3.35 \(\int \frac{F^{a+b x}}{x^{3/2}} \, dx\)

Optimal. Leaf size=54 \[ 2 \sqrt{\pi } \sqrt{b} F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{2 F^{a+b x}}{\sqrt{x}} \]

[Out]

(-2*F^(a + b*x))/Sqrt[x] + 2*Sqrt[b]*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Sqrt[Log[F]]

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Rubi [A]  time = 0.0454763, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2177, 2180, 2204} \[ 2 \sqrt{\pi } \sqrt{b} F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )-\frac{2 F^{a+b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*x)/x^(3/2),x]

[Out]

(-2*F^(a + b*x))/Sqrt[x] + 2*Sqrt[b]*F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]]*Sqrt[Log[F]]

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), 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] && LtQ[m, -1] && 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 \frac{F^{a+b x}}{x^{3/2}} \, dx &=-\frac{2 F^{a+b x}}{\sqrt{x}}+(2 b \log (F)) \int \frac{F^{a+b x}}{\sqrt{x}} \, dx\\ &=-\frac{2 F^{a+b x}}{\sqrt{x}}+(4 b \log (F)) \operatorname{Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 F^{a+b x}}{\sqrt{x}}+2 \sqrt{b} F^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right ) \sqrt{\log (F)}\\ \end{align*}

Mathematica [A]  time = 0.0151075, size = 38, normalized size = 0.7 \[ -\frac{2 F^a \left (F^{b x}-\sqrt{-b x \log (F)} \text{Gamma}\left (\frac{1}{2},-b x \log (F)\right )\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*x)/x^(3/2),x]

[Out]

(-2*F^a*(F^(b*x) - Gamma[1/2, -(b*x*Log[F])]*Sqrt[-(b*x*Log[F])]))/Sqrt[x]

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Maple [A]  time = 0.01, size = 64, normalized size = 1.2 \begin{align*} -{\frac{{F}^{a}}{b} \left ( -b \right ) ^{{\frac{3}{2}}}\sqrt{\ln \left ( F \right ) } \left ( -2\,{\frac{{{\rm e}^{b\ln \left ( F \right ) x}}}{\sqrt{x}\sqrt{-b}\sqrt{\ln \left ( F \right ) }}}+2\,{\frac{\sqrt{b}\sqrt{\pi }{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ) }{\sqrt{-b}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(b*x+a)/x^(3/2),x)

[Out]

-F^a*(-b)^(3/2)*ln(F)^(1/2)/b*(-2/x^(1/2)/(-b)^(1/2)/ln(F)^(1/2)*exp(b*ln(F)*x)+2/(-b)^(1/2)*b^(1/2)*Pi^(1/2)*
erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2)))

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Maxima [A]  time = 1.21438, size = 32, normalized size = 0.59 \begin{align*} -\frac{\sqrt{-b x \log \left (F\right )} F^{a} \Gamma \left (-\frac{1}{2}, -b x \log \left (F\right )\right )}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)/x^(3/2),x, algorithm="maxima")

[Out]

-sqrt(-b*x*log(F))*F^a*gamma(-1/2, -b*x*log(F))/sqrt(x)

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Fricas [A]  time = 1.54878, size = 122, normalized size = 2.26 \begin{align*} -\frac{2 \,{\left (\sqrt{\pi } \sqrt{-b \log \left (F\right )} F^{a} x \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} \sqrt{x}\right ) + F^{b x + a} \sqrt{x}\right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)/x^(3/2),x, algorithm="fricas")

[Out]

-2*(sqrt(pi)*sqrt(-b*log(F))*F^a*x*erf(sqrt(-b*log(F))*sqrt(x)) + F^(b*x + a)*sqrt(x))/x

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Sympy [A]  time = 6.77587, size = 34, normalized size = 0.63 \begin{align*} 4 F^{a} F^{b x} b \sqrt{x} \log{\left (F \right )} - \frac{2 F^{a} F^{b x}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(b*x+a)/x**(3/2),x)

[Out]

4*F**a*F**(b*x)*b*sqrt(x)*log(F) - 2*F**a*F**(b*x)/sqrt(x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{b x + a}}{x^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)/x^(3/2),x, algorithm="giac")

[Out]

integrate(F^(b*x + a)/x^(3/2), x)