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

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

[Out]

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

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Rubi [A]  time = 0.0635389, antiderivative size = 67, 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 = {2206, 2211, 2204} \[ \frac{(c+d x) F^{a+\frac{b}{(c+d x)^2}}}{d}-\frac{\sqrt{\pi } \sqrt{b} F^a \sqrt{\log (F)} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^2),x]

[Out]

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

Rule 2206

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

Rule 2211

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(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 F^{a+\frac{b}{(c+d x)^2}} \, dx &=\frac{F^{a+\frac{b}{(c+d x)^2}} (c+d x)}{d}+(2 b \log (F)) \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^2} \, dx\\ &=\frac{F^{a+\frac{b}{(c+d x)^2}} (c+d x)}{d}-\frac{(2 b \log (F)) \operatorname{Subst}\left (\int F^{a+b x^2} \, dx,x,\frac{1}{c+d x}\right )}{d}\\ &=\frac{F^{a+\frac{b}{(c+d x)^2}} (c+d x)}{d}-\frac{\sqrt{b} F^a \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right ) \sqrt{\log (F)}}{d}\\ \end{align*}

Mathematica [A]  time = 0.0339963, size = 63, normalized size = 0.94 \[ \frac{F^a \left ((c+d x) F^{\frac{b}{(c+d x)^2}}-\sqrt{\pi } \sqrt{b} \sqrt{\log (F)} \text{Erfi}\left (\frac{\sqrt{b} \sqrt{\log (F)}}{c+d x}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^2),x]

[Out]

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

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Maple [A]  time = 0.029, size = 74, normalized size = 1.1 \begin{align*}{F}^{a}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}x+{\frac{{F}^{a}c}{d}{F}^{{\frac{b}{ \left ( dx+c \right ) ^{2}}}}}-{\frac{{F}^{a}b\ln \left ( F \right ) \sqrt{\pi }}{d}{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-b\ln \left ( F \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),x)

[Out]

F^a*F^(b/(d*x+c)^2)*x+1/d*F^a*F^(b/(d*x+c)^2)*c-1/d*F^a*b*ln(F)*Pi^(1/2)/(-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*} 2 \, F^{a} b d \int \frac{F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} x}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\,{d x} \log \left (F\right ) + F^{a} F^{\frac{b}{d^{2} x^{2} + 2 \, c d x + c^{2}}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*F^a*b*d*integrate(F^(b/(d^2*x^2 + 2*c*d*x + c^2))*x/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)*log(F) + F
^a*F^(b/(d^2*x^2 + 2*c*d*x + c^2))*x

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Fricas [A]  time = 1.54225, size = 209, normalized size = 3.12 \begin{align*} \frac{\sqrt{\pi } F^{a} d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}} \operatorname{erf}\left (\frac{d \sqrt{-\frac{b \log \left (F\right )}{d^{2}}}}{d x + c}\right ) +{\left (d x + c\right )} F^{\frac{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + \frac{b}{{\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),x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^2), x)