3.31 \(\int F^{a+b x} x^{5/2} \, dx\)

Optimal. Leaf size=108 \[ -\frac{15 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{8 b^{7/2} \log ^{\frac{7}{2}}(F)}-\frac{5 x^{3/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{15 \sqrt{x} F^{a+b x}}{4 b^3 \log ^3(F)}+\frac{x^{5/2} F^{a+b x}}{b \log (F)} \]

[Out]

(-15*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(8*b^(7/2)*Log[F]^(7/2)) + (15*F^(a + b*x)*Sqrt[x])/(4*b
^3*Log[F]^3) - (5*F^(a + b*x)*x^(3/2))/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(5/2))/(b*Log[F])

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Rubi [A]  time = 0.0938183, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2176, 2180, 2204} \[ -\frac{15 \sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{8 b^{7/2} \log ^{\frac{7}{2}}(F)}-\frac{5 x^{3/2} F^{a+b x}}{2 b^2 \log ^2(F)}+\frac{15 \sqrt{x} F^{a+b x}}{4 b^3 \log ^3(F)}+\frac{x^{5/2} F^{a+b x}}{b \log (F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*x)*x^(5/2),x]

[Out]

(-15*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(8*b^(7/2)*Log[F]^(7/2)) + (15*F^(a + b*x)*Sqrt[x])/(4*b
^3*Log[F]^3) - (5*F^(a + b*x)*x^(3/2))/(2*b^2*Log[F]^2) + (F^(a + b*x)*x^(5/2))/(b*Log[F])

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 2204

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

Rubi steps

\begin{align*} \int F^{a+b x} x^{5/2} \, dx &=\frac{F^{a+b x} x^{5/2}}{b \log (F)}-\frac{5 \int F^{a+b x} x^{3/2} \, dx}{2 b \log (F)}\\ &=-\frac{5 F^{a+b x} x^{3/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{5/2}}{b \log (F)}+\frac{15 \int F^{a+b x} \sqrt{x} \, dx}{4 b^2 \log ^2(F)}\\ &=\frac{15 F^{a+b x} \sqrt{x}}{4 b^3 \log ^3(F)}-\frac{5 F^{a+b x} x^{3/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{5/2}}{b \log (F)}-\frac{15 \int \frac{F^{a+b x}}{\sqrt{x}} \, dx}{8 b^3 \log ^3(F)}\\ &=\frac{15 F^{a+b x} \sqrt{x}}{4 b^3 \log ^3(F)}-\frac{5 F^{a+b x} x^{3/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{5/2}}{b \log (F)}-\frac{15 \operatorname{Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt{x}\right )}{4 b^3 \log ^3(F)}\\ &=-\frac{15 F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{8 b^{7/2} \log ^{\frac{7}{2}}(F)}+\frac{15 F^{a+b x} \sqrt{x}}{4 b^3 \log ^3(F)}-\frac{5 F^{a+b x} x^{3/2}}{2 b^2 \log ^2(F)}+\frac{F^{a+b x} x^{5/2}}{b \log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0064971, size = 36, normalized size = 0.33 \[ \frac{\sqrt{x} F^a \text{Gamma}\left (\frac{7}{2},-b x \log (F)\right )}{b^3 \log ^3(F) \sqrt{-b x \log (F)}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*x)*x^(5/2),x]

[Out]

(F^a*Sqrt[x]*Gamma[7/2, -(b*x*Log[F])])/(b^3*Log[F]^3*Sqrt[-(b*x*Log[F])])

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Maple [A]  time = 0.01, size = 87, normalized size = 0.8 \begin{align*} -{\frac{{F}^{a}}{b} \left ({\frac{ \left ( 28\,{b}^{2}{x}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-70\,b\ln \left ( F \right ) x+105 \right ){{\rm e}^{b\ln \left ( F \right ) x}}}{28\,{b}^{3}}\sqrt{x} \left ( -b \right ) ^{{\frac{7}{2}}}\sqrt{\ln \left ( F \right ) }}-{\frac{15\,\sqrt{\pi }}{8} \left ( -b \right ) ^{{\frac{7}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){b}^{-{\frac{7}{2}}}} \right ) \left ( -b \right ) ^{-{\frac{5}{2}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(b*x+a)*x^(5/2),x)

[Out]

-F^a/(-b)^(5/2)/ln(F)^(7/2)/b*(1/28*x^(1/2)*(-b)^(7/2)*ln(F)^(1/2)*(28*b^2*x^2*ln(F)^2-70*b*ln(F)*x+105)/b^3*e
xp(b*ln(F)*x)-15/8*(-b)^(7/2)/b^(7/2)*Pi^(1/2)*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2)))

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Maxima [A]  time = 1.22414, size = 32, normalized size = 0.3 \begin{align*} -\frac{F^{a} x^{\frac{7}{2}} \Gamma \left (\frac{7}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)*x^(5/2),x, algorithm="maxima")

[Out]

-F^a*x^(7/2)*gamma(7/2, -b*x*log(F))/(-b*x*log(F))^(7/2)

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Fricas [A]  time = 1.51819, size = 219, normalized size = 2.03 \begin{align*} \frac{15 \, \sqrt{\pi } \sqrt{-b \log \left (F\right )} F^{a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) + 2 \,{\left (4 \, b^{3} x^{2} \log \left (F\right )^{3} - 10 \, b^{2} x \log \left (F\right )^{2} + 15 \, b \log \left (F\right )\right )} F^{b x + a} \sqrt{x}}{8 \, b^{4} \log \left (F\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)*x^(5/2),x, algorithm="fricas")

[Out]

1/8*(15*sqrt(pi)*sqrt(-b*log(F))*F^a*erf(sqrt(-b*log(F))*sqrt(x)) + 2*(4*b^3*x^2*log(F)^3 - 10*b^2*x*log(F)^2
+ 15*b*log(F))*F^(b*x + a)*sqrt(x))/(b^4*log(F)^4)

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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(F**(b*x+a)*x**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.25743, size = 111, normalized size = 1.03 \begin{align*} \frac{15 \, \sqrt{\pi } F^{a} \operatorname{erf}\left (-\sqrt{-b \log \left (F\right )} \sqrt{x}\right )}{8 \, \sqrt{-b \log \left (F\right )} b^{3} \log \left (F\right )^{3}} + \frac{{\left (4 \, b^{2} x^{\frac{5}{2}} \log \left (F\right )^{2} - 10 \, b x^{\frac{3}{2}} \log \left (F\right ) + 15 \, \sqrt{x}\right )} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\right )}}{4 \, b^{3} \log \left (F\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)*x^(5/2),x, algorithm="giac")

[Out]

15/8*sqrt(pi)*F^a*erf(-sqrt(-b*log(F))*sqrt(x))/(sqrt(-b*log(F))*b^3*log(F)^3) + 1/4*(4*b^2*x^(5/2)*log(F)^2 -
10*b*x^(3/2)*log(F) + 15*sqrt(x))*e^(b*x*log(F) + a*log(F))/(b^3*log(F)^3)