Optimal. Leaf size=100 \[ -\frac {F^{a+b (c+d x)^n} (c+d x)^{-2 n}}{2 d n}-\frac {b F^{a+b (c+d x)^n} (c+d x)^{-n} \log (F)}{2 d n}+\frac {b^2 F^a \text {Ei}\left (b (c+d x)^n \log (F)\right ) \log ^2(F)}{2 d n} \]
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Rubi [A]
time = 0.08, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2246, 2241}
\begin {gather*} \frac {b^2 F^a \log ^2(F) \text {Ei}\left (b (c+d x)^n \log (F)\right )}{2 d n}-\frac {(c+d x)^{-2 n} F^{a+b (c+d x)^n}}{2 d n}-\frac {b \log (F) (c+d x)^{-n} F^{a+b (c+d x)^n}}{2 d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2241
Rule 2246
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)^{-2 n}}{2 d n}+\frac {1}{2} (b \log (F)) \int F^{a+b (c+d x)^n} (c+d x)^{-1-n} \, dx\\ &=-\frac {F^{a+b (c+d x)^n} (c+d x)^{-2 n}}{2 d n}-\frac {b F^{a+b (c+d x)^n} (c+d x)^{-n} \log (F)}{2 d n}+\frac {1}{2} \left (b^2 \log ^2(F)\right ) \int \frac {F^{a+b (c+d x)^n}}{c+d x} \, dx\\ &=-\frac {F^{a+b (c+d x)^n} (c+d x)^{-2 n}}{2 d n}-\frac {b F^{a+b (c+d x)^n} (c+d x)^{-n} \log (F)}{2 d n}+\frac {b^2 F^a \text {Ei}\left (b (c+d x)^n \log (F)\right ) \log ^2(F)}{2 d n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.32 \begin {gather*} -\frac {b^2 F^a \Gamma \left (-2,-b (c+d x)^n \log (F)\right ) \log ^2(F)}{d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 99, normalized size = 0.99
method | result | size |
risch | \(-\frac {F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-2 n}}{2 n d}-\frac {\ln \left (F \right ) b \,F^{b \left (d x +c \right )^{n}} F^{a} \left (d x +c \right )^{-n}}{2 n d}-\frac {\ln \left (F \right )^{2} b^{2} F^{a} \expIntegral \left (1, -b \left (d x +c \right )^{n} \ln \left (F \right )\right )}{2 n d}\) | \(99\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 84, normalized size = 0.84 \begin {gather*} \frac {{\left (d x + c\right )}^{2 \, n} F^{a} b^{2} {\rm Ei}\left ({\left (d x + c\right )}^{n} b \log \left (F\right )\right ) \log \left (F\right )^{2} - {\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 )}}{2 \, {\left (d x + c\right )}^{2 \, n} d n} \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 = -0.01 \begin {gather*} \int \frac {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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