3.33 \(\int F^{a+b x} \sqrt{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0443172, antiderivative size = 62, 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 = {2176, 2180, 2204} \[ \frac{\sqrt{x} F^{a+b x}}{b \log (F)}-\frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{x} \sqrt{\log (F)}\right )}{2 b^{3/2} \log ^{\frac{3}{2}}(F)} \]

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]]])/(2*b^(3/2)*Log[F]^(3/2)) + (F^(a + b*x)*Sqrt[x])/(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} \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{\operatorname{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*}

Mathematica [A]  time = 0.0076998, size = 30, normalized size = 0.48 \[ -\frac{x^{3/2} F^a \text{Gamma}\left (\frac{3}{2},-b x \log (F)\right )}{(-b x \log (F))^{3/2}} \]

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 = 0.008, size = 66, normalized size = 1.1 \begin{align*} -{\frac{{F}^{a}}{b} \left ({\frac{{{\rm e}^{b\ln \left ( F \right ) x}}}{b}\sqrt{x} \left ( -b \right ) ^{{\frac{3}{2}}}\sqrt{\ln \left ( F \right ) }}-{\frac{\sqrt{\pi }}{2} \left ( -b \right ) ^{{\frac{3}{2}}}{\it erfi} \left ( \sqrt{b}\sqrt{x}\sqrt{\ln \left ( F \right ) } \right ){b}^{-{\frac{3}{2}}}} \right ){\frac{1}{\sqrt{-b}}} \left ( \ln \left ( F \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-F^a/(-b)^(1/2)/ln(F)^(3/2)/b*(x^(1/2)*(-b)^(3/2)*ln(F)^(1/2)/b*exp(b*ln(F)*x)-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 = 1.24986, size = 32, normalized size = 0.52 \begin{align*} -\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{align*}

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 = 1.5187, size = 153, normalized size = 2.47 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int F^{a + b x} \sqrt{x}\, dx \end{align*}

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 = 1.22632, size = 78, normalized size = 1.26 \begin{align*} \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{align*}

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))