3.1.33 \(\int F^{a+b x} \sqrt {x} \, dx\) [33]

Optimal. Leaf size=62 \[ -\frac {F^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{2 b^{3/2} \log ^{\frac {3}{2}}(F)}+\frac {F^{a+b x} \sqrt {x}}{b \log (F)} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 = {2207, 2211, 2235} \begin {gather*} \frac {\sqrt {x} F^{a+b x}}{b \log (F)}-\frac {\sqrt {\pi } F^a \text {Erfi}\left (\sqrt {b} \sqrt {x} \sqrt {\log (F)}\right )}{2 b^{3/2} \log ^{\frac {3}{2}}(F)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[F^(a + b*x)*Sqrt[x],x]

[Out]

-1/2*(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[x]*Sqrt[Log[F]]])/(b^(3/2)*Log[F]^(3/2)) + (F^(a + b*x)*Sqrt[x])/(b*Log[F
])

Rule 2207

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] &&  !TrueQ[$UseGamma]

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

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Mathematica [A]
time = 0.05, size = 30, normalized size = 0.48 \begin {gather*} -\frac {F^a x^{3/2} \Gamma \left (\frac {3}{2},-b x \log (F)\right )}{(-b x \log (F))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*x)*Sqrt[x],x]

[Out]

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

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Maple [A]
time = 0.01, size = 66, normalized size = 1.06

method	result	size
meijerg	\(-\frac {F^{a} \left (\frac {\sqrt {x}\, \left (-b \right )^{\frac {3}{2}} \sqrt {\ln \left (F \right )}\, {\mathrm e}^{x b \ln \left (F \right )}}{b}-\frac {\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, \erfi \left (\sqrt {b}\, \sqrt {x}\, \sqrt {\ln \left (F \right )}\right )}{2 b^{\frac {3}{2}}}\right )}{\sqrt {-b}\, \ln \left (F \right )^{\frac {3}{2}} b}\)	\(66\)

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

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Maxima [A]
time = 0.32, size = 24, normalized size = 0.39 \begin {gather*} -\frac {F^{a} x^{\frac {3}{2}} \Gamma \left (\frac {3}{2}, -b x \log \left (F\right )\right )}{\left (-b x \log \left (F\right )\right )^{\frac {3}{2}}} \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]

-F^a*x^(3/2)*gamma(3/2, -b*x*log(F))/(-b*x*log(F))^(3/2)

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Fricas [A]
time = 0.39, size = 51, normalized size = 0.82 \begin {gather*} \frac {2 \, F^{b x + a} b \sqrt {x} \log \left (F\right ) + \sqrt {\pi } \sqrt {-b \log \left (F\right )} F^{a} \operatorname {erf}\left (\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{2 \, b^{2} \log \left (F\right )^{2}} \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]

1/2*(2*F^(b*x + a)*b*sqrt(x)*log(F) + sqrt(pi)*sqrt(-b*log(F))*F^a*erf(sqrt(-b*log(F))*sqrt(x)))/(b^2*log(F)^2
)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 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 = 3.13, size = 58, normalized size = 0.94 \begin {gather*} \frac {\sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} \sqrt {x}\right )}{2 \, \sqrt {-b \log \left (F\right )} b \log \left (F\right )} + \frac {\sqrt {x} e^{\left (b x \log \left (F\right ) + a \log \left (F\right )\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="giac")

[Out]

1/2*sqrt(pi)*F^a*erf(-sqrt(-b*log(F))*sqrt(x))/(sqrt(-b*log(F))*b*log(F)) + sqrt(x)*e^(b*x*log(F) + a*log(F))/
(b*log(F))

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Mupad [B]
time = 3.41, size = 55, normalized size = 0.89 \begin {gather*} \frac {F^a\,F^{b\,x}\,\sqrt {x}}{b\,\ln \left (F\right )}+\frac {F^a\,\sqrt {x}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-b\,x\,\ln \left (F\right )}\right )}{2\,b\,\ln \left (F\right )\,\sqrt {-b\,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*F^(b*x)*x^(1/2))/(b*log(F)) + (F^a*x^(1/2)*pi^(1/2)*erfc((-b*x*log(F))^(1/2)))/(2*b*log(F)*(-b*x*log(F))^
(1/2))

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