Optimal. Leaf size=48 \[ \frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d n x^4-\frac{1}{25} b e n x^5 \]
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Rubi [A] time = 0.0516624, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {43, 2334, 12} \[ \frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b d n x^4-\frac{1}{25} b e n x^5 \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 12
Rubi steps
\begin{align*} \int x^3 (d+e x) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac{1}{20} x^3 (5 d+4 e x) \, dx\\ &=\frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{20} (b n) \int x^3 (5 d+4 e x) \, dx\\ &=\frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{20} (b n) \int \left (5 d x^3+4 e x^4\right ) \, dx\\ &=-\frac{1}{16} b d n x^4-\frac{1}{25} b e n x^5+\frac{1}{20} \left (5 d x^4+4 e x^5\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0244789, size = 48, normalized size = 1. \[ \frac{1}{400} x^4 \left (20 a (5 d+4 e x)+20 b (5 d+4 e x) \log \left (c x^n\right )-b n (25 d+16 e x)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.191, size = 264, normalized size = 5.5 \begin{align*}{\frac{b{x}^{4} \left ( 4\,ex+5\,d \right ) \ln \left ({x}^{n} \right ) }{20}}+{\frac{i}{10}}\pi \,be{x}^{5}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{10}}\pi \,be{x}^{5}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{10}}\pi \,be{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{10}}\pi \,be{x}^{5} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) be{x}^{5}}{5}}-{\frac{ben{x}^{5}}{25}}+{\frac{ae{x}^{5}}{5}}+{\frac{i}{8}}\pi \,bd{x}^{4}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-{\frac{i}{8}}\pi \,bd{x}^{4}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) -{\frac{i}{8}}\pi \,bd{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+{\frac{i}{8}}\pi \,bd{x}^{4} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{\ln \left ( c \right ) bd{x}^{4}}{4}}-{\frac{bdn{x}^{4}}{16}}+{\frac{ad{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15044, size = 77, normalized size = 1.6 \begin{align*} -\frac{1}{25} \, b e n x^{5} + \frac{1}{5} \, b e x^{5} \log \left (c x^{n}\right ) - \frac{1}{16} \, b d n x^{4} + \frac{1}{5} \, a e x^{5} + \frac{1}{4} \, b d x^{4} \log \left (c x^{n}\right ) + \frac{1}{4} \, a d x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.985886, size = 181, normalized size = 3.77 \begin{align*} -\frac{1}{25} \,{\left (b e n - 5 \, a e\right )} x^{5} - \frac{1}{16} \,{\left (b d n - 4 \, a d\right )} x^{4} + \frac{1}{20} \,{\left (4 \, b e x^{5} + 5 \, b d x^{4}\right )} \log \left (c\right ) + \frac{1}{20} \,{\left (4 \, b e n x^{5} + 5 \, b d n x^{4}\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.23149, size = 87, normalized size = 1.81 \begin{align*} \frac{a d x^{4}}{4} + \frac{a e x^{5}}{5} + \frac{b d n x^{4} \log{\left (x \right )}}{4} - \frac{b d n x^{4}}{16} + \frac{b d x^{4} \log{\left (c \right )}}{4} + \frac{b e n x^{5} \log{\left (x \right )}}{5} - \frac{b e n x^{5}}{25} + \frac{b e x^{5} \log{\left (c \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24925, size = 99, normalized size = 2.06 \begin{align*} \frac{1}{5} \, b n x^{5} e \log \left (x\right ) - \frac{1}{25} \, b n x^{5} e + \frac{1}{5} \, b x^{5} e \log \left (c\right ) + \frac{1}{4} \, b d n x^{4} \log \left (x\right ) - \frac{1}{16} \, b d n x^{4} + \frac{1}{5} \, a x^{5} e + \frac{1}{4} \, b d x^{4} \log \left (c\right ) + \frac{1}{4} \, a d x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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