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

Optimal. Leaf size=49 \[ \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {f^{a+b x^2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2214, 2204} \[ \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {Erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {f^{a+b x^2}}{x} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^2,x]

[Out]

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

Rule 2214

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)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 49, normalized size = 1.00 \[ \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )-\frac {f^{a+b x^2}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^2,x]

[Out]

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

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fricas [A]  time = 0.40, size = 40, normalized size = 0.82 \[ -\frac {\sqrt {\pi } \sqrt {-b \log \relax (f)} f^{a} x \operatorname {erf}\left (\sqrt {-b \log \relax (f)} x\right ) + f^{b x^{2} + a}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(b*x^2 + a)/x^2, x)

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maple [A]  time = 0.04, size = 44, normalized size = 0.90 \[ \frac {\sqrt {\pi }\, b \,f^{a} \erf \left (\sqrt {-b \ln \relax (f )}\, x \right ) \ln \relax (f )}{\sqrt {-b \ln \relax (f )}}-\frac {f^{a} f^{b \,x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.27, size = 28, normalized size = 0.57 \[ -\frac {\sqrt {-b x^{2} \log \relax (f)} f^{a} \Gamma \left (-\frac {1}{2}, -b x^{2} \log \relax (f)\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.47, size = 44, normalized size = 0.90 \[ \frac {b\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}\right )\,\ln \relax (f)}{\sqrt {b\,\ln \relax (f)}}-\frac {f^a\,f^{b\,x^2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b*x^2)/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x**2)/x**2, x)

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