Optimal. Leaf size=85 \[ \frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{3}{4} b d^2 e n x^2-\frac{b d^4 n \log (x)}{4 e}-b d^3 n x-\frac{1}{3} b d e^2 n x^3-\frac{1}{16} b e^3 n x^4 \]
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Rubi [A] time = 0.0434278, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {32, 2313, 12, 43} \[ \frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{3}{4} b d^2 e n x^2-\frac{b d^4 n \log (x)}{4 e}-b d^3 n x-\frac{1}{3} b d e^2 n x^3-\frac{1}{16} b e^3 n x^4 \]
Antiderivative was successfully verified.
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Rule 32
Rule 2313
Rule 12
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-(b n) \int \frac{(d+e x)^4}{4 e x} \, dx\\ &=\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{(b n) \int \frac{(d+e x)^4}{x} \, dx}{4 e}\\ &=\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}-\frac{(b n) \int \left (4 d^3 e+\frac{d^4}{x}+6 d^2 e^2 x+4 d e^3 x^2+e^4 x^3\right ) \, dx}{4 e}\\ &=-b d^3 n x-\frac{3}{4} b d^2 e n x^2-\frac{1}{3} b d e^2 n x^3-\frac{1}{16} b e^3 n x^4-\frac{b d^4 n \log (x)}{4 e}+\frac{(d+e x)^4 \left (a+b \log \left (c x^n\right )\right )}{4 e}\\ \end{align*}
Mathematica [A] time = 0.046026, size = 110, normalized size = 1.29 \[ \frac{1}{48} x \left (12 a \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+12 b \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right ) \log \left (c x^n\right )-b n \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.227, size = 571, normalized size = 6.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13255, size = 180, normalized size = 2.12 \begin{align*} -\frac{1}{16} \, b e^{3} n x^{4} + \frac{1}{4} \, b e^{3} x^{4} \log \left (c x^{n}\right ) - \frac{1}{3} \, b d e^{2} n x^{3} + \frac{1}{4} \, a e^{3} x^{4} + b d e^{2} x^{3} \log \left (c x^{n}\right ) - \frac{3}{4} \, b d^{2} e n x^{2} + a d e^{2} x^{3} + \frac{3}{2} \, b d^{2} e x^{2} \log \left (c x^{n}\right ) - b d^{3} n x + \frac{3}{2} \, a d^{2} e x^{2} + b d^{3} x \log \left (c x^{n}\right ) + a d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.998267, size = 360, normalized size = 4.24 \begin{align*} -\frac{1}{16} \,{\left (b e^{3} n - 4 \, a e^{3}\right )} x^{4} - \frac{1}{3} \,{\left (b d e^{2} n - 3 \, a d e^{2}\right )} x^{3} - \frac{3}{4} \,{\left (b d^{2} e n - 2 \, a d^{2} e\right )} x^{2} -{\left (b d^{3} n - a d^{3}\right )} x + \frac{1}{4} \,{\left (b e^{3} x^{4} + 4 \, b d e^{2} x^{3} + 6 \, b d^{2} e x^{2} + 4 \, b d^{3} x\right )} \log \left (c\right ) + \frac{1}{4} \,{\left (b e^{3} n x^{4} + 4 \, b d e^{2} n x^{3} + 6 \, b d^{2} e n x^{2} + 4 \, b d^{3} n x\right )} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.98225, size = 204, normalized size = 2.4 \begin{align*} a d^{3} x + \frac{3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac{a e^{3} x^{4}}{4} + b d^{3} n x \log{\left (x \right )} - b d^{3} n x + b d^{3} x \log{\left (c \right )} + \frac{3 b d^{2} e n x^{2} \log{\left (x \right )}}{2} - \frac{3 b d^{2} e n x^{2}}{4} + \frac{3 b d^{2} e x^{2} \log{\left (c \right )}}{2} + b d e^{2} n x^{3} \log{\left (x \right )} - \frac{b d e^{2} n x^{3}}{3} + b d e^{2} x^{3} \log{\left (c \right )} + \frac{b e^{3} n x^{4} \log{\left (x \right )}}{4} - \frac{b e^{3} n x^{4}}{16} + \frac{b e^{3} x^{4} \log{\left (c \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27096, size = 215, normalized size = 2.53 \begin{align*} \frac{1}{4} \, b n x^{4} e^{3} \log \left (x\right ) + b d n x^{3} e^{2} \log \left (x\right ) + \frac{3}{2} \, b d^{2} n x^{2} e \log \left (x\right ) - \frac{1}{16} \, b n x^{4} e^{3} - \frac{1}{3} \, b d n x^{3} e^{2} - \frac{3}{4} \, b d^{2} n x^{2} e + \frac{1}{4} \, b x^{4} e^{3} \log \left (c\right ) + b d x^{3} e^{2} \log \left (c\right ) + \frac{3}{2} \, b d^{2} x^{2} e \log \left (c\right ) + b d^{3} n x \log \left (x\right ) - b d^{3} n x + \frac{1}{4} \, a x^{4} e^{3} + a d x^{3} e^{2} + \frac{3}{2} \, a d^{2} x^{2} e + b d^{3} x \log \left (c\right ) + a d^{3} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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