3.192 \(\int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx\)

Optimal. Leaf size=66 \[ \frac {2 \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{n}-\frac {2 x^{-n/2} f^{a+b x^n}}{n} \]

[Out]

-2*f^(a+b*x^n)/n/(x^(1/2*n))+2*f^a*erfi(x^(1/2*n)*b^(1/2)*ln(f)^(1/2))*b^(1/2)*Pi^(1/2)*ln(f)^(1/2)/n

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Rubi [A]  time = 0.07, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2215, 2211, 2204} \[ \frac {2 \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {Erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{n}-\frac {2 x^{-n/2} f^{a+b x^n}}{n} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 - n/2),x]

[Out]

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

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]

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 2215

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)^Simplify[m + n]*F^(a +
b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[-4, Simpl
ify[(m + 1)/n], 5] &&  !RationalQ[m] && SumSimplerQ[m, n]

Rubi steps

\begin {align*} \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx &=-\frac {2 f^{a+b x^n} x^{-n/2}}{n}+(2 b \log (f)) \int f^{a+b x^n} x^{\frac {1}{2} (-2+n)} \, dx\\ &=-\frac {2 f^{a+b x^n} x^{-n/2}}{n}+\frac {(4 b \log (f)) \operatorname {Subst}\left (\int f^{a+b x^2} \, dx,x,x^{1+\frac {1}{2} (-2+n)}\right )}{n}\\ &=-\frac {2 f^{a+b x^n} x^{-n/2}}{n}+\frac {2 \sqrt {b} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right ) \sqrt {\log (f)}}{n}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 39, normalized size = 0.59 \[ -\frac {f^a x^{-n/2} \sqrt {-b \log (f) x^n} \Gamma \left (-\frac {1}{2},-b x^n \log (f)\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((f^a*Gamma[-1/2, -(b*x^n*Log[f])]*Sqrt[-(b*x^n*Log[f])])/(n*x^(n/2)))

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fricas [A]  time = 0.42, size = 83, normalized size = 1.26 \[ -\frac {2 \, {\left (\sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \relax (f)}}{x x^{-\frac {1}{2} \, n - 1}}\right ) + x x^{-\frac {1}{2} \, n - 1} e^{\left (\frac {a x^{2} x^{-n - 2} \log \relax (f) + b \log \relax (f)}{x^{2} x^{-n - 2}}\right )}\right )}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{b x^{n} + a} x^{-\frac {1}{2} \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 59, normalized size = 0.89 \[ \frac {2 \sqrt {\pi }\, b \,f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x^{\frac {n}{2}}\right ) \ln \relax (f )}{\sqrt {-b \ln \relax (f )}\, n}-\frac {2 f^{a} f^{b \,x^{n}} x^{-\frac {n}{2}}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/n*f^a/(x^(1/2*n))*f^(b*x^n)+2/n*f^a*ln(f)*b*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f))^(1/2)*x^(1/2*n))

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maxima [A]  time = 0.98, size = 35, normalized size = 0.53 \[ -\frac {\sqrt {-b x^{n} \log \relax (f)} f^{a} \Gamma \left (-\frac {1}{2}, -b x^{n} \log \relax (f)\right )}{n x^{\frac {1}{2} \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-b*x^n*log(f))*f^a*gamma(-1/2, -b*x^n*log(f))/(n*x^(1/2*n))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {f^{a+b\,x^n}}{x^{\frac {n}{2}+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^n)/x^(n/2 + 1),x)

[Out]

int(f^(a + b*x^n)/x^(n/2 + 1), x)

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

[Out]

Timed out

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