Optimal. Leaf size=75 \[ -\frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}-\frac{b d^2 n}{9 x^3}+\frac{b e^3 n \log (x)}{3 d}-\frac{b d e n}{2 x^2}-\frac{b e^2 n}{x} \]
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Rubi [A] time = 0.0709689, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {37, 2334, 12, 43} \[ -\frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}-\frac{b d^2 n}{9 x^3}+\frac{b e^3 n \log (x)}{3 d}-\frac{b d e n}{2 x^2}-\frac{b e^2 n}{x} \]
Antiderivative was successfully verified.
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Rule 37
Rule 2334
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \left (a+b \log \left (c x^n\right )\right )}{x^4} \, dx &=-\frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}-(b n) \int -\frac{(d+e x)^3}{3 d x^4} \, dx\\ &=-\frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac{(b n) \int \frac{(d+e x)^3}{x^4} \, dx}{3 d}\\ &=-\frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}+\frac{(b n) \int \left (\frac{d^3}{x^4}+\frac{3 d^2 e}{x^3}+\frac{3 d e^2}{x^2}+\frac{e^3}{x}\right ) \, dx}{3 d}\\ &=-\frac{b d^2 n}{9 x^3}-\frac{b d e n}{2 x^2}-\frac{b e^2 n}{x}+\frac{b e^3 n \log (x)}{3 d}-\frac{(d+e x)^3 \left (a+b \log \left (c x^n\right )\right )}{3 d x^3}\\ \end{align*}
Mathematica [A] time = 0.0385964, size = 76, normalized size = 1.01 \[ -\frac{6 a \left (d^2+3 d e x+3 e^2 x^2\right )+6 b \left (d^2+3 d e x+3 e^2 x^2\right ) \log \left (c x^n\right )+b n \left (2 d^2+9 d e x+18 e^2 x^2\right )}{18 x^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.121, size = 401, normalized size = 5.4 \begin{align*} -{\frac{b \left ( 3\,{e}^{2}{x}^{2}+3\,dex+{d}^{2} \right ) \ln \left ({x}^{n} \right ) }{3\,{x}^{3}}}-{\frac{-3\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}-9\,i\pi \,b{e}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+9\,i\pi \,b{e}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-9\,i\pi \,bdex{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +18\,\ln \left ( c \right ) b{e}^{2}{x}^{2}+18\,b{e}^{2}n{x}^{2}+18\,a{e}^{2}{x}^{2}+9\,i\pi \,bdex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -9\,i\pi \,b{e}^{2}{x}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +3\,i\pi \,b{d}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +9\,i\pi \,b{e}^{2}{x}^{2} \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +18\,\ln \left ( c \right ) bdex+9\,bdenx+18\,adex-9\,i\pi \,bdex \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}+3\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}-3\,i\pi \,b{d}^{2}{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) +9\,i\pi \,bdex{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}+6\,\ln \left ( c \right ) b{d}^{2}+2\,b{d}^{2}n+6\,a{d}^{2}}{18\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16077, size = 135, normalized size = 1.8 \begin{align*} -\frac{b e^{2} n}{x} - \frac{b e^{2} \log \left (c x^{n}\right )}{x} - \frac{b d e n}{2 \, x^{2}} - \frac{a e^{2}}{x} - \frac{b d e \log \left (c x^{n}\right )}{x^{2}} - \frac{b d^{2} n}{9 \, x^{3}} - \frac{a d e}{x^{2}} - \frac{b d^{2} \log \left (c x^{n}\right )}{3 \, x^{3}} - \frac{a d^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.97004, size = 246, normalized size = 3.28 \begin{align*} -\frac{2 \, b d^{2} n + 6 \, a d^{2} + 18 \,{\left (b e^{2} n + a e^{2}\right )} x^{2} + 9 \,{\left (b d e n + 2 \, a d e\right )} x + 6 \,{\left (3 \, b e^{2} x^{2} + 3 \, b d e x + b d^{2}\right )} \log \left (c\right ) + 6 \,{\left (3 \, b e^{2} n x^{2} + 3 \, b d e n x + b d^{2} n\right )} \log \left (x\right )}{18 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.58323, size = 134, normalized size = 1.79 \begin{align*} - \frac{a d^{2}}{3 x^{3}} - \frac{a d e}{x^{2}} - \frac{a e^{2}}{x} - \frac{b d^{2} n \log{\left (x \right )}}{3 x^{3}} - \frac{b d^{2} n}{9 x^{3}} - \frac{b d^{2} \log{\left (c \right )}}{3 x^{3}} - \frac{b d e n \log{\left (x \right )}}{x^{2}} - \frac{b d e n}{2 x^{2}} - \frac{b d e \log{\left (c \right )}}{x^{2}} - \frac{b e^{2} n \log{\left (x \right )}}{x} - \frac{b e^{2} n}{x} - \frac{b e^{2} \log{\left (c \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28461, size = 146, normalized size = 1.95 \begin{align*} -\frac{18 \, b n x^{2} e^{2} \log \left (x\right ) + 18 \, b d n x e \log \left (x\right ) + 18 \, b n x^{2} e^{2} + 9 \, b d n x e + 18 \, b x^{2} e^{2} \log \left (c\right ) + 18 \, b d x e \log \left (c\right ) + 6 \, b d^{2} n \log \left (x\right ) + 2 \, b d^{2} n + 18 \, a x^{2} e^{2} + 18 \, a d x e + 6 \, b d^{2} \log \left (c\right ) + 6 \, a d^{2}}{18 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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