^{-b x} x^{13/2} \, dx\) [78]

Optimal result

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

-135135/64*x^(1/2)/b^7/exp(b*x)-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*Pi^(1/2)*erf(b^(1/ 
2)*x^(1/2))/b^(15/2)
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.16 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {\sqrt {b x} \Gamma \left (\frac {15}{2},b x\right )}{b^8 \sqrt {x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.24, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {2607, 2607, 2607, 2607, 2607, 2607, 2607, 2611, 2634}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2607

\(\displaystyle \frac {13 \int e^{-b x} x^{11/2}dx}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {13 \left (\frac {11 \int e^{-b x} x^{9/2}dx}{2 b}-\frac {x^{11/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {13 \left (\frac {11 \left (\frac {9 \int e^{-b x} x^{7/2}dx}{2 b}-\frac {x^{9/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{11/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \int e^{-b x} x^{5/2}dx}{2 b}-\frac {x^{7/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{9/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{11/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \int e^{-b x} x^{3/2}dx}{2 b}-\frac {x^{5/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{7/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{9/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{11/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int e^{-b x} \sqrt {x}dx}{2 b}-\frac {x^{3/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{5/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{7/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{9/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{11/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {e^{-b x}}{\sqrt {x}}dx}{2 b}-\frac {\sqrt {x} e^{-b x}}{b}\right )}{2 b}-\frac {x^{3/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{5/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{7/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{9/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{11/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\int e^{-b x}d\sqrt {x}}{b}-\frac {\sqrt {x} e^{-b x}}{b}\right )}{2 b}-\frac {x^{3/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{5/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{7/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{9/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{11/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {13 \left (\frac {11 \left (\frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{2 b^{3/2}}-\frac {\sqrt {x} e^{-b x}}{b}\right )}{2 b}-\frac {x^{3/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{5/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{7/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{9/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{11/2} e^{-b x}}{b}\right )}{2 b}-\frac {x^{13/2} e^{-b x}}{b}\)

Input:

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

Output:

-(x^(13/2)/(b*E^(b*x))) + (13*(-(x^(11/2)/(b*E^(b*x))) + (11*(-(x^(9/2)/(b 
*E^(b*x))) + (9*(-(x^(7/2)/(b*E^(b*x))) + (7*(-(x^(5/2)/(b*E^(b*x))) + (5* 
(-(x^(3/2)/(b*E^(b*x))) + (3*(-(Sqrt[x]/(b*E^(b*x))) + (Sqrt[Pi]*Erf[Sqrt[ 
b]*Sqrt[x]])/(2*b^(3/2))))/(2*b)))/(2*b)))/(2*b)))/(2*b)))/(2*b)))/(2*b)
 

Defintions of rubi rules used

rule 2607

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] - Simp[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 2611

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : 
> Simp[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 2634

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] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

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	\(-\frac {x^{\frac {13}{2}} {\mathrm e}^{-b x}}{b}+\frac {-\frac {13 x^{\frac {11}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {13 \left (-\frac {11 x^{\frac {9}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {11 \left (-\frac {9 x^{\frac {7}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {9 \left (-\frac {7 x^{\frac {5}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {7 \left (-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{b}}{b}\)	\(145\)

Input:

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

Output:

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+1351350*b*x+2027025)*exp(-b*x)+135135/128*P 
i^(1/2)*erf(b^(1/2)*x^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.54 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {2 \, {\left (64 \, b^{7} x^{6} + 416 \, b^{6} x^{5} + 2288 \, b^{5} x^{4} + 10296 \, b^{4} x^{3} + 36036 \, b^{3} x^{2} + 90090 \, b^{2} x + 135135 \, b\right )} \sqrt {x} e^{\left (-b x\right )} - 135135 \, \sqrt {\pi } \sqrt {b} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 \, b^{8}} \] Input:

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

Output:

-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)*sqrt(x)*e^(-b*x) - 135135*sqrt(pi)*sqrt 
(b)*erf(sqrt(b)*sqrt(x)))/b^8
 

Sympy [A] (verification not implemented)

