3.201 \(\int \frac{f^{c (a+b x)^2}}{x^3} \, dx\)

Optimal. Leaf size=135 \[ 2 a^2 b^2 c^2 \log ^2(f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^2}}{x},x\right )+b^2 c \log (f) \text{Unintegrable}\left (\frac{f^{c (a+b x)^2}}{x},x\right )+\sqrt{\pi } a b^2 c^{3/2} \log ^{\frac{3}{2}}(f) \text{Erfi}\left (\sqrt{c} \sqrt{\log (f)} (a+b x)\right )-\frac{f^{c (a+b x)^2}}{2 x^2}-\frac{a b c \log (f) f^{c (a+b x)^2}}{x} \]

[Out]

-f^(c*(a + b*x)^2)/(2*x^2) - (a*b*c*f^(c*(a + b*x)^2)*Log[f])/x + a*b^2*c^(3/2)*Sqrt[Pi]*Erfi[Sqrt[c]*(a + b*x
)*Sqrt[Log[f]]]*Log[f]^(3/2) + b^2*c*Log[f]*Unintegrable[f^(c*(a + b*x)^2)/x, x] + 2*a^2*b^2*c^2*Log[f]^2*Unin
tegrable[f^(c*(a + b*x)^2)/x, x]

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Rubi [A]  time = 0.0971859, 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 = {} \[ \int \frac{f^{c (a+b x)^2}}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[f^(c*(a + b*x)^2)/x^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.408973, size = 0, normalized size = 0. \[ \int \frac{f^{c (a+b x)^2}}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[f^(c*(a + b*x)^2)/x^3,x]

[Out]

Integrate[f^(c*(a + b*x)^2)/x^3, x]

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Maple [A]  time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{{f}^{c \left ( bx+a \right ) ^{2}}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(c*(b*x+a)^2)/x^3,x)

[Out]

int(f^(c*(b*x+a)^2)/x^3,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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^2)/x^3,x, algorithm="maxima")

[Out]

integrate(f^((b*x + a)^2*c)/x^3, 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^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^2)/x^3,x, algorithm="fricas")

[Out]

integral(f^(b^2*c*x^2 + 2*a*b*c*x + a^2*c)/x^3, 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^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(c*(b*x+a)**2)/x**3,x)

[Out]

Integral(f**(c*(a + b*x)**2)/x**3, 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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f^(c*(b*x+a)^2)/x^3,x, algorithm="giac")

[Out]

integrate(f^((b*x + a)^2*c)/x^3, x)