Optimal. Leaf size=119 \[ 3 d^2 e \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+3 d e^2 x \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{3}{2} b d^2 e n \log ^2(x)-\frac{b d^3 n}{x}-3 b d e^2 n x-\frac{1}{4} b e^3 n x^2 \]
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Rubi [A] time = 0.0877142, antiderivative size = 92, 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} \[ -\frac{1}{2} \left (-6 d^2 e \log (x)+\frac{2 d^3}{x}-6 d e^2 x-e^3 x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{3}{2} b d^2 e n \log ^2(x)-\frac{b d^3 n}{x}-3 b d e^2 n x-\frac{1}{4} b e^3 n x^2 \]
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^2} \, dx &=-\frac{1}{2} \left (\frac{2 d^3}{x}-6 d e^2 x-e^3 x^2-6 d^2 e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (3 d e^2-\frac{d^3}{x^2}+\frac{e^3 x}{2}+\frac{3 d^2 e \log (x)}{x}\right ) \, dx\\ &=-\frac{b d^3 n}{x}-3 b d e^2 n x-\frac{1}{4} b e^3 n x^2-\frac{1}{2} \left (\frac{2 d^3}{x}-6 d e^2 x-e^3 x^2-6 d^2 e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )-\left (3 b d^2 e n\right ) \int \frac{\log (x)}{x} \, dx\\ &=-\frac{b d^3 n}{x}-3 b d e^2 n x-\frac{1}{4} b e^3 n x^2-\frac{3}{2} b d^2 e n \log ^2(x)-\frac{1}{2} \left (\frac{2 d^3}{x}-6 d e^2 x-e^3 x^2-6 d^2 e \log (x)\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0802664, size = 118, normalized size = 0.99 \[ \frac{3 d^2 e \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}-\frac{d^3 \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{1}{2} e^3 x^2 \left (a+b \log \left (c x^n\right )\right )+3 a d e^2 x+3 b d e^2 x \log \left (c x^n\right )-\frac{b d^3 n}{x}-3 b d e^2 n x-\frac{1}{4} b e^3 n x^2 \]
Antiderivative was successfully verified.
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Maple [C] time = 0.276, size = 588, normalized size = 4.9 \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 = 1.11752, size = 171, normalized size = 1.44 \begin{align*} -\frac{1}{4} \, b e^{3} n x^{2} + \frac{1}{2} \, b e^{3} x^{2} \log \left (c x^{n}\right ) - 3 \, b d e^{2} n x + \frac{1}{2} \, a e^{3} x^{2} + 3 \, b d e^{2} x \log \left (c x^{n}\right ) + 3 \, a d e^{2} x + \frac{3 \, b d^{2} e \log \left (c x^{n}\right )^{2}}{2 \, n} + 3 \, a d^{2} e \log \left (x\right ) - \frac{b d^{3} n}{x} - \frac{b d^{3} \log \left (c x^{n}\right )}{x} - \frac{a d^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03515, size = 338, normalized size = 2.84 \begin{align*} \frac{6 \, b d^{2} e n x \log \left (x\right )^{2} - 4 \, b d^{3} n - 4 \, a d^{3} -{\left (b e^{3} n - 2 \, a e^{3}\right )} x^{3} - 12 \,{\left (b d e^{2} n - a d e^{2}\right )} x^{2} + 2 \,{\left (b e^{3} x^{3} + 6 \, b d e^{2} x^{2} - 2 \, b d^{3}\right )} \log \left (c\right ) + 2 \,{\left (b e^{3} n x^{3} + 6 \, b d e^{2} n x^{2} + 6 \, b d^{2} e x \log \left (c\right ) - 2 \, b d^{3} n + 6 \, a d^{2} e x\right )} \log \left (x\right )}{4 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.76585, size = 182, normalized size = 1.53 \begin{align*} - \frac{a d^{3}}{x} + 3 a d^{2} e \log{\left (x \right )} + 3 a d e^{2} x + \frac{a e^{3} x^{2}}{2} - \frac{b d^{3} n \log{\left (x \right )}}{x} - \frac{b d^{3} n}{x} - \frac{b d^{3} \log{\left (c \right )}}{x} + \frac{3 b d^{2} e n \log{\left (x \right )}^{2}}{2} + 3 b d^{2} e \log{\left (c \right )} \log{\left (x \right )} + 3 b d e^{2} n x \log{\left (x \right )} - 3 b d e^{2} n x + 3 b d e^{2} x \log{\left (c \right )} + \frac{b e^{3} n x^{2} \log{\left (x \right )}}{2} - \frac{b e^{3} n x^{2}}{4} + \frac{b e^{3} x^{2} \log{\left (c \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31787, size = 208, normalized size = 1.75 \begin{align*} \frac{6 \, b d^{2} n x e \log \left (x\right )^{2} + 2 \, b n x^{3} e^{3} \log \left (x\right ) + 12 \, b d n x^{2} e^{2} \log \left (x\right ) + 12 \, b d^{2} x e \log \left (c\right ) \log \left (x\right ) - b n x^{3} e^{3} - 12 \, b d n x^{2} e^{2} + 2 \, b x^{3} e^{3} \log \left (c\right ) + 12 \, b d x^{2} e^{2} \log \left (c\right ) - 4 \, b d^{3} n \log \left (x\right ) + 12 \, a d^{2} x e \log \left (x\right ) - 4 \, b d^{3} n + 2 \, a x^{3} e^{3} + 12 \, a d x^{2} e^{2} - 4 \, b d^{3} \log \left (c\right ) - 4 \, a d^{3}}{4 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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