Optimal. Leaf size=100 \[ -\frac{2 (c+d x)^n F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac{2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}+\frac{(c+d x)^{2 n} F^{a+b (c+d x)^n}}{b d n \log (F)} \]
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Rubi [A] time = 0.119527, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2213, 2209} \[ -\frac{2 (c+d x)^n F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac{2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}+\frac{(c+d x)^{2 n} F^{a+b (c+d x)^n}}{b d n \log (F)} \]
Antiderivative was successfully verified.
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Rule 2213
Rule 2209
Rubi steps
\begin{align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1+3 n} \, dx &=\frac{F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)}-\frac{2 \int F^{a+b (c+d x)^n} (c+d x)^{-1+2 n} \, dx}{b \log (F)}\\ &=-\frac{2 F^{a+b (c+d x)^n} (c+d x)^n}{b^2 d n \log ^2(F)}+\frac{F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)}+\frac{2 \int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx}{b^2 \log ^2(F)}\\ &=\frac{2 F^{a+b (c+d x)^n}}{b^3 d n \log ^3(F)}-\frac{2 F^{a+b (c+d x)^n} (c+d x)^n}{b^2 d n \log ^2(F)}+\frac{F^{a+b (c+d x)^n} (c+d x)^{2 n}}{b d n \log (F)}\\ \end{align*}
Mathematica [C] time = 0.0065167, size = 31, normalized size = 0.31 \[ \frac{F^a \text{Gamma}\left (3,-b \log (F) (c+d x)^n\right )}{b^3 d n \log ^3(F)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 59, normalized size = 0.6 \begin{align*}{\frac{ \left ( \left ( \left ( dx+c \right ) ^{n} \right ) ^{2}{b}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}-2\,b \left ( dx+c \right ) ^{n}\ln \left ( F \right ) +2 \right ){F}^{a+b \left ( dx+c \right ) ^{n}}}{ \left ( \ln \left ( F \right ) \right ) ^{3}{b}^{3}nd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04473, size = 89, normalized size = 0.89 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{2 \, n} F^{a} b^{2} \log \left (F\right )^{2} - 2 \,{\left (d x + c\right )}^{n} F^{a} b \log \left (F\right ) + 2 \, F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{3} d n \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56674, size = 157, normalized size = 1.57 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{2 \, n} b^{2} \log \left (F\right )^{2} - 2 \,{\left (d x + c\right )}^{n} b \log \left (F\right ) + 2\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{3} d n \log \left (F\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (d x + c\right )}^{3 \, n - 1} F^{{\left (d x + c\right )}^{n} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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