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]

-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)

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Rubi [A]  time = 0.15, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 0.16 \[ -\frac {\sqrt {b x} \Gamma \left (\frac {15}{2},b x\right )}{b^8 \sqrt {x}} \]

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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fricas [A]  time = 0.45, size = 82, normalized size = 0.54 \[ -\frac {2 \, {\left (64 \, b^{7} x^{6} + 416 \, b^{6} x^{5} + 2288 \, b^{5} x^{4} + 10296 \, b^{4} x^{3} + 36036 \, b^{3} x^{2} + 90090 \, b^{2} x + 135135 \, b\right )} \sqrt {x} e^{\left (-b x\right )} - 135135 \, \sqrt {\pi } \sqrt {b} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 \, b^{8}} \]

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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giac [A]  time = 0.36, size = 80, normalized size = 0.53 \[ -\frac {{\left (64 \, b^{6} x^{\frac {13}{2}} + 416 \, b^{5} x^{\frac {11}{2}} + 2288 \, b^{4} x^{\frac {9}{2}} + 10296 \, b^{3} x^{\frac {7}{2}} + 36036 \, b^{2} x^{\frac {5}{2}} + 90090 \, b x^{\frac {3}{2}} + 135135 \, \sqrt {x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} - \frac {135135 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} \sqrt {x}\right )}{128 \, b^{\frac {15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/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)

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maple [A]  time = 0.14, size = 145, normalized size = 0.96 \[ -\frac {x^{\frac {13}{2}} {\mathrm e}^{-b x}}{b}+\frac {-\frac {13 x^{\frac {11}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {13 \left (-\frac {11 x^{\frac {9}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {11 \left (-\frac {9 x^{\frac {7}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {9 \left (-\frac {7 x^{\frac {5}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {7 \left (-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \erf \left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{2 b}\right )}{b}}{b} \]

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 = 0.84, size = 79, normalized size = 0.52 \[ -\frac {{\left (64 \, b^{6} x^{\frac {13}{2}} + 416 \, b^{5} x^{\frac {11}{2}} + 2288 \, b^{4} x^{\frac {9}{2}} + 10296 \, b^{3} x^{\frac {7}{2}} + 36036 \, b^{2} x^{\frac {5}{2}} + 90090 \, b x^{\frac {3}{2}} + 135135 \, \sqrt {x}\right )} e^{\left (-b x\right )}}{64 \, b^{7}} + \frac {135135 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right )}{128 \, b^{\frac {15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(13/2)/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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mupad [B]  time = 3.43, size = 89, normalized size = 0.59 \[ -\frac {135135\,x^{13/2}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{128\,b\,{\left (b\,x\right )}^{13/2}}-\frac {x^{13/2}\,{\mathrm {e}}^{-b\,x}\,\left (\frac {135135\,\sqrt {b\,x}}{64}+\frac {45045\,{\left (b\,x\right )}^{3/2}}{32}+\frac {9009\,{\left (b\,x\right )}^{5/2}}{16}+\frac {1287\,{\left (b\,x\right )}^{7/2}}{8}+\frac {143\,{\left (b\,x\right )}^{9/2}}{4}+\frac {13\,{\left (b\,x\right )}^{11/2}}{2}+{\left (b\,x\right )}^{13/2}\right )}{b\,{\left (b\,x\right )}^{13/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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