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

Optimal. Leaf size=93 \[ b \log (f) \text{Unintegrable}\left (\frac{f^{a+b x+c x^2}}{x},x\right )+\sqrt{\pi } \sqrt{c} \sqrt{\log (f)} f^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{\sqrt{\log (f)} (b+2 c x)}{2 \sqrt{c}}\right )-\frac{f^{a+b x+c x^2}}{x} \]

[Out]

-(f^(a + b*x + c*x^2)/x) + Sqrt[c]*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqr
t[Log[f]] + b*Log[f]*Unintegrable[f^(a + b*x + c*x^2)/x, x]

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Rubi [A]  time = 0.0759386, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{f^{a+b x+c x^2}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[f^(a + b*x + c*x^2)/x^2,x]

[Out]

-(f^(a + b*x + c*x^2)/x) + Sqrt[c]*f^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[((b + 2*c*x)*Sqrt[Log[f]])/(2*Sqrt[c])]*Sqr
t[Log[f]] + b*Log[f]*Defer[Int][f^(a + b*x + c*x^2)/x, x]

Rubi steps

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

Mathematica [A]  time = 0.257055, size = 0, normalized size = 0. \[ \int \frac{f^{a+b x+c x^2}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[f^(a + b*x + c*x^2)/x^2,x]

[Out]

Integrate[f^(a + b*x + c*x^2)/x^2, x]

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Maple [A]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c{x}^{2}+bx+a}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c*x^2 + b*x + a)/x^2, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{f^{c x^{2} + b x + a}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(f^(c*x^2 + b*x + a)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(a + b*x + c*x**2)/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(c*x^2 + b*x + a)/x^2, x)