3.88 \(\int f^{a+b x^2} \, dx\)

Optimal. Leaf size=37 \[ \frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2204} \[ \frac {\sqrt {\pi } f^a \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.00, size = 37, normalized size = 1.00 \[ \frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2),x]

[Out]

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

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fricas [A]  time = 0.42, size = 32, normalized size = 0.86 \[ -\frac {\sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{2 \, b \log \relax (f)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.28, size = 26, normalized size = 0.70 \[ -\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \relax (f)} x\right )}{2 \, \sqrt {-b \log \relax (f)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 26, normalized size = 0.70 \[ \frac {\sqrt {\pi }\, f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right )}{2 \sqrt {-b \ln \relax (f )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a),x)

[Out]

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

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maxima [A]  time = 0.92, size = 25, normalized size = 0.68 \[ \frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right )}{2 \, \sqrt {-b \log \relax (f)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.55, size = 26, normalized size = 0.70 \[ \frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )}{2\,\sqrt {b\,\ln \relax (f)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^2),x)

[Out]

(f^a*pi^(1/2)*erfi((b*x*log(f))/(b*log(f))^(1/2)))/(2*(b*log(f))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + b x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a),x)

[Out]

Integral(f**(a + b*x**2), x)

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