Optimal. Leaf size=118 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+3 d e^2 \log (x) \left (a+b \log \left (c x^n\right )\right )+e^3 x \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{x}-\frac{b d^3 n}{4 x^2}-\frac{3}{2} b d e^2 n \log ^2(x)-b e^3 n x \]
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Rubi [A] time = 0.0909902, antiderivative size = 91, normalized size of antiderivative = 0.77, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {43, 2334, 2301} \[ -\frac{1}{2} \left (\frac{6 d^2 e}{x}+\frac{d^3}{x^2}-6 d e^2 \log (x)-2 e^3 x\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3 b d^2 e n}{x}-\frac{b d^3 n}{4 x^2}-\frac{3}{2} b d e^2 n \log ^2(x)-b e^3 n x \]
Antiderivative was successfully verified.
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Rule 43
Rule 2334
Rule 2301
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac{1}{2} \left (\frac{d^3}{x^2}+\frac{6 d^2 e}{x}-2 e^3 x-6 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (e^3-\frac{d^3}{2 x^3}-\frac{3 d^2 e}{x^2}+\frac{3 d e^2 \log (x)}{x}\right ) \, dx\\ &=-\frac{b d^3 n}{4 x^2}-\frac{3 b d^2 e n}{x}-b e^3 n x-\frac{1}{2} \left (\frac{d^3}{x^2}+\frac{6 d^2 e}{x}-2 e^3 x-6 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (3 b d e^2 n\right ) \int \frac{\log (x)}{x} \, dx\\ &=-\frac{b d^3 n}{4 x^2}-\frac{3 b d^2 e n}{x}-b e^3 n x-\frac{3}{2} b d e^2 n \log ^2(x)-\frac{1}{2} \left (\frac{d^3}{x^2}+\frac{6 d^2 e}{x}-2 e^3 x-6 d e^2 \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0778604, size = 115, normalized size = 0.97 \[ -\frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}+\frac{3 d e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+a e^3 x+b e^3 x \log \left (c x^n\right )-\frac{3 b d^2 e n}{x}-\frac{b d^3 n}{4 x^2}-b e^3 n x \]
Antiderivative was successfully verified.
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Maple [C] time = 0.263, size = 586, normalized size = 5. \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 = 1.09208, size = 169, normalized size = 1.43 \begin{align*} -b e^{3} n x + b e^{3} x \log \left (c x^{n}\right ) + a e^{3} x + \frac{3 \, b d e^{2} \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d e^{2} \log \left (x\right ) - \frac{3 \, b d^{2} e n}{x} - \frac{3 \, b d^{2} e \log \left (c x^{n}\right )}{x} - \frac{b d^{3} n}{4 \, x^{2}} - \frac{3 \, a d^{2} e}{x} - \frac{b d^{3} \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{a d^{3}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.01498, size = 338, normalized size = 2.86 \begin{align*} \frac{6 \, b d e^{2} n x^{2} \log \left (x\right )^{2} - b d^{3} n - 2 \, a d^{3} - 4 \,{\left (b e^{3} n - a e^{3}\right )} x^{3} - 12 \,{\left (b d^{2} e n + a d^{2} e\right )} x + 2 \,{\left (2 \, b e^{3} x^{3} - 6 \, b d^{2} e x - b d^{3}\right )} \log \left (c\right ) + 2 \,{\left (2 \, b e^{3} n x^{3} + 6 \, b d e^{2} x^{2} \log \left (c\right ) - 6 \, b d^{2} e n x + 6 \, a d e^{2} x^{2} - b d^{3} n\right )} \log \left (x\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.94532, size = 182, normalized size = 1.54 \begin{align*} - \frac{a d^{3}}{2 x^{2}} - \frac{3 a d^{2} e}{x} + 3 a d e^{2} \log{\left (x \right )} + a e^{3} x - \frac{b d^{3} n \log{\left (x \right )}}{2 x^{2}} - \frac{b d^{3} n}{4 x^{2}} - \frac{b d^{3} \log{\left (c \right )}}{2 x^{2}} - \frac{3 b d^{2} e n \log{\left (x \right )}}{x} - \frac{3 b d^{2} e n}{x} - \frac{3 b d^{2} e \log{\left (c \right )}}{x} + \frac{3 b d e^{2} n \log{\left (x \right )}^{2}}{2} + 3 b d e^{2} \log{\left (c \right )} \log{\left (x \right )} + b e^{3} n x \log{\left (x \right )} - b e^{3} n x + b e^{3} x \log{\left (c \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33339, size = 208, normalized size = 1.76 \begin{align*} \frac{6 \, b d n x^{2} e^{2} \log \left (x\right )^{2} + 4 \, b n x^{3} e^{3} \log \left (x\right ) - 12 \, b d^{2} n x e \log \left (x\right ) + 12 \, b d x^{2} e^{2} \log \left (c\right ) \log \left (x\right ) - 4 \, b n x^{3} e^{3} - 12 \, b d^{2} n x e + 4 \, b x^{3} e^{3} \log \left (c\right ) - 12 \, b d^{2} x e \log \left (c\right ) - 2 \, b d^{3} n \log \left (x\right ) + 12 \, a d x^{2} e^{2} \log \left (x\right ) - b d^{3} n + 4 \, a x^{3} e^{3} - 12 \, a d^{2} x e - 2 \, b d^{3} \log \left (c\right ) - 2 \, a d^{3}}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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