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]

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

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Rubi [A]  time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 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]

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
)]

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*}

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Mathematica [A]  time = 0.01, size = 47, normalized size = 1.00 \[ \frac {\sqrt {\pi } \text {erf}\left (\sqrt {c} \sqrt {\log (F)} (a+b x)^{n/2}\right )}{b \sqrt {c} n \sqrt {\log (F)}} \]

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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fricas [A]  time = 0.47, size = 47, normalized size = 1.00 \[ \frac {\sqrt {\pi } \sqrt {c \log \relax (F)} \operatorname {erf}\left ({\left (b x + a\right )} \sqrt {c \log \relax (F)} {\left (b x + a\right )}^{\frac {1}{2} \, n - 1}\right )}{b c n \log \relax (F)} \]

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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giac [A]  time = 0.46, size = 35, normalized size = 0.74 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {c \log \relax (F)} \sqrt {{\left (b x + a\right )}^{n}}\right )}{\sqrt {c \log \relax (F)} b n} \]

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]

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

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maple [A]  time = 0.16, size = 34, normalized size = 0.72 \[ \frac {\sqrt {\pi }\, \erf \left (\sqrt {c \ln \relax (F )}\, \left (b x +a \right )^{\frac {n}{2}}\right )}{\sqrt {c \ln \relax (F )}\, b n} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{2} \, n - 1}}{F^{{\left (b x + a\right )}^{n} c}}\,{d x} \]

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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mupad [B]  time = 4.04, size = 35, normalized size = 0.74 \[ \frac {\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {c}\,\sqrt {\ln \relax (F)}\,{\left (a+b\,x\right )}^{n/2}\right )}{b\,\sqrt {c}\,n\,\sqrt {\ln \relax (F)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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