3.274 \(\int \frac{F^{a+b (c+d x)^2}}{(c+d x)^2} \, dx\)

Optimal. Leaf size=67 \[ \frac{\sqrt{\pi } \sqrt{b} F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{d}-\frac{F^{a+b (c+d x)^2}}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.0792093, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2214, 2204} \[ \frac{\sqrt{\pi } \sqrt{b} F^a \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )}{d}-\frac{F^{a+b (c+d x)^2}}{d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^2)/(c + d*x)^2,x]

[Out]

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

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

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

Mathematica [A]  time = 0.0447884, size = 63, normalized size = 0.94 \[ \frac{F^a \left (\sqrt{\pi } \sqrt{b} \sqrt{\log (F)} \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} (c+d x)\right )-\frac{F^{b (c+d x)^2}}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^2)/(c + d*x)^2,x]

[Out]

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

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Maple [A]  time = 0.049, size = 62, normalized size = 0.9 \begin{align*} -{\frac{{F}^{b \left ( dx+c \right ) ^{2}}{F}^{a}}{ \left ( dx+c \right ) d}}+{\frac{b\ln \left ( F \right ) \sqrt{\pi }{F}^{a}}{d}{\it Erf} \left ( \sqrt{-b\ln \left ( F \right ) } \left ( dx+c \right ) \right ){\frac{1}{\sqrt{-b\ln \left ( F \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d/(d*x+c)*F^(b*(d*x+c)^2)*F^a+1/d*b*ln(F)*Pi^(1/2)*F^a/(-b*ln(F))^(1/2)*erf((-b*ln(F))^(1/2)*(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^2, x)

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Fricas [A]  time = 1.52966, size = 192, normalized size = 2.87 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )} F^{a} \operatorname{erf}\left (\frac{\sqrt{-b d^{2} \log \left (F\right )}{\left (d x + c\right )}}{d}\right ) + F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a} d}{d^{3} x + c d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(pi)*sqrt(-b*d^2*log(F))*(d*x + c)*F^a*erf(sqrt(-b*d^2*log(F))*(d*x + c)/d) + F^(b*d^2*x^2 + 2*b*c*d*x +
b*c^2 + a)*d)/(d^3*x + c*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + b \left (c + d x\right )^{2}}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(F**(a + b*(c + d*x)**2)/(c + d*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^2, x)