Optimal. Leaf size=22 \[ e F^{c (a+b x)} x^2 \log ^{1+n}(d x) \]
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Rubi [A]
time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2233}
\begin {gather*} e x^2 \log ^{n+1}(d x) F^{c (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2233
Rubi steps
\begin {align*} \int F^{c (a+b x)} x \log ^n(d x) (e+e n+e (2+b c x \log (F)) \log (d x)) \, dx &=e F^{c (a+b x)} x^2 \log ^{1+n}(d x)\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 23, normalized size = 1.05 \begin {gather*} e F^{a c+b c x} x^2 \log ^{1+n}(d x) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.12, size = 198, normalized size = 9.00
method | result | size |
risch | \(\left (-\frac {i \pi e \,x^{2} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right ) F^{c \left (b x +a \right )}}{2}+\frac {i \pi e \,x^{2} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2} F^{c \left (b x +a \right )}}{2}+\frac {i \pi e \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2} F^{c \left (b x +a \right )}}{2}-\frac {i \pi e \,x^{2} \mathrm {csgn}\left (i d x \right )^{3} F^{c \left (b x +a \right )}}{2}+\ln \left (d \right ) e \,x^{2} F^{c \left (b x +a \right )}+e \,x^{2} F^{c \left (b x +a \right )} \ln \left (x \right )\right ) \left (\ln \left (d \right )+\ln \left (x \right )-\frac {i \pi \,\mathrm {csgn}\left (i d x \right ) \left (-\mathrm {csgn}\left (i d x \right )+\mathrm {csgn}\left (i d \right )\right ) \left (-\mathrm {csgn}\left (i d x \right )+\mathrm {csgn}\left (i x \right )\right )}{2}\right )^{n}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 44, normalized size = 2.00 \begin {gather*} {\left (F^{a c} x^{2} e \log \left (d\right ) + F^{a c} x^{2} e \log \left (x\right )\right )} e^{\left (b c x \log \left (F\right ) + n \log \left (\log \left (d\right ) + \log \left (x\right )\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 26, normalized size = 1.18 \begin {gather*} F^{b c x + a c} x^{2} \log \left (d x\right )^{n} e \log \left (d x\right ) \end {gather*}
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 \begin {gather*} e \left (\int F^{a c} F^{b c x} x \log {\left (d x \right )}^{n}\, dx + \int F^{a c} F^{b c x} n x \log {\left (d x \right )}^{n}\, dx + \int 2 F^{a c} F^{b c x} x \log {\left (d x \right )} \log {\left (d x \right )}^{n}\, dx + \int F^{a c} F^{b c x} b c x^{2} \log {\left (F \right )} \log {\left (d x \right )} \log {\left (d x \right )}^{n}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.39, size = 23, normalized size = 1.05 \begin {gather*} F^{a\,c+b\,c\,x}\,e\,x^2\,{\ln \left (d\,x\right )}^{n+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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