Optimal. Leaf size=71 \[ \frac {\left ((d+e x)^2\right )^m F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{-2 m} \Gamma \left (2 m+1,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
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Rubi [A] time = 0.06, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2188, 2181} \[ \frac {\left ((d+e x)^2\right )^m F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{-2 m} \text {Gamma}\left (2 m+1,-\frac {b c \log (F) (d+e x)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 2188
Rubi steps
\begin {align*} \int F^{c (a+b x)} \left (d^2+2 d e x+e^2 x^2\right )^m \, dx &=(d+e x)^{-2 m} \left ((d+e x)^2\right )^m \int F^{c (a+b x)} (d+e x)^{2 m} \, dx\\ &=\frac {F^{c \left (a-\frac {b d}{e}\right )} \left ((d+e x)^2\right )^m \Gamma \left (1+2 m,-\frac {b c (d+e x) \log (F)}{e}\right ) \left (-\frac {b c (d+e x) \log (F)}{e}\right )^{-2 m}}{b c \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 71, normalized size = 1.00 \[ \frac {\left ((d+e x)^2\right )^m F^{c \left (a-\frac {b d}{e}\right )} \left (-\frac {b c \log (F) (d+e x)}{e}\right )^{-2 m} \Gamma \left (2 m+1,-\frac {b c (d+e x) \log (F)}{e}\right )}{b c \log (F)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m} F^{b c x + a c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int F^{\left (b x +a \right ) c} \left (e^{2} x^{2}+2 d e x +d^{2}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}^{m} F^{{\left (b x + a\right )} c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int F^{c\,\left (a+b\,x\right )}\,{\left (d^2+2\,d\,e\,x+e^2\,x^2\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \left (\left (d + e x\right )^{2}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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