Optimal. Leaf size=139 \[ -\frac{2 e f x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+2,-b d x \log (F))}{b^2 d^2 \log ^2(F)}+\frac{f^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+3,-b d x \log (F))}{b^3 d^3 \log ^3(F)}+\frac{e^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+1,-b d x \log (F))}{b d \log (F)} \]
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Rubi [A] time = 0.309197, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2199, 2181} \[ -\frac{2 e f x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+2,-b d x \log (F))}{b^2 d^2 \log ^2(F)}+\frac{f^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+3,-b d x \log (F))}{b^3 d^3 \log ^3(F)}+\frac{e^2 x^m F^{a+b c} (-b d x \log (F))^{-m} \text{Gamma}(m+1,-b d x \log (F))}{b d \log (F)} \]
Antiderivative was successfully verified.
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Rule 2199
Rule 2181
Rubi steps
\begin{align*} \int F^{a+b (c+d x)} x^m (e+f x)^2 \, dx &=\int \left (e^2 F^{a+b c+b d x} x^m+2 e f F^{a+b c+b d x} x^{1+m}+f^2 F^{a+b c+b d x} x^{2+m}\right ) \, dx\\ &=e^2 \int F^{a+b c+b d x} x^m \, dx+(2 e f) \int F^{a+b c+b d x} x^{1+m} \, dx+f^2 \int F^{a+b c+b d x} x^{2+m} \, dx\\ &=\frac{f^2 F^{a+b c} x^m \Gamma (3+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b^3 d^3 \log ^3(F)}-\frac{2 e f F^{a+b c} x^m \Gamma (2+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b^2 d^2 \log ^2(F)}+\frac{e^2 F^{a+b c} x^m \Gamma (1+m,-b d x \log (F)) (-b d x \log (F))^{-m}}{b d \log (F)}\\ \end{align*}
Mathematica [A] time = 0.124451, size = 86, normalized size = 0.62 \[ \frac{x^m F^{a+b c} (-b d x \log (F))^{-m} \left (b d e \log (F) (b d e \log (F) \text{Gamma}(m+1,-b d x \log (F))-2 f \text{Gamma}(m+2,-b d x \log (F)))+f^2 \text{Gamma}(m+3,-b d x \log (F))\right )}{b^3 d^3 \log ^3(F)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.097, size = 433, normalized size = 3.1 \begin{align*} -{\frac{ \left ( \ln \left ( F \right ) \right ) ^{-3-m} \left ( -bd \right ) ^{-m}{F}^{bc+a}{f}^{2} \left ({x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}m \left ({m}^{2}+3\,m+2 \right ) \Gamma \left ( m \right ) \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m} \left ({b}^{2}{d}^{2}{x}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-mbdx\ln \left ( F \right ) +{m}^{2}-2\,bdx\ln \left ( F \right ) +3\,m+2 \right ){{\rm e}^{bdx\ln \left ( F \right ) }}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}m \left ({m}^{2}+3\,m+2 \right ) \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}\Gamma \left ( m,-bdx\ln \left ( F \right ) \right ) \right ) }{{b}^{3}{d}^{3}}}+2\,{\frac{ \left ( \ln \left ( F \right ) \right ) ^{-2-m} \left ( -bd \right ) ^{-m}{F}^{bc+a}fe \left ({x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m} \left ( 1+m \right ) m\Gamma \left ( m \right ) \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}+{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m} \left ( bdx\ln \left ( F \right ) -1-m \right ){{\rm e}^{bdx\ln \left ( F \right ) }}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m} \left ( 1+m \right ) m \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}\Gamma \left ( m,-bdx\ln \left ( F \right ) \right ) \right ) }{{b}^{2}{d}^{2}}}-{\frac{{F}^{bc+a} \left ( -bd \right ) ^{-m} \left ( \ln \left ( F \right ) \right ) ^{-m-1}{e}^{2} \left ({x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}m\Gamma \left ( m \right ) \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}{{\rm e}^{bdx\ln \left ( F \right ) }}-{x}^{m} \left ( -bd \right ) ^{m} \left ( \ln \left ( F \right ) \right ) ^{m}m \left ( -bdx\ln \left ( F \right ) \right ) ^{-m}\Gamma \left ( m,-bdx\ln \left ( F \right ) \right ) \right ) }{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27436, size = 166, normalized size = 1.19 \begin{align*} -\left (-b d x \log \left (F\right )\right )^{-m - 3} F^{b c + a} f^{2} x^{m + 3} \Gamma \left (m + 3, -b d x \log \left (F\right )\right ) - 2 \, \left (-b d x \log \left (F\right )\right )^{-m - 2} F^{b c + a} e f x^{m + 2} \Gamma \left (m + 2, -b d x \log \left (F\right )\right ) - \left (-b d x \log \left (F\right )\right )^{-m - 1} F^{b c + a} e^{2} x^{m + 1} \Gamma \left (m + 1, -b d x \log \left (F\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60851, size = 386, normalized size = 2.78 \begin{align*} -\frac{{\left ({\left (b d f^{2} m + 2 \, b d f^{2}\right )} x \log \left (F\right ) -{\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} d^{2} e f x\right )} \log \left (F\right )^{2}\right )} F^{b d x + b c + a} x^{m} -{\left (b^{2} d^{2} e^{2} \log \left (F\right )^{2} + f^{2} m^{2} + 3 \, f^{2} m + 2 \, f^{2} - 2 \,{\left (b d e f m + b d e f\right )} \log \left (F\right )\right )} e^{\left (-m \log \left (-b d \log \left (F\right )\right ) +{\left (b c + a\right )} \log \left (F\right )\right )} \Gamma \left (m + 1, -b d x \log \left (F\right )\right )}{b^{3} d^{3} \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + b \left (c + d x\right )} x^{m} \left (e + f x\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2} F^{{\left (d x + c\right )} b + a} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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