3.1.34 \(\int \frac {F^{a+b x}}{\sqrt {x}} \, dx\) [34]

Optimal. Leaf size=38 \[ \frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{\sqrt {b} \sqrt {\log (F)}} \]

[Out]

F^a*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(1/2)/ln(F)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2211, 2235} \begin {gather*} \frac {\sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{\sqrt {b} \sqrt {\log (F)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b*x)/Sqrt[x],x]

[Out]

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

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 2235

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}}{\sqrt {x}} \, dx &=2 \text {Subst}\left (\int F^{a+b x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{\sqrt {b} \sqrt {\log (F)}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 30, normalized size = 0.79 \begin {gather*} -\frac {F^a \sqrt {x} \Gamma \left (\frac {1}{2},-b x \log (F)\right )}{\sqrt {-b x \log (F)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*x)/Sqrt[x],x]

[Out]

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

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Maple [A]
time = 0.02, size = 27, normalized size = 0.71

method	result	size
meijerg	\(\frac {F^{a} \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \left (F \right )}\right ) \sqrt {\pi }}{\sqrt {b}\, \sqrt {\ln \left (F \right )}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(b*x+a)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

F^a*erfi(b^(1/2)*x^(1/2)*ln(F)^(1/2))*Pi^(1/2)/b^(1/2)/ln(F)^(1/2)

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Maxima [A]
time = 0.32, size = 29, normalized size = 0.76 \begin {gather*} \frac {\sqrt {\pi } F^{a} \sqrt {x} {\left (\operatorname {erf}\left (\sqrt {-b x \log \left (F\right )}\right ) - 1\right )}}{\sqrt {-b x \log \left (F\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)/x^(1/2),x, algorithm="maxima")

[Out]

sqrt(pi)*F^a*sqrt(x)*(erf(sqrt(-b*x*log(F))) - 1)/sqrt(-b*x*log(F))

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Fricas [A]
time = 0.40, size = 34, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b \log \left (F\right )} F^{a} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{b \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)/x^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {F^{a + b x}}{\sqrt {x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(b*x+a)/x**(1/2),x)

[Out]

Integral(F**(a + b*x)/sqrt(x), x)

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Giac [A]
time = 3.09, size = 28, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{\sqrt {-b \log \left (F\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(b*x+a)/x^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 3.50, size = 32, normalized size = 0.84 \begin {gather*} \frac {F^a\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \left (F\right )}\right )\,\sqrt {-\pi \,b\,x\,\ln \left (F\right )}}{b\,\sqrt {x}\,\ln \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b*x)/x^(1/2),x)

[Out]

(F^a*erfc((-b*x*log(F))^(1/2))*(-b*x*pi*log(F))^(1/2))/(b*x^(1/2)*log(F))

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