Optimal. Leaf size=63 \[ -\frac {F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^n}{b d n \log (F)} \]
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Rubi [A]
time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2244, 2240}
\begin {gather*} \frac {(c+d x)^n F^{a+b (c+d x)^n}}{b d n \log (F)}-\frac {F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2240
Rule 2244
Rubi steps
\begin {align*} \int F^{a+b (c+d x)^n} (c+d x)^{-1+2 n} \, dx &=\frac {F^{a+b (c+d x)^n} (c+d x)^n}{b d n \log (F)}-\frac {\int F^{a+b (c+d x)^n} (c+d x)^{-1+n} \, dx}{b \log (F)}\\ &=-\frac {F^{a+b (c+d x)^n}}{b^2 d n \log ^2(F)}+\frac {F^{a+b (c+d x)^n} (c+d x)^n}{b d n \log (F)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.01, size = 32, normalized size = 0.51 \begin {gather*} -\frac {F^a \Gamma \left (2,-b (c+d x)^n \log (F)\right )}{b^2 d n \log ^2(F)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 41, normalized size = 0.65
method | result | size |
risch | \(\frac {\left (b \left (d x +c \right )^{n} \ln \left (F \right )-1\right ) F^{a +b \left (d x +c \right )^{n}}}{b^{2} n d \ln \left (F \right )^{2}}\) | \(41\) |
norman | \(\frac {{\mathrm e}^{n \ln \left (d x +c \right )} {\mathrm e}^{\left (a +b \,{\mathrm e}^{n \ln \left (d x +c \right )}\right ) \ln \left (F \right )}}{d b n \ln \left (F \right )}-\frac {{\mathrm e}^{\left (a +b \,{\mathrm e}^{n \ln \left (d x +c \right )}\right ) \ln \left (F \right )}}{b^{2} n d \ln \left (F \right )^{2}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 45, normalized size = 0.71 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{n} F^{a} b \log \left (F\right ) - F^{a}\right )} F^{{\left (d x + c\right )}^{n} b}}{b^{2} d n \log \left (F\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 44, normalized size = 0.70 \begin {gather*} \frac {{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) - 1\right )} e^{\left ({\left (d x + c\right )}^{n} b \log \left (F\right ) + a \log \left (F\right )\right )}}{b^{2} d n \log \left (F\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int F^{a+b\,{\left (c+d\,x\right )}^n}\,{\left (c+d\,x\right )}^{2\,n-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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