Optimal. Leaf size=46 \[ -\frac {b n (f x)^{1+m}}{f (1+m)^2}+\frac {(f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)} \]
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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2341}
\begin {gather*} \frac {(f x)^{m+1} \left (a+b \log \left (c x^n\right )\right )}{f (m+1)}-\frac {b n (f x)^{m+1}}{f (m+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rubi steps
\begin {align*} \int (f x)^m \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {b n (f x)^{1+m}}{f (1+m)^2}+\frac {(f x)^{1+m} \left (a+b \log \left (c x^n\right )\right )}{f (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.70 \begin {gather*} \frac {x (f x)^m \left (a+a m-b n+b (1+m) \log \left (c x^n\right )\right )}{(1+m)^2} \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 = 0.03, size = 371, normalized size = 8.07
method | result | size |
risch | \(\frac {b x \,{\mathrm e}^{\frac {m \left (-i \pi \mathrm {csgn}\left (i f x \right )^{3}+i \pi \mathrm {csgn}\left (i f x \right )^{2} \mathrm {csgn}\left (i f \right )+i \pi \mathrm {csgn}\left (i f x \right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i f x \right ) \mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right )+2 \ln \left (x \right )+2 \ln \left (f \right )\right )}{2}} \ln \left (x^{n}\right )}{1+m}-\frac {\left (i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) m -i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} m -i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} m +i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} m +i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 b \ln \left (c \right ) m -2 b \ln \left (c \right )-2 a m +2 b n -2 a \right ) x \,{\mathrm e}^{\frac {m \left (-i \pi \mathrm {csgn}\left (i f x \right )^{3}+i \pi \mathrm {csgn}\left (i f x \right )^{2} \mathrm {csgn}\left (i f \right )+i \pi \mathrm {csgn}\left (i f x \right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i f x \right ) \mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x \right )+2 \ln \left (x \right )+2 \ln \left (f \right )\right )}{2}}}{2 \left (1+m \right )^{2}}\) | \(371\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 57, normalized size = 1.24 \begin {gather*} -\frac {b f^{m} n x x^{m}}{{\left (m + 1\right )}^{2}} + \frac {\left (f x\right )^{m + 1} b \log \left (c x^{n}\right )}{f {\left (m + 1\right )}} + \frac {\left (f x\right )^{m + 1} a}{f {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 52, normalized size = 1.13 \begin {gather*} \frac {{\left ({\left (b m + b\right )} n x \log \left (x\right ) + {\left (b m + b\right )} x \log \left (c\right ) + {\left (a m - b n + a\right )} x\right )} e^{\left (m \log \left (f\right ) + m \log \left (x\right )\right )}}{m^{2} + 2 \, m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (37) = 74\).
time = 4.94, size = 141, normalized size = 3.07 \begin {gather*} \begin {cases} \frac {a m x \left (f x\right )^{m}}{m^{2} + 2 m + 1} + \frac {a x \left (f x\right )^{m}}{m^{2} + 2 m + 1} + \frac {b m x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1} - \frac {b n x \left (f x\right )^{m}}{m^{2} + 2 m + 1} + \frac {b x \left (f x\right )^{m} \log {\left (c x^{n} \right )}}{m^{2} + 2 m + 1} & \text {for}\: m \neq -1 \\\frac {\begin {cases} a \log {\left (x \right )} & \text {for}\: b = 0 \\- \left (- a - b \log {\left (c \right )}\right ) \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\left (- a - b \log {\left (c x^{n} \right )}\right )^{2}}{2 b n} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (46) = 92\).
time = 1.83, size = 95, normalized size = 2.07 \begin {gather*} \frac {b f^{m} m n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} + \frac {b f^{m} n x x^{m} \log \left (x\right )}{m^{2} + 2 \, m + 1} - \frac {b f^{m} n x x^{m}}{m^{2} + 2 \, m + 1} + \frac {\left (f x\right )^{m} b x \log \left (c\right )}{m + 1} + \frac {\left (f x\right )^{m} a x}{m + 1} \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 {\left (f\,x\right )}^m\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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