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

Optimal. Leaf size=77 \[ 2 a b c \log (f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^2}}{x},x\right )+\sqrt{\pi } b \sqrt{c} \sqrt{\log (f)} \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )-\frac{f^{c (a+b x)^2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0579473, 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 = {} \[ \int \frac{f^{c (a+b x)^2}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[f^(c*(a + b*x)^2)/x^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.307016, size = 0, normalized size = 0. \[ \int \frac{f^{c (a+b x)^2}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[f^(c*(a + b*x)^2)/x^2,x]

[Out]

Integrate[f^(c*(a + b*x)^2)/x^2, x]

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Maple [A]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c \left ( bx+a \right ) ^{2}}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^((b*x + a)^2*c)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(f**(c*(a + b*x)**2)/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^((b*x + a)^2*c)/x^2, x)