Optimal. Leaf size=54 \[ -\frac {F^a (c+d x)^3 \left (-b \log (F) (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-b (c+d x)^n \log (F)\right )}{d n} \]
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Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac {F^a (c+d x)^3 \left (-b \log (F) (c+d x)^n\right )^{-3/n} \text {Gamma}\left (\frac {3}{n},-b \log (F) (c+d x)^n\right )}{d n} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^2 \, dx &=-\frac {F^a (c+d x)^3 \Gamma \left (\frac {3}{n},-b (c+d x)^n \log (F)\right ) \left (-b (c+d x)^n \log (F)\right )^{-3/n}}{d n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 54, normalized size = 1.00 \[ -\frac {F^a (c+d x)^3 \left (-b \log (F) (c+d x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-b (c+d x)^n \log (F)\right )}{d n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} F^{{\left (d x + c\right )}^{n} b + a}, x\right ) \]
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 \[ \int {\left (d x + c\right )}^{2} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{2} F^{b \left (d x +c \right )^{n}+a}\, dx \]
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 \[ \int {\left (d x + c\right )}^{2} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 73, normalized size = 1.35 \[ \frac {F^a\,{\mathrm {e}}^{\frac {b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n}{2}}\,{\left (c+d\,x\right )}^3\,{\mathrm {M}}_{\frac {1}{2}-\frac {3}{2\,n},\frac {3}{2\,n}}\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n\right )}{3\,d\,{\left (b\,\ln \relax (F)\,{\left (c+d\,x\right )}^n\right )}^{\frac {3}{2\,n}+\frac {1}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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