3.381 \(\int F^{-c (a+b x)^n} (a+b x)^{-1+\frac{n}{2}} \, dx\)

Optimal. Leaf size=47 \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{c} \sqrt{\log (F)} (a+b x)^{n/2}\right )}{b \sqrt{c} n \sqrt{\log (F)}} \]

[Out]

(Sqrt[Pi]*Erf[Sqrt[c]*(a + b*x)^(n/2)*Sqrt[Log[F]]])/(b*Sqrt[c]*n*Sqrt[Log[F]])

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Rubi [A]  time = 0.0484519, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2211, 2205} \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{c} \sqrt{\log (F)} (a+b x)^{n/2}\right )}{b \sqrt{c} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-1 + n/2)/F^(c*(a + b*x)^n),x]

[Out]

(Sqrt[Pi]*Erf[Sqrt[c]*(a + b*x)^(n/2)*Sqrt[Log[F]]])/(b*Sqrt[c]*n*Sqrt[Log[F]])

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 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int F^{-c (a+b x)^n} (a+b x)^{-1+\frac{n}{2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int F^{-c x^2} \, dx,x,(a+b x)^{n/2}\right )}{b n}\\ &=\frac{\sqrt{\pi } \text{erf}\left (\sqrt{c} (a+b x)^{n/2} \sqrt{\log (F)}\right )}{b \sqrt{c} n \sqrt{\log (F)}}\\ \end{align*}

Mathematica [A]  time = 0.00731, size = 47, normalized size = 1. \[ \frac{\sqrt{\pi } \text{Erf}\left (\sqrt{c} \sqrt{\log (F)} (a+b x)^{n/2}\right )}{b \sqrt{c} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-1 + n/2)/F^(c*(a + b*x)^n),x]

[Out]

(Sqrt[Pi]*Erf[Sqrt[c]*(a + b*x)^(n/2)*Sqrt[Log[F]]])/(b*Sqrt[c]*n*Sqrt[Log[F]])

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Maple [A]  time = 0.079, size = 34, normalized size = 0.7 \begin{align*}{\frac{\sqrt{\pi }}{bn}{\it Erf} \left ( \sqrt{c\ln \left ( F \right ) } \left ( bx+a \right ) ^{{\frac{n}{2}}} \right ){\frac{1}{\sqrt{c\ln \left ( F \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{2} \, n - 1}}{F^{{\left (b x + a\right )}^{n} c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^(1/2*n - 1)/F^((b*x + a)^n*c), x)

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Fricas [A]  time = 1.60274, size = 124, normalized size = 2.64 \begin{align*} \frac{\sqrt{\pi } \sqrt{c \log \left (F\right )} \operatorname{erf}\left ({\left (b x + a\right )} \sqrt{c \log \left (F\right )}{\left (b x + a\right )}^{\frac{1}{2} \, n - 1}\right )}{b c n \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((b*x+a)**(-1+1/2*n)/(F**(c*(b*x+a)**n)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{\frac{1}{2} \, n - 1}}{F^{{\left (b x + a\right )}^{n} c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^(1/2*n - 1)/F^((b*x + a)^n*c), x)