3.90 \(\int \frac{f^{a+b x^2}}{x^4} \, dx\)

Optimal. Leaf size=73 \[ \frac{2}{3} \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )-\frac{f^{a+b x^2}}{3 x^3}-\frac{2 b \log (f) f^{a+b x^2}}{3 x} \]

[Out]

-f^(a + b*x^2)/(3*x^3) - (2*b*f^(a + b*x^2)*Log[f])/(3*x) + (2*b^(3/2)*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]
]]*Log[f]^(3/2))/3

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Rubi [A]  time = 0.0548927, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2214, 2204} \[ \frac{2}{3} \sqrt{\pi } b^{3/2} f^a \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )-\frac{f^{a+b x^2}}{3 x^3}-\frac{2 b \log (f) f^{a+b x^2}}{3 x} \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b*x^2)/x^4,x]

[Out]

-f^(a + b*x^2)/(3*x^3) - (2*b*f^(a + b*x^2)*Log[f])/(3*x) + (2*b^(3/2)*f^a*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]
]]*Log[f]^(3/2))/3

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]))

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 \frac{f^{a+b x^2}}{x^4} \, dx &=-\frac{f^{a+b x^2}}{3 x^3}+\frac{1}{3} (2 b \log (f)) \int \frac{f^{a+b x^2}}{x^2} \, dx\\ &=-\frac{f^{a+b x^2}}{3 x^3}-\frac{2 b f^{a+b x^2} \log (f)}{3 x}+\frac{1}{3} \left (4 b^2 \log ^2(f)\right ) \int f^{a+b x^2} \, dx\\ &=-\frac{f^{a+b x^2}}{3 x^3}-\frac{2 b f^{a+b x^2} \log (f)}{3 x}+\frac{2}{3} b^{3/2} f^a \sqrt{\pi } \text{erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right ) \log ^{\frac{3}{2}}(f)\\ \end{align*}

Mathematica [A]  time = 0.0408832, size = 62, normalized size = 0.85 \[ \frac{1}{3} f^a \left (2 \sqrt{\pi } b^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\sqrt{b} x \sqrt{\log (f)}\right )-\frac{f^{b x^2} \left (2 b x^2 \log (f)+1\right )}{x^3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b*x^2)/x^4,x]

[Out]

(f^a*(2*b^(3/2)*Sqrt[Pi]*Erfi[Sqrt[b]*x*Sqrt[Log[f]]]*Log[f]^(3/2) - (f^(b*x^2)*(1 + 2*b*x^2*Log[f]))/x^3))/3

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Maple [A]  time = 0.027, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{f}^{a}{f}^{b{x}^{2}}}{3\,{x}^{3}}}-{\frac{2\,{f}^{a}\ln \left ( f \right ) b{f}^{b{x}^{2}}}{3\,x}}+{\frac{2\,{f}^{a} \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}\sqrt{\pi }}{3}{\it Erf} \left ( \sqrt{-b\ln \left ( f \right ) }x \right ){\frac{1}{\sqrt{-b\ln \left ( f \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(b*x^2+a)/x^4,x)

[Out]

-1/3*f^a/x^3*f^(b*x^2)-2/3*f^a*ln(f)*b/x*f^(b*x^2)+2/3*f^a*ln(f)^2*b^2*Pi^(1/2)/(-b*ln(f))^(1/2)*erf((-b*ln(f)
)^(1/2)*x)

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Maxima [A]  time = 1.20686, size = 38, normalized size = 0.52 \begin{align*} -\frac{\left (-b x^{2} \log \left (f\right )\right )^{\frac{3}{2}} f^{a} \Gamma \left (-\frac{3}{2}, -b x^{2} \log \left (f\right )\right )}{2 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^4,x, algorithm="maxima")

[Out]

-1/2*(-b*x^2*log(f))^(3/2)*f^a*gamma(-3/2, -b*x^2*log(f))/x^3

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Fricas [A]  time = 1.78261, size = 157, normalized size = 2.15 \begin{align*} -\frac{2 \, \sqrt{\pi } \sqrt{-b \log \left (f\right )} b f^{a} x^{3} \operatorname{erf}\left (\sqrt{-b \log \left (f\right )} x\right ) \log \left (f\right ) +{\left (2 \, b x^{2} \log \left (f\right ) + 1\right )} f^{b x^{2} + a}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^4,x, algorithm="fricas")

[Out]

-1/3*(2*sqrt(pi)*sqrt(-b*log(f))*b*f^a*x^3*erf(sqrt(-b*log(f))*x)*log(f) + (2*b*x^2*log(f) + 1)*f^(b*x^2 + a))
/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + b x^{2}}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(b*x**2+a)/x**4,x)

[Out]

Integral(f**(a + b*x**2)/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{b x^{2} + a}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(b*x^2+a)/x^4,x, algorithm="giac")

[Out]

integrate(f^(b*x^2 + a)/x^4, x)