3.2.91 \(\int f^{a+b x^n} x^{-1+\frac {n}{2}} \, dx\) [191]

Optimal. Leaf size=43 \[ \frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{\sqrt {b} n \sqrt {\log (f)}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2242, 2235} \begin {gather*} \frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{\sqrt {b} n \sqrt {\log (f)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^n)*x^(-1 + n/2),x]

[Out]

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

Rule 2235

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 2242

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

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Mathematica [A]
time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} \frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right )}{\sqrt {b} n \sqrt {\log (f)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 32, normalized size = 0.74

method	result	size
meijerg	\(\frac {f^{a} \erfi \left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right ) \sqrt {\pi }}{n \sqrt {b}\, \sqrt {\ln \left (f \right )}}\)	\(32\)
risch	\(\frac {f^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{n \sqrt {-b \ln \left (f \right )}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 38, normalized size = 0.88 \begin {gather*} \frac {\sqrt {\pi } f^{a} x^{\frac {1}{2} \, n} {\left (\operatorname {erf}\left (\sqrt {-b x^{n} \log \left (f\right )}\right ) - 1\right )}}{\sqrt {-b x^{n} \log \left (f\right )} n} \end {gather*}

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

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Fricas [A]
time = 0.39, size = 42, normalized size = 0.98 \begin {gather*} -\frac {\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x x^{\frac {1}{2} \, n - 1}\right )}{b n \log \left (f\right )} \end {gather*}

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]

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

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Sympy [A]
time = 85.97, size = 53, normalized size = 1.23 \begin {gather*} \begin {cases} - \frac {4 b f^{a} f^{b x^{n}} x^{\frac {3 n}{2}} \log {\left (f \right )}}{3 n} + \frac {2 f^{a} f^{b x^{n}} x^{\frac {n}{2}}}{n} & \text {for}\: n \neq 0 \\f^{a + b} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-4*b*f**a*f**(b*x**n)*x**(3*n/2)*log(f)/(3*n) + 2*f**a*f**(b*x**n)*x**(n/2)/n, Ne(n, 0)), (f**(a +
b)*log(x), True))

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Giac [A]
time = 6.06, size = 33, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (f\right )} \sqrt {x^{n}}\right )}{\sqrt {-b \log \left (f\right )} n} \end {gather*}

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]

-sqrt(pi)*f^a*erf(-sqrt(-b*log(f))*sqrt(x^n))/(sqrt(-b*log(f))*n)

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

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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