Time = 124.40 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int e^{-b x} x^{13/2} \, dx=- \frac {x^{\frac {13}{2}} e^{- b x}}{b} - \frac {13 x^{\frac {11}{2}} e^{- b x}}{2 b^{2}} - \frac {143 x^{\frac {9}{2}} e^{- b x}}{4 b^{3}} - \frac {1287 x^{\frac {7}{2}} e^{- b x}}{8 b^{4}} - \frac {9009 x^{\frac {5}{2}} e^{- b x}}{16 b^{5}} - \frac {45045 x^{\frac {3}{2}} e^{- b x}}{32 b^{6}} - \frac {135135 \sqrt {x} e^{- b x}}{64 b^{7}} + \frac {135135 \sqrt {\pi } \operatorname {erf}{\left (\sqrt {b} \sqrt {x} \right )}}{128 b^{\frac {15}{2}}} \] Input:

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

Output:

-x**(13/2)*exp(-b*x)/b - 13*x**(11/2)*exp(-b*x)/(2*b**2) - 143*x**(9/2)*ex 
p(-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)

Time = 0.03 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.52 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {{\left (64 \, b^{6} x^{\frac {13}{2}} + 416 \, b^{5} x^{\frac {11}{2}} + 2288 \, b^{4} x^{\frac {9}{2}} + 10296 \, b^{3} x^{\frac {7}{2}} + 36036 \, b^{2} x^{\frac {5}{2}} + 90090 \, b x^{\frac {3}{2}} + 135135 \, \sqrt {x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} + \frac {135135 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 \, b^{\frac {15}{2}}} \] Input:

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

Output:

-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)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.53 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {{\left (64 \, b^{6} x^{\frac {13}{2}} + 416 \, b^{5} x^{\frac {11}{2}} + 2288 \, b^{4} x^{\frac {9}{2}} + 10296 \, b^{3} x^{\frac {7}{2}} + 36036 \, b^{2} x^{\frac {5}{2}} + 90090 \, b x^{\frac {3}{2}} + 135135 \, \sqrt {x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} - \frac {135135 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} \sqrt {x}\right )}{128 \, b^{\frac {15}{2}}} \] Input:

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

Output:

-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 = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.59 \[ \int e^{-b x} x^{13/2} \, dx=-\frac {135135\,x^{13/2}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{128\,b\,{\left (b\,x\right )}^{13/2}}-\frac {x^{13/2}\,{\mathrm {e}}^{-b\,x}\,\left (\frac {135135\,\sqrt {b\,x}}{64}+\frac {45045\,{\left (b\,x\right )}^{3/2}}{32}+\frac {9009\,{\left (b\,x\right )}^{5/2}}{16}+\frac {1287\,{\left (b\,x\right )}^{7/2}}{8}+\frac {143\,{\left (b\,x\right )}^{9/2}}{4}+\frac {13\,{\left (b\,x\right )}^{11/2}}{2}+{\left (b\,x\right )}^{13/2}\right )}{b\,{\left (b\,x\right )}^{13/2}} \] Input:

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

Output:

- (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))
 

Reduce [F]

\[ \int e^{-b x} x^{13/2} \, dx=\frac {135135 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right )-128 \sqrt {x}\, b^{6} x^{6}-832 \sqrt {x}\, b^{5} x^{5}-4576 \sqrt {x}\, b^{4} x^{4}-20592 \sqrt {x}\, b^{3} x^{3}-72072 \sqrt {x}\, b^{2} x^{2}-180180 \sqrt {x}\, b x -270270 \sqrt {x}}{128 e^{b x} b^{7}} \] Input:

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

Output:

(135135*e**(b*x)*int(sqrt(x)/(e**(b*x)*x),x) - 128*sqrt(x)*b**6*x**6 - 832 
*sqrt(x)*b**5*x**5 - 4576*sqrt(x)*b**4*x**4 - 20592*sqrt(x)*b**3*x**3 - 72 
072*sqrt(x)*b**2*x**2 - 180180*sqrt(x)*b*x - 270270*sqrt(x))/(128*e**(b*x) 
*b**7)