Optimal. Leaf size=137 \[ 2 a^2 b^2 c^2 \log ^2(f) \text {Int}\left (\frac {f^{c (a+b x)^2}}{x},x\right )+b^2 c \log (f) \text {Int}\left (\frac {f^{c (a+b x)^2}}{x},x\right )+\sqrt {\pi } a b^2 c^{3/2} \log ^{\frac {3}{2}}(f) \text {erfi}\left (\sqrt {c} \sqrt {\log (f)} (a+b x)\right )-\frac {f^{c (a+b x)^2}}{2 x^2}-\frac {a b c \log (f) f^{c (a+b x)^2}}{x} \]
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Rubi [A] time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {f^{c (a+b x)^2}}{x^3} \, dx \]
Verification is Not applicable to the result.
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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*}
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Mathematica [A] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {f^{c (a+b x)^2}}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {f^{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}}{x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{{\left (b x + a\right )}^{2} c}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {f^{\left (b x +a \right )^{2} c}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{{\left (b x + a\right )}^{2} c}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {f^{c\,{\left (a+b\,x\right )}^2}}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{c \left (a + b x\right )^{2}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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